functional identities and their applications to graded...
TRANSCRIPT
![Page 1: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/1.jpg)
Examples of functional identities General FI Theory Applications
Functional identities and their applications tograded algebras
Matej Bresar,University of Ljubljana, University of Maribor, Slovenia
July 2009
Matej Bresar Functional identities and their applications to graded algebras
![Page 2: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/2.jpg)
Examples of functional identities General FI Theory Applications
Example 1
R: a ringf : R → R
f (x)y = 0 for all x , y ∈ R
=⇒
f = 0 or R “very special” (its left annihilator is nonzero: aR = 0with a 6= 0)
Matej Bresar Functional identities and their applications to graded algebras
![Page 3: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/3.jpg)
Examples of functional identities General FI Theory Applications
Example 1
R: a ring
f : R → R
f (x)y = 0 for all x , y ∈ R
=⇒
f = 0 or R “very special” (its left annihilator is nonzero: aR = 0with a 6= 0)
Matej Bresar Functional identities and their applications to graded algebras
![Page 4: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/4.jpg)
Examples of functional identities General FI Theory Applications
Example 1
R: a ringf : R → R
f (x)y = 0 for all x , y ∈ R
=⇒
f = 0 or R “very special” (its left annihilator is nonzero: aR = 0with a 6= 0)
Matej Bresar Functional identities and their applications to graded algebras
![Page 5: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/5.jpg)
Examples of functional identities General FI Theory Applications
Example 1
R: a ringf : R → R
f (x)y = 0 for all x , y ∈ R
=⇒
f = 0 or R “very special” (its left annihilator is nonzero: aR = 0with a 6= 0)
Matej Bresar Functional identities and their applications to graded algebras
![Page 6: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/6.jpg)
Examples of functional identities General FI Theory Applications
Example 1
R: a ringf : R → R
f (x)y = 0 for all x , y ∈ R
=⇒
f = 0 or R “very special” (its left annihilator is nonzero: aR = 0with a 6= 0)
Matej Bresar Functional identities and their applications to graded algebras
![Page 7: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/7.jpg)
Examples of functional identities General FI Theory Applications
Example 1
R: a ringf : R → R
f (x)y = 0 for all x , y ∈ R
=⇒
f = 0
or R “very special” (its left annihilator is nonzero: aR = 0with a 6= 0)
Matej Bresar Functional identities and their applications to graded algebras
![Page 8: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/8.jpg)
Examples of functional identities General FI Theory Applications
Example 1
R: a ringf : R → R
f (x)y = 0 for all x , y ∈ R
=⇒
f = 0 or R “very special”
(its left annihilator is nonzero: aR = 0with a 6= 0)
Matej Bresar Functional identities and their applications to graded algebras
![Page 9: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/9.jpg)
Examples of functional identities General FI Theory Applications
Example 1
R: a ringf : R → R
f (x)y = 0 for all x , y ∈ R
=⇒
f = 0 or R “very special” (its left annihilator is nonzero: aR = 0with a 6= 0)
Matej Bresar Functional identities and their applications to graded algebras
![Page 10: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/10.jpg)
Examples of functional identities General FI Theory Applications
Example 2
R prime (I , J ideals: IJ = 0 =⇒ I = 0 or J = 0)f , g : R → R
f (x)y + g(y)x = 0 for all x , y ∈ R
=⇒
f = g = 0 or (note: xy + (−y)x = 0 is an example)R iscommutative
Proof.
f (x)(yz)w = −g(yz)xw = f (xw)yz = −g(y)xwz =f (x)ywz =⇒ f (R)R [R, R ] = 0 =⇒ f = 0 or [R, R ] = 0.
Matej Bresar Functional identities and their applications to graded algebras
![Page 11: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/11.jpg)
Examples of functional identities General FI Theory Applications
Example 2
R prime (I , J ideals: IJ = 0 =⇒ I = 0 or J = 0)f , g : R → R
f (x)y + g(y)x = 0 for all x , y ∈ R
=⇒
f = g = 0 or (note: xy + (−y)x = 0 is an example)R iscommutative
Proof.
f (x)(yz)w = −g(yz)xw = f (xw)yz = −g(y)xwz =f (x)ywz =⇒ f (R)R [R, R ] = 0 =⇒ f = 0 or [R, R ] = 0.
Matej Bresar Functional identities and their applications to graded algebras
![Page 12: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/12.jpg)
Examples of functional identities General FI Theory Applications
Example 2
R prime (I , J ideals: IJ = 0 =⇒ I = 0 or J = 0)f , g : R → R
f (x)y + g(y)x = 0 for all x , y ∈ R
=⇒
f = g = 0 or (note: xy + (−y)x = 0 is an example)R iscommutative
Proof.
f (x)(yz)w = −g(yz)xw = f (xw)yz = −g(y)xwz =f (x)ywz =⇒ f (R)R [R, R ] = 0 =⇒ f = 0 or [R, R ] = 0.
Matej Bresar Functional identities and their applications to graded algebras
![Page 13: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/13.jpg)
Examples of functional identities General FI Theory Applications
Example 2
R prime (I , J ideals: IJ = 0 =⇒ I = 0 or J = 0)f , g : R → R
f (x)y + g(y)x = 0 for all x , y ∈ R
=⇒
f = g = 0 or (note: xy + (−y)x = 0 is an example)R iscommutative
Proof.
f (x)(yz)w = −g(yz)xw = f (xw)yz = −g(y)xwz =f (x)ywz =⇒ f (R)R [R, R ] = 0 =⇒ f = 0 or [R, R ] = 0.
Matej Bresar Functional identities and their applications to graded algebras
![Page 14: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/14.jpg)
Examples of functional identities General FI Theory Applications
Example 2
R prime (I , J ideals: IJ = 0 =⇒ I = 0 or J = 0)f , g : R → R
f (x)y + g(y)x = 0 for all x , y ∈ R
=⇒
f = g = 0
or (note: xy + (−y)x = 0 is an example)R iscommutative
Proof.
f (x)(yz)w = −g(yz)xw = f (xw)yz = −g(y)xwz =f (x)ywz =⇒ f (R)R [R, R ] = 0 =⇒ f = 0 or [R, R ] = 0.
Matej Bresar Functional identities and their applications to graded algebras
![Page 15: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/15.jpg)
Examples of functional identities General FI Theory Applications
Example 2
R prime (I , J ideals: IJ = 0 =⇒ I = 0 or J = 0)f , g : R → R
f (x)y + g(y)x = 0 for all x , y ∈ R
=⇒
f = g = 0 or (note: xy + (−y)x = 0 is an example)
R iscommutative
Proof.
f (x)(yz)w = −g(yz)xw = f (xw)yz = −g(y)xwz =f (x)ywz =⇒ f (R)R [R, R ] = 0 =⇒ f = 0 or [R, R ] = 0.
Matej Bresar Functional identities and their applications to graded algebras
![Page 16: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/16.jpg)
Examples of functional identities General FI Theory Applications
Example 2
R prime (I , J ideals: IJ = 0 =⇒ I = 0 or J = 0)f , g : R → R
f (x)y + g(y)x = 0 for all x , y ∈ R
=⇒
f = g = 0 or (note: xy + (−y)x = 0 is an example)R iscommutative
Proof.
f (x)(yz)w = −g(yz)xw = f (xw)yz = −g(y)xwz =f (x)ywz =⇒ f (R)R [R, R ] = 0 =⇒ f = 0 or [R, R ] = 0.
Matej Bresar Functional identities and their applications to graded algebras
![Page 17: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/17.jpg)
Examples of functional identities General FI Theory Applications
Example 2
R prime (I , J ideals: IJ = 0 =⇒ I = 0 or J = 0)f , g : R → R
f (x)y + g(y)x = 0 for all x , y ∈ R
=⇒
f = g = 0 or (note: xy + (−y)x = 0 is an example)R iscommutative
Proof.
f (x)(yz)w = −g(yz)xw
= f (xw)yz = −g(y)xwz =f (x)ywz =⇒ f (R)R [R, R ] = 0 =⇒ f = 0 or [R, R ] = 0.
Matej Bresar Functional identities and their applications to graded algebras
![Page 18: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/18.jpg)
Examples of functional identities General FI Theory Applications
Example 2
R prime (I , J ideals: IJ = 0 =⇒ I = 0 or J = 0)f , g : R → R
f (x)y + g(y)x = 0 for all x , y ∈ R
=⇒
f = g = 0 or (note: xy + (−y)x = 0 is an example)R iscommutative
Proof.
f (x)(yz)w = −g(yz)xw = f (xw)yz = −g(y)xwz =f (x)ywz
=⇒ f (R)R [R, R ] = 0 =⇒ f = 0 or [R, R ] = 0.
Matej Bresar Functional identities and their applications to graded algebras
![Page 19: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/19.jpg)
Examples of functional identities General FI Theory Applications
Example 2
R prime (I , J ideals: IJ = 0 =⇒ I = 0 or J = 0)f , g : R → R
f (x)y + g(y)x = 0 for all x , y ∈ R
=⇒
f = g = 0 or (note: xy + (−y)x = 0 is an example)R iscommutative
Proof.
f (x)(yz)w = −g(yz)xw = f (xw)yz = −g(y)xwz =f (x)ywz =⇒ f (R)R [R, R ] = 0
=⇒ f = 0 or [R, R ] = 0.
Matej Bresar Functional identities and their applications to graded algebras
![Page 20: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/20.jpg)
Examples of functional identities General FI Theory Applications
Example 2
R prime (I , J ideals: IJ = 0 =⇒ I = 0 or J = 0)f , g : R → R
f (x)y + g(y)x = 0 for all x , y ∈ R
=⇒
f = g = 0 or (note: xy + (−y)x = 0 is an example)R iscommutative
Proof.
f (x)(yz)w = −g(yz)xw = f (xw)yz = −g(y)xwz =f (x)ywz =⇒ f (R)R [R, R ] = 0 =⇒ f = 0 or [R, R ] = 0.
Matej Bresar Functional identities and their applications to graded algebras
![Page 21: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/21.jpg)
Examples of functional identities General FI Theory Applications
Example 3
Z : center of R, R primef , g : R → R
f (x)y + g(y)x ∈ Z for all x , y ∈ R
=⇒
f = g = 0 or (note: x2 − tr(x)x ∈ Z on R = M2(F ),hencef (x)y + f (y)x ∈ Z with f (x) = x − tr(x))R embeds in M2(F )
Proof: Algebraic manipulations + structure theory of PI-rings
Matej Bresar Functional identities and their applications to graded algebras
![Page 22: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/22.jpg)
Examples of functional identities General FI Theory Applications
Example 3
Z : center of R, R prime
f , g : R → R
f (x)y + g(y)x ∈ Z for all x , y ∈ R
=⇒
f = g = 0 or (note: x2 − tr(x)x ∈ Z on R = M2(F ),hencef (x)y + f (y)x ∈ Z with f (x) = x − tr(x))R embeds in M2(F )
Proof: Algebraic manipulations + structure theory of PI-rings
Matej Bresar Functional identities and their applications to graded algebras
![Page 23: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/23.jpg)
Examples of functional identities General FI Theory Applications
Example 3
Z : center of R, R primef , g : R → R
f (x)y + g(y)x ∈ Z for all x , y ∈ R
=⇒
f = g = 0 or (note: x2 − tr(x)x ∈ Z on R = M2(F ),hencef (x)y + f (y)x ∈ Z with f (x) = x − tr(x))R embeds in M2(F )
Proof: Algebraic manipulations + structure theory of PI-rings
Matej Bresar Functional identities and their applications to graded algebras
![Page 24: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/24.jpg)
Examples of functional identities General FI Theory Applications
Example 3
Z : center of R, R primef , g : R → R
f (x)y + g(y)x ∈ Z for all x , y ∈ R
=⇒
f = g = 0 or (note: x2 − tr(x)x ∈ Z on R = M2(F ),hencef (x)y + f (y)x ∈ Z with f (x) = x − tr(x))R embeds in M2(F )
Proof: Algebraic manipulations + structure theory of PI-rings
Matej Bresar Functional identities and their applications to graded algebras
![Page 25: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/25.jpg)
Examples of functional identities General FI Theory Applications
Example 3
Z : center of R, R primef , g : R → R
f (x)y + g(y)x ∈ Z for all x , y ∈ R
=⇒
f = g = 0 or (note: x2 − tr(x)x ∈ Z on R = M2(F ),hencef (x)y + f (y)x ∈ Z with f (x) = x − tr(x))R embeds in M2(F )
Proof: Algebraic manipulations + structure theory of PI-rings
Matej Bresar Functional identities and their applications to graded algebras
![Page 26: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/26.jpg)
Examples of functional identities General FI Theory Applications
Example 3
Z : center of R, R primef , g : R → R
f (x)y + g(y)x ∈ Z for all x , y ∈ R
=⇒
f = g = 0
or (note: x2 − tr(x)x ∈ Z on R = M2(F ),hencef (x)y + f (y)x ∈ Z with f (x) = x − tr(x))R embeds in M2(F )
Proof: Algebraic manipulations + structure theory of PI-rings
Matej Bresar Functional identities and their applications to graded algebras
![Page 27: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/27.jpg)
Examples of functional identities General FI Theory Applications
Example 3
Z : center of R, R primef , g : R → R
f (x)y + g(y)x ∈ Z for all x , y ∈ R
=⇒
f = g = 0 or (note: x2 − tr(x)x ∈ Z on R = M2(F ),
hencef (x)y + f (y)x ∈ Z with f (x) = x − tr(x))R embeds in M2(F )
Proof: Algebraic manipulations + structure theory of PI-rings
Matej Bresar Functional identities and their applications to graded algebras
![Page 28: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/28.jpg)
Examples of functional identities General FI Theory Applications
Example 3
Z : center of R, R primef , g : R → R
f (x)y + g(y)x ∈ Z for all x , y ∈ R
=⇒
f = g = 0 or (note: x2 − tr(x)x ∈ Z on R = M2(F ),hencef (x)y + f (y)x ∈ Z with f (x) = x − tr(x))
R embeds in M2(F )
Proof: Algebraic manipulations + structure theory of PI-rings
Matej Bresar Functional identities and their applications to graded algebras
![Page 29: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/29.jpg)
Examples of functional identities General FI Theory Applications
Example 3
Z : center of R, R primef , g : R → R
f (x)y + g(y)x ∈ Z for all x , y ∈ R
=⇒
f = g = 0 or (note: x2 − tr(x)x ∈ Z on R = M2(F ),hencef (x)y + f (y)x ∈ Z with f (x) = x − tr(x))R embeds in M2(F )
Proof: Algebraic manipulations + structure theory of PI-rings
Matej Bresar Functional identities and their applications to graded algebras
![Page 30: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/30.jpg)
Examples of functional identities General FI Theory Applications
Example 3
Z : center of R, R primef , g : R → R
f (x)y + g(y)x ∈ Z for all x , y ∈ R
=⇒
f = g = 0 or (note: x2 − tr(x)x ∈ Z on R = M2(F ),hencef (x)y + f (y)x ∈ Z with f (x) = x − tr(x))R embeds in M2(F )
Proof:
Algebraic manipulations + structure theory of PI-rings
Matej Bresar Functional identities and their applications to graded algebras
![Page 31: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/31.jpg)
Examples of functional identities General FI Theory Applications
Example 3
Z : center of R, R primef , g : R → R
f (x)y + g(y)x ∈ Z for all x , y ∈ R
=⇒
f = g = 0 or (note: x2 − tr(x)x ∈ Z on R = M2(F ),hencef (x)y + f (y)x ∈ Z with f (x) = x − tr(x))R embeds in M2(F )
Proof: Algebraic manipulations + structure theory of PI-rings
Matej Bresar Functional identities and their applications to graded algebras
![Page 32: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/32.jpg)
Examples of functional identities General FI Theory Applications
Example 4
f1, f2, . . . , fn : Rn−1 → R, R prime
f1(x2, . . . , xn)x1 + f2(x1, x3 . . . , xn)x2 + . . . + fn(x1, . . . , xn−1)xn ∈ Z
=⇒
f1 = f2 = . . . = fn = 0 or R embeds in Mn(F )
Note: A multilinear PI (polynomial identity) is such an FI with fi“polynomials”. FI theory: a complement to PI theory.
Matej Bresar Functional identities and their applications to graded algebras
![Page 33: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/33.jpg)
Examples of functional identities General FI Theory Applications
Example 4
f1, f2, . . . , fn : Rn−1 → R, R prime
f1(x2, . . . , xn)x1 + f2(x1, x3 . . . , xn)x2 + . . . + fn(x1, . . . , xn−1)xn ∈ Z
=⇒
f1 = f2 = . . . = fn = 0 or R embeds in Mn(F )
Note: A multilinear PI (polynomial identity) is such an FI with fi“polynomials”. FI theory: a complement to PI theory.
Matej Bresar Functional identities and their applications to graded algebras
![Page 34: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/34.jpg)
Examples of functional identities General FI Theory Applications
Example 4
f1, f2, . . . , fn : Rn−1 → R, R prime
f1(x2, . . . , xn)x1 + f2(x1, x3 . . . , xn)x2 + . . . + fn(x1, . . . , xn−1)xn ∈ Z
=⇒
f1 = f2 = . . . = fn = 0 or R embeds in Mn(F )
Note: A multilinear PI (polynomial identity) is such an FI with fi“polynomials”. FI theory: a complement to PI theory.
Matej Bresar Functional identities and their applications to graded algebras
![Page 35: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/35.jpg)
Examples of functional identities General FI Theory Applications
Example 4
f1, f2, . . . , fn : Rn−1 → R, R prime
f1(x2, . . . , xn)x1 + f2(x1, x3 . . . , xn)x2 + . . . + fn(x1, . . . , xn−1)xn ∈ Z
=⇒
f1 = f2 = . . . = fn = 0 or R embeds in Mn(F )
Note: A multilinear PI (polynomial identity) is such an FI with fi“polynomials”. FI theory: a complement to PI theory.
Matej Bresar Functional identities and their applications to graded algebras
![Page 36: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/36.jpg)
Examples of functional identities General FI Theory Applications
Example 4
f1, f2, . . . , fn : Rn−1 → R, R prime
f1(x2, . . . , xn)x1 + f2(x1, x3 . . . , xn)x2 + . . . + fn(x1, . . . , xn−1)xn ∈ Z
=⇒
f1 = f2 = . . . = fn = 0
or R embeds in Mn(F )
Note: A multilinear PI (polynomial identity) is such an FI with fi“polynomials”. FI theory: a complement to PI theory.
Matej Bresar Functional identities and their applications to graded algebras
![Page 37: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/37.jpg)
Examples of functional identities General FI Theory Applications
Example 4
f1, f2, . . . , fn : Rn−1 → R, R prime
f1(x2, . . . , xn)x1 + f2(x1, x3 . . . , xn)x2 + . . . + fn(x1, . . . , xn−1)xn ∈ Z
=⇒
f1 = f2 = . . . = fn = 0 or R embeds in Mn(F )
Note: A multilinear PI (polynomial identity) is such an FI with fi“polynomials”. FI theory: a complement to PI theory.
Matej Bresar Functional identities and their applications to graded algebras
![Page 38: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/38.jpg)
Examples of functional identities General FI Theory Applications
Example 4
f1, f2, . . . , fn : Rn−1 → R, R prime
f1(x2, . . . , xn)x1 + f2(x1, x3 . . . , xn)x2 + . . . + fn(x1, . . . , xn−1)xn ∈ Z
=⇒
f1 = f2 = . . . = fn = 0 or R embeds in Mn(F )
Note: A multilinear PI (polynomial identity) is such an FI with fi“polynomials”.
FI theory: a complement to PI theory.
Matej Bresar Functional identities and their applications to graded algebras
![Page 39: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/39.jpg)
Examples of functional identities General FI Theory Applications
Example 4
f1, f2, . . . , fn : Rn−1 → R, R prime
f1(x2, . . . , xn)x1 + f2(x1, x3 . . . , xn)x2 + . . . + fn(x1, . . . , xn−1)xn ∈ Z
=⇒
f1 = f2 = . . . = fn = 0 or R embeds in Mn(F )
Note: A multilinear PI (polynomial identity) is such an FI with fi“polynomials”. FI theory
: a complement to PI theory.
Matej Bresar Functional identities and their applications to graded algebras
![Page 40: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/40.jpg)
Examples of functional identities General FI Theory Applications
Example 4
f1, f2, . . . , fn : Rn−1 → R, R prime
f1(x2, . . . , xn)x1 + f2(x1, x3 . . . , xn)x2 + . . . + fn(x1, . . . , xn−1)xn ∈ Z
=⇒
f1 = f2 = . . . = fn = 0 or R embeds in Mn(F )
Note: A multilinear PI (polynomial identity) is such an FI with fi“polynomials”. FI theory: a complement to PI theory.
Matej Bresar Functional identities and their applications to graded algebras
![Page 41: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/41.jpg)
Examples of functional identities General FI Theory Applications
Example 5
f , g : R → R
f (x)y = xg(y) for all x , y ∈ R
Expected solution:f (x) = xa, g(y) = ay for some a ∈ R.If 1 ∈ R: a = f (1) = g(1).Without 1?E.g., R is an ideal of S : then a may belong to S .In general: rings of quotients have to be involved.In the context of prime rings,the maximal (left or right) ring ofquotients is suitable.
Matej Bresar Functional identities and their applications to graded algebras
![Page 42: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/42.jpg)
Examples of functional identities General FI Theory Applications
Example 5
f , g : R → R
f (x)y = xg(y) for all x , y ∈ R
Expected solution:f (x) = xa, g(y) = ay for some a ∈ R.If 1 ∈ R: a = f (1) = g(1).Without 1?E.g., R is an ideal of S : then a may belong to S .In general: rings of quotients have to be involved.In the context of prime rings,the maximal (left or right) ring ofquotients is suitable.
Matej Bresar Functional identities and their applications to graded algebras
![Page 43: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/43.jpg)
Examples of functional identities General FI Theory Applications
Example 5
f , g : R → R
f (x)y = xg(y) for all x , y ∈ R
Expected solution:f (x) = xa, g(y) = ay for some a ∈ R.If 1 ∈ R: a = f (1) = g(1).Without 1?E.g., R is an ideal of S : then a may belong to S .In general: rings of quotients have to be involved.In the context of prime rings,the maximal (left or right) ring ofquotients is suitable.
Matej Bresar Functional identities and their applications to graded algebras
![Page 44: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/44.jpg)
Examples of functional identities General FI Theory Applications
Example 5
f , g : R → R
f (x)y = xg(y) for all x , y ∈ R
Expected solution:
f (x) = xa, g(y) = ay for some a ∈ R.If 1 ∈ R: a = f (1) = g(1).Without 1?E.g., R is an ideal of S : then a may belong to S .In general: rings of quotients have to be involved.In the context of prime rings,the maximal (left or right) ring ofquotients is suitable.
Matej Bresar Functional identities and their applications to graded algebras
![Page 45: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/45.jpg)
Examples of functional identities General FI Theory Applications
Example 5
f , g : R → R
f (x)y = xg(y) for all x , y ∈ R
Expected solution:f (x) = xa, g(y) = ay for some a ∈ R.
If 1 ∈ R: a = f (1) = g(1).Without 1?E.g., R is an ideal of S : then a may belong to S .In general: rings of quotients have to be involved.In the context of prime rings,the maximal (left or right) ring ofquotients is suitable.
Matej Bresar Functional identities and their applications to graded algebras
![Page 46: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/46.jpg)
Examples of functional identities General FI Theory Applications
Example 5
f , g : R → R
f (x)y = xg(y) for all x , y ∈ R
Expected solution:f (x) = xa, g(y) = ay for some a ∈ R.If 1 ∈ R: a = f (1) = g(1).
Without 1?E.g., R is an ideal of S : then a may belong to S .In general: rings of quotients have to be involved.In the context of prime rings,the maximal (left or right) ring ofquotients is suitable.
Matej Bresar Functional identities and their applications to graded algebras
![Page 47: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/47.jpg)
Examples of functional identities General FI Theory Applications
Example 5
f , g : R → R
f (x)y = xg(y) for all x , y ∈ R
Expected solution:f (x) = xa, g(y) = ay for some a ∈ R.If 1 ∈ R: a = f (1) = g(1).Without 1?
E.g., R is an ideal of S : then a may belong to S .In general: rings of quotients have to be involved.In the context of prime rings,the maximal (left or right) ring ofquotients is suitable.
Matej Bresar Functional identities and their applications to graded algebras
![Page 48: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/48.jpg)
Examples of functional identities General FI Theory Applications
Example 5
f , g : R → R
f (x)y = xg(y) for all x , y ∈ R
Expected solution:f (x) = xa, g(y) = ay for some a ∈ R.If 1 ∈ R: a = f (1) = g(1).Without 1?E.g., R is an ideal of S :
then a may belong to S .In general: rings of quotients have to be involved.In the context of prime rings,the maximal (left or right) ring ofquotients is suitable.
Matej Bresar Functional identities and their applications to graded algebras
![Page 49: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/49.jpg)
Examples of functional identities General FI Theory Applications
Example 5
f , g : R → R
f (x)y = xg(y) for all x , y ∈ R
Expected solution:f (x) = xa, g(y) = ay for some a ∈ R.If 1 ∈ R: a = f (1) = g(1).Without 1?E.g., R is an ideal of S : then a may belong to S .
In general: rings of quotients have to be involved.In the context of prime rings,the maximal (left or right) ring ofquotients is suitable.
Matej Bresar Functional identities and their applications to graded algebras
![Page 50: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/50.jpg)
Examples of functional identities General FI Theory Applications
Example 5
f , g : R → R
f (x)y = xg(y) for all x , y ∈ R
Expected solution:f (x) = xa, g(y) = ay for some a ∈ R.If 1 ∈ R: a = f (1) = g(1).Without 1?E.g., R is an ideal of S : then a may belong to S .In general:
rings of quotients have to be involved.In the context of prime rings,the maximal (left or right) ring ofquotients is suitable.
Matej Bresar Functional identities and their applications to graded algebras
![Page 51: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/51.jpg)
Examples of functional identities General FI Theory Applications
Example 5
f , g : R → R
f (x)y = xg(y) for all x , y ∈ R
Expected solution:f (x) = xa, g(y) = ay for some a ∈ R.If 1 ∈ R: a = f (1) = g(1).Without 1?E.g., R is an ideal of S : then a may belong to S .In general: rings of quotients have to be involved.
In the context of prime rings,the maximal (left or right) ring ofquotients is suitable.
Matej Bresar Functional identities and their applications to graded algebras
![Page 52: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/52.jpg)
Examples of functional identities General FI Theory Applications
Example 5
f , g : R → R
f (x)y = xg(y) for all x , y ∈ R
Expected solution:f (x) = xa, g(y) = ay for some a ∈ R.If 1 ∈ R: a = f (1) = g(1).Without 1?E.g., R is an ideal of S : then a may belong to S .In general: rings of quotients have to be involved.In the context of prime rings,
the maximal (left or right) ring ofquotients is suitable.
Matej Bresar Functional identities and their applications to graded algebras
![Page 53: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/53.jpg)
Examples of functional identities General FI Theory Applications
Example 5
f , g : R → R
f (x)y = xg(y) for all x , y ∈ R
Expected solution:f (x) = xa, g(y) = ay for some a ∈ R.If 1 ∈ R: a = f (1) = g(1).Without 1?E.g., R is an ideal of S : then a may belong to S .In general: rings of quotients have to be involved.In the context of prime rings,the maximal (left or right) ring ofquotients is suitable.
Matej Bresar Functional identities and their applications to graded algebras
![Page 54: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/54.jpg)
Examples of functional identities General FI Theory Applications
Example 6
f : R → R additive, R prime:
f (x)x = xf (x) for all x ∈ R
=⇒
f (x) = λx + µ(x),
where λ ∈ C , the extended centroid of R,and µ : R → C .(M. B., 1990)
Matej Bresar Functional identities and their applications to graded algebras
![Page 55: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/55.jpg)
Examples of functional identities General FI Theory Applications
Example 6
f : R → R
additive, R prime:
f (x)x = xf (x) for all x ∈ R
=⇒
f (x) = λx + µ(x),
where λ ∈ C , the extended centroid of R,and µ : R → C .(M. B., 1990)
Matej Bresar Functional identities and their applications to graded algebras
![Page 56: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/56.jpg)
Examples of functional identities General FI Theory Applications
Example 6
f : R → R additive,
R prime:
f (x)x = xf (x) for all x ∈ R
=⇒
f (x) = λx + µ(x),
where λ ∈ C , the extended centroid of R,and µ : R → C .(M. B., 1990)
Matej Bresar Functional identities and their applications to graded algebras
![Page 57: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/57.jpg)
Examples of functional identities General FI Theory Applications
Example 6
f : R → R additive, R prime:
f (x)x = xf (x) for all x ∈ R
=⇒
f (x) = λx + µ(x),
where λ ∈ C , the extended centroid of R,and µ : R → C .(M. B., 1990)
Matej Bresar Functional identities and their applications to graded algebras
![Page 58: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/58.jpg)
Examples of functional identities General FI Theory Applications
Example 6
f : R → R additive, R prime:
f (x)x = xf (x) for all x ∈ R
=⇒
f (x) = λx + µ(x),
where λ ∈ C , the extended centroid of R,and µ : R → C .(M. B., 1990)
Matej Bresar Functional identities and their applications to graded algebras
![Page 59: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/59.jpg)
Examples of functional identities General FI Theory Applications
Example 6
f : R → R additive, R prime:
f (x)x = xf (x) for all x ∈ R
=⇒
f (x) = λx + µ(x),
where λ ∈ C , the extended centroid of R,and µ : R → C .(M. B., 1990)
Matej Bresar Functional identities and their applications to graded algebras
![Page 60: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/60.jpg)
Examples of functional identities General FI Theory Applications
Example 6
f : R → R additive, R prime:
f (x)x = xf (x) for all x ∈ R
=⇒
f (x) = λx + µ(x),
where λ ∈ C , the extended centroid of R,and µ : R → C .(M. B., 1990)
Matej Bresar Functional identities and their applications to graded algebras
![Page 61: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/61.jpg)
Examples of functional identities General FI Theory Applications
Example 6
f : R → R additive, R prime:
f (x)x = xf (x) for all x ∈ R
=⇒
f (x) = λx + µ(x),
where λ ∈ C , the extended centroid of R,
and µ : R → C .(M. B., 1990)
Matej Bresar Functional identities and their applications to graded algebras
![Page 62: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/62.jpg)
Examples of functional identities General FI Theory Applications
Example 6
f : R → R additive, R prime:
f (x)x = xf (x) for all x ∈ R
=⇒
f (x) = λx + µ(x),
where λ ∈ C , the extended centroid of R,and µ : R → C .
(M. B., 1990)
Matej Bresar Functional identities and their applications to graded algebras
![Page 63: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/63.jpg)
Examples of functional identities General FI Theory Applications
Example 6
f : R → R additive, R prime:
f (x)x = xf (x) for all x ∈ R
=⇒
f (x) = λx + µ(x),
where λ ∈ C , the extended centroid of R,and µ : R → C .(M. B., 1990)
Matej Bresar Functional identities and their applications to graded algebras
![Page 64: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/64.jpg)
Examples of functional identities General FI Theory Applications
Example 7
f : R × R → R biadditive, R prime:
f (x , x)x = xf (x , x) for all x ∈ R
=⇒
f (x , x) = λx2 + µ(x)x + ν(x),
where λ ∈ C , and µ, ν : R → C with µ additive.(M. B., 1990)Applications! Hint: interprete f (x , x)x = xf (x , x) as
(x ∗ x)x = x(x ∗ x)
where ∗ is another (nonassociative) product on R.
Matej Bresar Functional identities and their applications to graded algebras
![Page 65: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/65.jpg)
Examples of functional identities General FI Theory Applications
Example 7
f : R × R → R
biadditive, R prime:
f (x , x)x = xf (x , x) for all x ∈ R
=⇒
f (x , x) = λx2 + µ(x)x + ν(x),
where λ ∈ C , and µ, ν : R → C with µ additive.(M. B., 1990)Applications! Hint: interprete f (x , x)x = xf (x , x) as
(x ∗ x)x = x(x ∗ x)
where ∗ is another (nonassociative) product on R.
Matej Bresar Functional identities and their applications to graded algebras
![Page 66: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/66.jpg)
Examples of functional identities General FI Theory Applications
Example 7
f : R × R → R biadditive,
R prime:
f (x , x)x = xf (x , x) for all x ∈ R
=⇒
f (x , x) = λx2 + µ(x)x + ν(x),
where λ ∈ C , and µ, ν : R → C with µ additive.(M. B., 1990)Applications! Hint: interprete f (x , x)x = xf (x , x) as
(x ∗ x)x = x(x ∗ x)
where ∗ is another (nonassociative) product on R.
Matej Bresar Functional identities and their applications to graded algebras
![Page 67: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/67.jpg)
Examples of functional identities General FI Theory Applications
Example 7
f : R × R → R biadditive, R prime:
f (x , x)x = xf (x , x) for all x ∈ R
=⇒
f (x , x) = λx2 + µ(x)x + ν(x),
where λ ∈ C , and µ, ν : R → C with µ additive.(M. B., 1990)Applications! Hint: interprete f (x , x)x = xf (x , x) as
(x ∗ x)x = x(x ∗ x)
where ∗ is another (nonassociative) product on R.
Matej Bresar Functional identities and their applications to graded algebras
![Page 68: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/68.jpg)
Examples of functional identities General FI Theory Applications
Example 7
f : R × R → R biadditive, R prime:
f (x , x)x = xf (x , x) for all x ∈ R
=⇒
f (x , x) = λx2 + µ(x)x + ν(x),
where λ ∈ C , and µ, ν : R → C with µ additive.(M. B., 1990)Applications! Hint: interprete f (x , x)x = xf (x , x) as
(x ∗ x)x = x(x ∗ x)
where ∗ is another (nonassociative) product on R.
Matej Bresar Functional identities and their applications to graded algebras
![Page 69: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/69.jpg)
Examples of functional identities General FI Theory Applications
Example 7
f : R × R → R biadditive, R prime:
f (x , x)x = xf (x , x) for all x ∈ R
=⇒
f (x , x) = λx2 + µ(x)x + ν(x),
where λ ∈ C , and µ, ν : R → C with µ additive.(M. B., 1990)Applications! Hint: interprete f (x , x)x = xf (x , x) as
(x ∗ x)x = x(x ∗ x)
where ∗ is another (nonassociative) product on R.
Matej Bresar Functional identities and their applications to graded algebras
![Page 70: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/70.jpg)
Examples of functional identities General FI Theory Applications
Example 7
f : R × R → R biadditive, R prime:
f (x , x)x = xf (x , x) for all x ∈ R
=⇒
f (x , x) = λx2 + µ(x)x + ν(x),
where λ ∈ C , and µ, ν : R → C with µ additive.(M. B., 1990)Applications! Hint: interprete f (x , x)x = xf (x , x) as
(x ∗ x)x = x(x ∗ x)
where ∗ is another (nonassociative) product on R.
Matej Bresar Functional identities and their applications to graded algebras
![Page 71: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/71.jpg)
Examples of functional identities General FI Theory Applications
Example 7
f : R × R → R biadditive, R prime:
f (x , x)x = xf (x , x) for all x ∈ R
=⇒
f (x , x) = λx2 + µ(x)x + ν(x),
where λ ∈ C , and µ, ν : R → C
with µ additive.(M. B., 1990)Applications! Hint: interprete f (x , x)x = xf (x , x) as
(x ∗ x)x = x(x ∗ x)
where ∗ is another (nonassociative) product on R.
Matej Bresar Functional identities and their applications to graded algebras
![Page 72: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/72.jpg)
Examples of functional identities General FI Theory Applications
Example 7
f : R × R → R biadditive, R prime:
f (x , x)x = xf (x , x) for all x ∈ R
=⇒
f (x , x) = λx2 + µ(x)x + ν(x),
where λ ∈ C , and µ, ν : R → C with µ additive.
(M. B., 1990)Applications! Hint: interprete f (x , x)x = xf (x , x) as
(x ∗ x)x = x(x ∗ x)
where ∗ is another (nonassociative) product on R.
Matej Bresar Functional identities and their applications to graded algebras
![Page 73: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/73.jpg)
Examples of functional identities General FI Theory Applications
Example 7
f : R × R → R biadditive, R prime:
f (x , x)x = xf (x , x) for all x ∈ R
=⇒
f (x , x) = λx2 + µ(x)x + ν(x),
where λ ∈ C , and µ, ν : R → C with µ additive.(M. B., 1990)
Applications! Hint: interprete f (x , x)x = xf (x , x) as
(x ∗ x)x = x(x ∗ x)
where ∗ is another (nonassociative) product on R.
Matej Bresar Functional identities and their applications to graded algebras
![Page 74: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/74.jpg)
Examples of functional identities General FI Theory Applications
Example 7
f : R × R → R biadditive, R prime:
f (x , x)x = xf (x , x) for all x ∈ R
=⇒
f (x , x) = λx2 + µ(x)x + ν(x),
where λ ∈ C , and µ, ν : R → C with µ additive.(M. B., 1990)Applications!
Hint: interprete f (x , x)x = xf (x , x) as
(x ∗ x)x = x(x ∗ x)
where ∗ is another (nonassociative) product on R.
Matej Bresar Functional identities and their applications to graded algebras
![Page 75: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/75.jpg)
Examples of functional identities General FI Theory Applications
Example 7
f : R × R → R biadditive, R prime:
f (x , x)x = xf (x , x) for all x ∈ R
=⇒
f (x , x) = λx2 + µ(x)x + ν(x),
where λ ∈ C , and µ, ν : R → C with µ additive.(M. B., 1990)Applications! Hint: interprete f (x , x)x = xf (x , x)
as
(x ∗ x)x = x(x ∗ x)
where ∗ is another (nonassociative) product on R.
Matej Bresar Functional identities and their applications to graded algebras
![Page 76: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/76.jpg)
Examples of functional identities General FI Theory Applications
Example 7
f : R × R → R biadditive, R prime:
f (x , x)x = xf (x , x) for all x ∈ R
=⇒
f (x , x) = λx2 + µ(x)x + ν(x),
where λ ∈ C , and µ, ν : R → C with µ additive.(M. B., 1990)Applications! Hint: interprete f (x , x)x = xf (x , x) as
(x ∗ x)x = x(x ∗ x)
where ∗ is another (nonassociative) product on R.
Matej Bresar Functional identities and their applications to graded algebras
![Page 77: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/77.jpg)
Examples of functional identities General FI Theory Applications
Example 7
f : R × R → R biadditive, R prime:
f (x , x)x = xf (x , x) for all x ∈ R
=⇒
f (x , x) = λx2 + µ(x)x + ν(x),
where λ ∈ C , and µ, ν : R → C with µ additive.(M. B., 1990)Applications! Hint: interprete f (x , x)x = xf (x , x) as
(x ∗ x)x = x(x ∗ x)
where ∗ is another (nonassociative) product on R.
Matej Bresar Functional identities and their applications to graded algebras
![Page 78: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/78.jpg)
Examples of functional identities General FI Theory Applications
Defining d-free sets
X : a subset of a ring Q with center C“Definition” (K. Beidar, M. Chebotar, 2000):X is a d-free subset of Q if FI’s such as
d
∑i=1
Ei (x1, . . . , xi−1, xi+1, . . . , xd )xi + xiFi (x1, . . . , xi−1, xi+1, . . . , xd ) = 0
have only standard solutions, i.e.,
Ei =d
∑j=1j 6=i
xjpij + λi , Fj = −d
∑i=1i 6=j
pijxi − λj ,
where
pij = pij (x1, . . . , xi−1, xi+1, . . . , xj−i , xj+1, . . . , xd ) ∈ Q,
λi = λi (x1, . . . , xi−1, xi+1, . . . , xd ) ∈ C .
Matej Bresar Functional identities and their applications to graded algebras
![Page 79: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/79.jpg)
Examples of functional identities General FI Theory Applications
Defining d-free sets
X : a subset of a ring Q with center C
“Definition” (K. Beidar, M. Chebotar, 2000):X is a d-free subset of Q if FI’s such as
d
∑i=1
Ei (x1, . . . , xi−1, xi+1, . . . , xd )xi + xiFi (x1, . . . , xi−1, xi+1, . . . , xd ) = 0
have only standard solutions, i.e.,
Ei =d
∑j=1j 6=i
xjpij + λi , Fj = −d
∑i=1i 6=j
pijxi − λj ,
where
pij = pij (x1, . . . , xi−1, xi+1, . . . , xj−i , xj+1, . . . , xd ) ∈ Q,
λi = λi (x1, . . . , xi−1, xi+1, . . . , xd ) ∈ C .
Matej Bresar Functional identities and their applications to graded algebras
![Page 80: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/80.jpg)
Examples of functional identities General FI Theory Applications
Defining d-free sets
X : a subset of a ring Q with center C“Definition” (K. Beidar, M. Chebotar, 2000):
X is a d-free subset of Q if FI’s such as
d
∑i=1
Ei (x1, . . . , xi−1, xi+1, . . . , xd )xi + xiFi (x1, . . . , xi−1, xi+1, . . . , xd ) = 0
have only standard solutions, i.e.,
Ei =d
∑j=1j 6=i
xjpij + λi , Fj = −d
∑i=1i 6=j
pijxi − λj ,
where
pij = pij (x1, . . . , xi−1, xi+1, . . . , xj−i , xj+1, . . . , xd ) ∈ Q,
λi = λi (x1, . . . , xi−1, xi+1, . . . , xd ) ∈ C .
Matej Bresar Functional identities and their applications to graded algebras
![Page 81: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/81.jpg)
Examples of functional identities General FI Theory Applications
Defining d-free sets
X : a subset of a ring Q with center C“Definition” (K. Beidar, M. Chebotar, 2000):X is a d-free subset of Q if FI’s such as
d
∑i=1
Ei (x1, . . . , xi−1, xi+1, . . . , xd )xi + xiFi (x1, . . . , xi−1, xi+1, . . . , xd ) = 0
have only standard solutions, i.e.,
Ei =d
∑j=1j 6=i
xjpij + λi , Fj = −d
∑i=1i 6=j
pijxi − λj ,
where
pij = pij (x1, . . . , xi−1, xi+1, . . . , xj−i , xj+1, . . . , xd ) ∈ Q,
λi = λi (x1, . . . , xi−1, xi+1, . . . , xd ) ∈ C .
Matej Bresar Functional identities and their applications to graded algebras
![Page 82: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/82.jpg)
Examples of functional identities General FI Theory Applications
Defining d-free sets
X : a subset of a ring Q with center C“Definition” (K. Beidar, M. Chebotar, 2000):X is a d-free subset of Q if FI’s such as
d
∑i=1
Ei (x1, . . . , xi−1, xi+1, . . . , xd )xi + xiFi (x1, . . . , xi−1, xi+1, . . . , xd ) = 0
have only standard solutions, i.e.,
Ei =d
∑j=1j 6=i
xjpij + λi , Fj = −d
∑i=1i 6=j
pijxi − λj ,
where
pij = pij (x1, . . . , xi−1, xi+1, . . . , xj−i , xj+1, . . . , xd ) ∈ Q,
λi = λi (x1, . . . , xi−1, xi+1, . . . , xd ) ∈ C .
Matej Bresar Functional identities and their applications to graded algebras
![Page 83: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/83.jpg)
Examples of functional identities General FI Theory Applications
Defining d-free sets
X : a subset of a ring Q with center C“Definition” (K. Beidar, M. Chebotar, 2000):X is a d-free subset of Q if FI’s such as
d
∑i=1
Ei (x1, . . . , xi−1, xi+1, . . . , xd )xi + xiFi (x1, . . . , xi−1, xi+1, . . . , xd ) = 0
have only standard solutions,
i.e.,
Ei =d
∑j=1j 6=i
xjpij + λi , Fj = −d
∑i=1i 6=j
pijxi − λj ,
where
pij = pij (x1, . . . , xi−1, xi+1, . . . , xj−i , xj+1, . . . , xd ) ∈ Q,
λi = λi (x1, . . . , xi−1, xi+1, . . . , xd ) ∈ C .
Matej Bresar Functional identities and their applications to graded algebras
![Page 84: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/84.jpg)
Examples of functional identities General FI Theory Applications
Defining d-free sets
X : a subset of a ring Q with center C“Definition” (K. Beidar, M. Chebotar, 2000):X is a d-free subset of Q if FI’s such as
d
∑i=1
Ei (x1, . . . , xi−1, xi+1, . . . , xd )xi + xiFi (x1, . . . , xi−1, xi+1, . . . , xd ) = 0
have only standard solutions, i.e.,
Ei =d
∑j=1j 6=i
xjpij + λi ,
Fj = −d
∑i=1i 6=j
pijxi − λj ,
where
pij = pij (x1, . . . , xi−1, xi+1, . . . , xj−i , xj+1, . . . , xd ) ∈ Q,
λi = λi (x1, . . . , xi−1, xi+1, . . . , xd ) ∈ C .
Matej Bresar Functional identities and their applications to graded algebras
![Page 85: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/85.jpg)
Examples of functional identities General FI Theory Applications
Defining d-free sets
X : a subset of a ring Q with center C“Definition” (K. Beidar, M. Chebotar, 2000):X is a d-free subset of Q if FI’s such as
d
∑i=1
Ei (x1, . . . , xi−1, xi+1, . . . , xd )xi + xiFi (x1, . . . , xi−1, xi+1, . . . , xd ) = 0
have only standard solutions, i.e.,
Ei =d
∑j=1j 6=i
xjpij + λi , Fj = −d
∑i=1i 6=j
pijxi − λj ,
where
pij = pij (x1, . . . , xi−1, xi+1, . . . , xj−i , xj+1, . . . , xd ) ∈ Q,
λi = λi (x1, . . . , xi−1, xi+1, . . . , xd ) ∈ C .
Matej Bresar Functional identities and their applications to graded algebras
![Page 86: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/86.jpg)
Examples of functional identities General FI Theory Applications
Defining d-free sets
X : a subset of a ring Q with center C“Definition” (K. Beidar, M. Chebotar, 2000):X is a d-free subset of Q if FI’s such as
d
∑i=1
Ei (x1, . . . , xi−1, xi+1, . . . , xd )xi + xiFi (x1, . . . , xi−1, xi+1, . . . , xd ) = 0
have only standard solutions, i.e.,
Ei =d
∑j=1j 6=i
xjpij + λi , Fj = −d
∑i=1i 6=j
pijxi − λj ,
where
pij = pij (x1, . . . , xi−1, xi+1, . . . , xj−i , xj+1, . . . , xd ) ∈ Q,
λi = λi (x1, . . . , xi−1, xi+1, . . . , xd ) ∈ C .
Matej Bresar Functional identities and their applications to graded algebras
![Page 87: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/87.jpg)
Examples of functional identities General FI Theory Applications
Defining d-free sets
X : a subset of a ring Q with center C“Definition” (K. Beidar, M. Chebotar, 2000):X is a d-free subset of Q if FI’s such as
d
∑i=1
Ei (x1, . . . , xi−1, xi+1, . . . , xd )xi + xiFi (x1, . . . , xi−1, xi+1, . . . , xd ) = 0
have only standard solutions, i.e.,
Ei =d
∑j=1j 6=i
xjpij + λi , Fj = −d
∑i=1i 6=j
pijxi − λj ,
where
pij = pij (x1, . . . , xi−1, xi+1, . . . , xj−i , xj+1, . . . , xd ) ∈ Q,
λi = λi (x1, . . . , xi−1, xi+1, . . . , xd ) ∈ C .
Matej Bresar Functional identities and their applications to graded algebras
![Page 88: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/88.jpg)
Examples of functional identities General FI Theory Applications
d-free sets exist!
K. Beidar (1998): A prime ring R is a d-free subset of its maximalleft ring of quotients Q
⇐⇒R does not satisfy a PI of degree 2d − 2 (i.e., R cannot beembedded in Md−1(F )).
Other important examples of d-free sets:
Lie ideals of prime rings
symmetric elements of prime rings with involution
skew elements of prime rings with involution (and their Lieideals)
semiprime rings
Mn(B), B any unital ring
etc.
Matej Bresar Functional identities and their applications to graded algebras
![Page 89: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/89.jpg)
Examples of functional identities General FI Theory Applications
d-free sets exist!
K. Beidar (1998): A prime ring R is a d-free subset of its maximalleft ring of quotients Q
⇐⇒R does not satisfy a PI of degree 2d − 2 (i.e., R cannot beembedded in Md−1(F )).
Other important examples of d-free sets:
Lie ideals of prime rings
symmetric elements of prime rings with involution
skew elements of prime rings with involution (and their Lieideals)
semiprime rings
Mn(B), B any unital ring
etc.
Matej Bresar Functional identities and their applications to graded algebras
![Page 90: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/90.jpg)
Examples of functional identities General FI Theory Applications
d-free sets exist!
K. Beidar (1998): A prime ring R is a d-free subset of its maximalleft ring of quotients Q
⇐⇒R does not satisfy a PI of degree 2d − 2 (i.e., R cannot beembedded in Md−1(F )).
Other important examples of d-free sets:
Lie ideals of prime rings
symmetric elements of prime rings with involution
skew elements of prime rings with involution (and their Lieideals)
semiprime rings
Mn(B), B any unital ring
etc.
Matej Bresar Functional identities and their applications to graded algebras
![Page 91: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/91.jpg)
Examples of functional identities General FI Theory Applications
d-free sets exist!
K. Beidar (1998): A prime ring R is a d-free subset of its maximalleft ring of quotients Q
⇐⇒R does not satisfy a PI of degree 2d − 2 (i.e., R cannot beembedded in Md−1(F )).
Other important examples of d-free sets:
Lie ideals of prime rings
symmetric elements of prime rings with involution
skew elements of prime rings with involution (and their Lieideals)
semiprime rings
Mn(B), B any unital ring
etc.
Matej Bresar Functional identities and their applications to graded algebras
![Page 92: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/92.jpg)
Examples of functional identities General FI Theory Applications
d-free sets exist!
K. Beidar (1998): A prime ring R is a d-free subset of its maximalleft ring of quotients Q
⇐⇒R does not satisfy a PI of degree 2d − 2 (i.e., R cannot beembedded in Md−1(F )).
Other important examples of d-free sets:
Lie ideals of prime rings
symmetric elements of prime rings with involution
skew elements of prime rings with involution (and their Lieideals)
semiprime rings
Mn(B), B any unital ring
etc.
Matej Bresar Functional identities and their applications to graded algebras
![Page 93: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/93.jpg)
Examples of functional identities General FI Theory Applications
d-free sets exist!
K. Beidar (1998): A prime ring R is a d-free subset of its maximalleft ring of quotients Q
⇐⇒R does not satisfy a PI of degree 2d − 2 (i.e., R cannot beembedded in Md−1(F )).
Other important examples of d-free sets:
Lie ideals of prime rings
symmetric elements of prime rings with involution
skew elements of prime rings with involution (and their Lieideals)
semiprime rings
Mn(B), B any unital ring
etc.
Matej Bresar Functional identities and their applications to graded algebras
![Page 94: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/94.jpg)
Examples of functional identities General FI Theory Applications
d-free sets exist!
K. Beidar (1998): A prime ring R is a d-free subset of its maximalleft ring of quotients Q
⇐⇒R does not satisfy a PI of degree 2d − 2 (i.e., R cannot beembedded in Md−1(F )).
Other important examples of d-free sets:
Lie ideals of prime rings
symmetric elements of prime rings with involution
skew elements of prime rings with involution (and their Lieideals)
semiprime rings
Mn(B), B any unital ring
etc.
Matej Bresar Functional identities and their applications to graded algebras
![Page 95: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/95.jpg)
Examples of functional identities General FI Theory Applications
d-free sets exist!
K. Beidar (1998): A prime ring R is a d-free subset of its maximalleft ring of quotients Q
⇐⇒R does not satisfy a PI of degree 2d − 2 (i.e., R cannot beembedded in Md−1(F )).
Other important examples of d-free sets:
Lie ideals of prime rings
symmetric elements of prime rings with involution
skew elements of prime rings with involution (and their Lieideals)
semiprime rings
Mn(B), B any unital ring
etc.
Matej Bresar Functional identities and their applications to graded algebras
![Page 96: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/96.jpg)
Examples of functional identities General FI Theory Applications
d-free sets exist!
K. Beidar (1998): A prime ring R is a d-free subset of its maximalleft ring of quotients Q
⇐⇒R does not satisfy a PI of degree 2d − 2 (i.e., R cannot beembedded in Md−1(F )).
Other important examples of d-free sets:
Lie ideals of prime rings
symmetric elements of prime rings with involution
skew elements of prime rings with involution (and their Lieideals)
semiprime rings
Mn(B), B any unital ring
etc.
Matej Bresar Functional identities and their applications to graded algebras
![Page 97: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/97.jpg)
Examples of functional identities General FI Theory Applications
d-free sets exist!
K. Beidar (1998): A prime ring R is a d-free subset of its maximalleft ring of quotients Q
⇐⇒R does not satisfy a PI of degree 2d − 2 (i.e., R cannot beembedded in Md−1(F )).
Other important examples of d-free sets:
Lie ideals of prime rings
symmetric elements of prime rings with involution
skew elements of prime rings with involution (and their Lieideals)
semiprime rings
Mn(B), B any unital ring
etc.Matej Bresar Functional identities and their applications to graded algebras
![Page 98: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/98.jpg)
Examples of functional identities General FI Theory Applications
FI’s on d-free sets
S an arbitrary set,α : S → Q a (fixed) map such that α(S) is d-free;then one can handle FI’s such as
∑t
α(xi1) . . . α(xip )Ft(xj1 , . . . , xjq )α(xk1) . . . α(xkr ) = 0.
M. Bresar, M. A. Chebotar, W. S. Martindale, Functional Identities,Birkhauser Verlag, 2007.
Matej Bresar Functional identities and their applications to graded algebras
![Page 99: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/99.jpg)
Examples of functional identities General FI Theory Applications
FI’s on d-free sets
S an arbitrary set,
α : S → Q a (fixed) map such that α(S) is d-free;then one can handle FI’s such as
∑t
α(xi1) . . . α(xip )Ft(xj1 , . . . , xjq )α(xk1) . . . α(xkr ) = 0.
M. Bresar, M. A. Chebotar, W. S. Martindale, Functional Identities,Birkhauser Verlag, 2007.
Matej Bresar Functional identities and their applications to graded algebras
![Page 100: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/100.jpg)
Examples of functional identities General FI Theory Applications
FI’s on d-free sets
S an arbitrary set,α : S → Q a (fixed) map such that α(S) is d-free;
then one can handle FI’s such as
∑t
α(xi1) . . . α(xip )Ft(xj1 , . . . , xjq )α(xk1) . . . α(xkr ) = 0.
M. Bresar, M. A. Chebotar, W. S. Martindale, Functional Identities,Birkhauser Verlag, 2007.
Matej Bresar Functional identities and their applications to graded algebras
![Page 101: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/101.jpg)
Examples of functional identities General FI Theory Applications
FI’s on d-free sets
S an arbitrary set,α : S → Q a (fixed) map such that α(S) is d-free;then one can handle FI’s such as
∑t
α(xi1) . . . α(xip )Ft(xj1 , . . . , xjq )α(xk1) . . . α(xkr ) = 0.
M. Bresar, M. A. Chebotar, W. S. Martindale, Functional Identities,Birkhauser Verlag, 2007.
Matej Bresar Functional identities and their applications to graded algebras
![Page 102: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/102.jpg)
Examples of functional identities General FI Theory Applications
FI’s on d-free sets
S an arbitrary set,α : S → Q a (fixed) map such that α(S) is d-free;then one can handle FI’s such as
∑t
α(xi1) . . . α(xip )Ft(xj1 , . . . , xjq )α(xk1) . . . α(xkr ) = 0.
M. Bresar, M. A. Chebotar, W. S. Martindale, Functional Identities,Birkhauser Verlag, 2007.
Matej Bresar Functional identities and their applications to graded algebras
![Page 103: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/103.jpg)
Examples of functional identities General FI Theory Applications
Lie homomorphisms of associative rings
S , R rings; α : S → R is a Lie homomorphism if
α(x + y) = α(x) + α(y), α([x , y ]) = [α(x), α(y)].
Here, [x , y ] = xy − yx . ((R, +, [ . ]) is a Lie ring.)Examples:
α = homomorphism
α = −antihomomorphism
α = τ : S → Z , τ([S , S ]) = 0
Herstein’s problem (1961): Is every Lie isomorphism α betweenprime rings of the form α = ϕ + τ, ϕ a homomorphism or thenegative of an antihomorphism?M. B. (1990): Yes (modulo technicalities).
Matej Bresar Functional identities and their applications to graded algebras
![Page 104: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/104.jpg)
Examples of functional identities General FI Theory Applications
Lie homomorphisms of associative rings
S , R rings; α : S → R is a Lie homomorphism if
α(x + y) = α(x) + α(y), α([x , y ]) = [α(x), α(y)].
Here, [x , y ] = xy − yx . ((R, +, [ . ]) is a Lie ring.)Examples:
α = homomorphism
α = −antihomomorphism
α = τ : S → Z , τ([S , S ]) = 0
Herstein’s problem (1961): Is every Lie isomorphism α betweenprime rings of the form α = ϕ + τ, ϕ a homomorphism or thenegative of an antihomorphism?M. B. (1990): Yes (modulo technicalities).
Matej Bresar Functional identities and their applications to graded algebras
![Page 105: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/105.jpg)
Examples of functional identities General FI Theory Applications
Lie homomorphisms of associative rings
S , R rings; α : S → R is a Lie homomorphism if
α(x + y) = α(x) + α(y), α([x , y ]) = [α(x), α(y)].
Here, [x , y ] = xy − yx .
((R, +, [ . ]) is a Lie ring.)Examples:
α = homomorphism
α = −antihomomorphism
α = τ : S → Z , τ([S , S ]) = 0
Herstein’s problem (1961): Is every Lie isomorphism α betweenprime rings of the form α = ϕ + τ, ϕ a homomorphism or thenegative of an antihomorphism?M. B. (1990): Yes (modulo technicalities).
Matej Bresar Functional identities and their applications to graded algebras
![Page 106: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/106.jpg)
Examples of functional identities General FI Theory Applications
Lie homomorphisms of associative rings
S , R rings; α : S → R is a Lie homomorphism if
α(x + y) = α(x) + α(y), α([x , y ]) = [α(x), α(y)].
Here, [x , y ] = xy − yx . ((R, +, [ . ]) is a Lie ring.)
Examples:
α = homomorphism
α = −antihomomorphism
α = τ : S → Z , τ([S , S ]) = 0
Herstein’s problem (1961): Is every Lie isomorphism α betweenprime rings of the form α = ϕ + τ, ϕ a homomorphism or thenegative of an antihomorphism?M. B. (1990): Yes (modulo technicalities).
Matej Bresar Functional identities and their applications to graded algebras
![Page 107: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/107.jpg)
Examples of functional identities General FI Theory Applications
Lie homomorphisms of associative rings
S , R rings; α : S → R is a Lie homomorphism if
α(x + y) = α(x) + α(y), α([x , y ]) = [α(x), α(y)].
Here, [x , y ] = xy − yx . ((R, +, [ . ]) is a Lie ring.)Examples:
α = homomorphism
α = −antihomomorphism
α = τ : S → Z , τ([S , S ]) = 0
Herstein’s problem (1961): Is every Lie isomorphism α betweenprime rings of the form α = ϕ + τ, ϕ a homomorphism or thenegative of an antihomorphism?M. B. (1990): Yes (modulo technicalities).
Matej Bresar Functional identities and their applications to graded algebras
![Page 108: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/108.jpg)
Examples of functional identities General FI Theory Applications
Lie homomorphisms of associative rings
S , R rings; α : S → R is a Lie homomorphism if
α(x + y) = α(x) + α(y), α([x , y ]) = [α(x), α(y)].
Here, [x , y ] = xy − yx . ((R, +, [ . ]) is a Lie ring.)Examples:
α = homomorphism
α = −antihomomorphism
α = τ : S → Z , τ([S , S ]) = 0
Herstein’s problem (1961): Is every Lie isomorphism α betweenprime rings of the form α = ϕ + τ, ϕ a homomorphism or thenegative of an antihomorphism?M. B. (1990): Yes (modulo technicalities).
Matej Bresar Functional identities and their applications to graded algebras
![Page 109: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/109.jpg)
Examples of functional identities General FI Theory Applications
Lie homomorphisms of associative rings
S , R rings; α : S → R is a Lie homomorphism if
α(x + y) = α(x) + α(y), α([x , y ]) = [α(x), α(y)].
Here, [x , y ] = xy − yx . ((R, +, [ . ]) is a Lie ring.)Examples:
α = homomorphism
α = −antihomomorphism
α = τ : S → Z , τ([S , S ]) = 0
Herstein’s problem (1961): Is every Lie isomorphism α betweenprime rings of the form α = ϕ + τ, ϕ a homomorphism or thenegative of an antihomorphism?M. B. (1990): Yes (modulo technicalities).
Matej Bresar Functional identities and their applications to graded algebras
![Page 110: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/110.jpg)
Examples of functional identities General FI Theory Applications
Lie homomorphisms of associative rings
S , R rings; α : S → R is a Lie homomorphism if
α(x + y) = α(x) + α(y), α([x , y ]) = [α(x), α(y)].
Here, [x , y ] = xy − yx . ((R, +, [ . ]) is a Lie ring.)Examples:
α = homomorphism
α = −antihomomorphism
α = τ : S → Z , τ([S , S ]) = 0
Herstein’s problem (1961): Is every Lie isomorphism α betweenprime rings of the form α = ϕ + τ, ϕ a homomorphism or thenegative of an antihomorphism?M. B. (1990): Yes (modulo technicalities).
Matej Bresar Functional identities and their applications to graded algebras
![Page 111: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/111.jpg)
Examples of functional identities General FI Theory Applications
Lie homomorphisms of associative rings
S , R rings; α : S → R is a Lie homomorphism if
α(x + y) = α(x) + α(y), α([x , y ]) = [α(x), α(y)].
Here, [x , y ] = xy − yx . ((R, +, [ . ]) is a Lie ring.)Examples:
α = homomorphism
α = −antihomomorphism
α = τ : S → Z , τ([S , S ]) = 0
Herstein’s problem (1961): Is every Lie isomorphism α betweenprime rings of the form α = ϕ + τ, ϕ a homomorphism or thenegative of an antihomorphism?
M. B. (1990): Yes (modulo technicalities).
Matej Bresar Functional identities and their applications to graded algebras
![Page 112: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/112.jpg)
Examples of functional identities General FI Theory Applications
Lie homomorphisms of associative rings
S , R rings; α : S → R is a Lie homomorphism if
α(x + y) = α(x) + α(y), α([x , y ]) = [α(x), α(y)].
Here, [x , y ] = xy − yx . ((R, +, [ . ]) is a Lie ring.)Examples:
α = homomorphism
α = −antihomomorphism
α = τ : S → Z , τ([S , S ]) = 0
Herstein’s problem (1961): Is every Lie isomorphism α betweenprime rings of the form α = ϕ + τ, ϕ a homomorphism or thenegative of an antihomorphism?M. B. (1990): Yes
(modulo technicalities).
Matej Bresar Functional identities and their applications to graded algebras
![Page 113: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/113.jpg)
Examples of functional identities General FI Theory Applications
Lie homomorphisms of associative rings
S , R rings; α : S → R is a Lie homomorphism if
α(x + y) = α(x) + α(y), α([x , y ]) = [α(x), α(y)].
Here, [x , y ] = xy − yx . ((R, +, [ . ]) is a Lie ring.)Examples:
α = homomorphism
α = −antihomomorphism
α = τ : S → Z , τ([S , S ]) = 0
Herstein’s problem (1961): Is every Lie isomorphism α betweenprime rings of the form α = ϕ + τ, ϕ a homomorphism or thenegative of an antihomorphism?M. B. (1990): Yes (modulo technicalities).
Matej Bresar Functional identities and their applications to graded algebras
![Page 114: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/114.jpg)
Examples of functional identities General FI Theory Applications
Idea of proof
[α(y 2), α(y)] = α([y 2, y ]) = 0
x = α(y) =⇒ [α(α−1(x)2), x ] = 0
i.e., [f (x , x), x ] = 0 as in Example 7
=⇒ f (x , x) = α(α−1(x)2) = λx2 + µ(x)x + ν(x)
=⇒ α(y 2) = λα(y)2 + µ′(y)α(y) + ν′(y).
Matej Bresar Functional identities and their applications to graded algebras
![Page 115: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/115.jpg)
Examples of functional identities General FI Theory Applications
Idea of proof
[α(y 2), α(y)] = α([y 2, y ]) = 0
x = α(y) =⇒ [α(α−1(x)2), x ] = 0
i.e., [f (x , x), x ] = 0 as in Example 7
=⇒ f (x , x) = α(α−1(x)2) = λx2 + µ(x)x + ν(x)
=⇒ α(y 2) = λα(y)2 + µ′(y)α(y) + ν′(y).
Matej Bresar Functional identities and their applications to graded algebras
![Page 116: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/116.jpg)
Examples of functional identities General FI Theory Applications
Idea of proof
[α(y 2), α(y)] = α([y 2, y ]) = 0
x = α(y) =⇒ [α(α−1(x)2), x ] = 0
i.e., [f (x , x), x ] = 0 as in Example 7
=⇒ f (x , x) = α(α−1(x)2) = λx2 + µ(x)x + ν(x)
=⇒ α(y 2) = λα(y)2 + µ′(y)α(y) + ν′(y).
Matej Bresar Functional identities and their applications to graded algebras
![Page 117: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/117.jpg)
Examples of functional identities General FI Theory Applications
Idea of proof
[α(y 2), α(y)] = α([y 2, y ]) = 0
x = α(y) =⇒ [α(α−1(x)2), x ] = 0
i.e., [f (x , x), x ] = 0 as in Example 7
=⇒ f (x , x) = α(α−1(x)2) = λx2 + µ(x)x + ν(x)
=⇒ α(y 2) = λα(y)2 + µ′(y)α(y) + ν′(y).
Matej Bresar Functional identities and their applications to graded algebras
![Page 118: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/118.jpg)
Examples of functional identities General FI Theory Applications
Idea of proof
[α(y 2), α(y)] = α([y 2, y ]) = 0
x = α(y) =⇒ [α(α−1(x)2), x ] = 0
i.e., [f (x , x), x ] = 0 as in Example 7
=⇒ f (x , x) = α(α−1(x)2) = λx2 + µ(x)x + ν(x)
=⇒ α(y 2) = λα(y)2 + µ′(y)α(y) + ν′(y).
Matej Bresar Functional identities and their applications to graded algebras
![Page 119: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/119.jpg)
Examples of functional identities General FI Theory Applications
Idea of proof
[α(y 2), α(y)] = α([y 2, y ]) = 0
x = α(y) =⇒ [α(α−1(x)2), x ] = 0
i.e., [f (x , x), x ] = 0 as in Example 7
=⇒ f (x , x) = α(α−1(x)2) = λx2 + µ(x)x + ν(x)
=⇒ α(y 2) = λα(y)2 + µ′(y)α(y) + ν′(y).
Matej Bresar Functional identities and their applications to graded algebras
![Page 120: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/120.jpg)
Examples of functional identities General FI Theory Applications
Idea of proof
[α(y 2), α(y)] = α([y 2, y ]) = 0
x = α(y) =⇒ [α(α−1(x)2), x ] = 0
i.e., [f (x , x), x ] = 0 as in Example 7
=⇒ f (x , x) = α(α−1(x)2) = λx2 + µ(x)x + ν(x)
=⇒ α(y 2) = λα(y)2 + µ′(y)α(y) + ν′(y).
Matej Bresar Functional identities and their applications to graded algebras
![Page 121: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/121.jpg)
Examples of functional identities General FI Theory Applications
Lie homomorphisms of skew elements
Other Herstein’s questions on Lie homomorphismsR a ring with involution ∗,
K = {x ∈ R | x∗ = −x}
skew elements of R.Can a Lie isomorphism α : [K , K ]→ [K , K ] be extended to ahomomorphism?K. Beidar, M. Bresar, M. Chebotar, W. Martindale (series ofpapers 1994-2002): Yes, unless R ⊆ M21(F ).A detailed analysis shows: Mn(F ) with n = 1, 2, 3, 4, 5, 6, 8 mustreally be excluded.Here, R is simple or even prime, or satisfies some abstracttechnical conditions.
Matej Bresar Functional identities and their applications to graded algebras
![Page 122: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/122.jpg)
Examples of functional identities General FI Theory Applications
Lie homomorphisms of skew elements
Other Herstein’s questions on Lie homomorphisms
R a ring with involution ∗,
K = {x ∈ R | x∗ = −x}
skew elements of R.Can a Lie isomorphism α : [K , K ]→ [K , K ] be extended to ahomomorphism?K. Beidar, M. Bresar, M. Chebotar, W. Martindale (series ofpapers 1994-2002): Yes, unless R ⊆ M21(F ).A detailed analysis shows: Mn(F ) with n = 1, 2, 3, 4, 5, 6, 8 mustreally be excluded.Here, R is simple or even prime, or satisfies some abstracttechnical conditions.
Matej Bresar Functional identities and their applications to graded algebras
![Page 123: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/123.jpg)
Examples of functional identities General FI Theory Applications
Lie homomorphisms of skew elements
Other Herstein’s questions on Lie homomorphismsR a ring with involution ∗,
K = {x ∈ R | x∗ = −x}
skew elements of R.Can a Lie isomorphism α : [K , K ]→ [K , K ] be extended to ahomomorphism?K. Beidar, M. Bresar, M. Chebotar, W. Martindale (series ofpapers 1994-2002): Yes, unless R ⊆ M21(F ).A detailed analysis shows: Mn(F ) with n = 1, 2, 3, 4, 5, 6, 8 mustreally be excluded.Here, R is simple or even prime, or satisfies some abstracttechnical conditions.
Matej Bresar Functional identities and their applications to graded algebras
![Page 124: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/124.jpg)
Examples of functional identities General FI Theory Applications
Lie homomorphisms of skew elements
Other Herstein’s questions on Lie homomorphismsR a ring with involution ∗,
K = {x ∈ R | x∗ = −x}
skew elements of R.
Can a Lie isomorphism α : [K , K ]→ [K , K ] be extended to ahomomorphism?K. Beidar, M. Bresar, M. Chebotar, W. Martindale (series ofpapers 1994-2002): Yes, unless R ⊆ M21(F ).A detailed analysis shows: Mn(F ) with n = 1, 2, 3, 4, 5, 6, 8 mustreally be excluded.Here, R is simple or even prime, or satisfies some abstracttechnical conditions.
Matej Bresar Functional identities and their applications to graded algebras
![Page 125: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/125.jpg)
Examples of functional identities General FI Theory Applications
Lie homomorphisms of skew elements
Other Herstein’s questions on Lie homomorphismsR a ring with involution ∗,
K = {x ∈ R | x∗ = −x}
skew elements of R.Can a Lie isomorphism α : [K , K ]→ [K , K ] be extended to ahomomorphism?
K. Beidar, M. Bresar, M. Chebotar, W. Martindale (series ofpapers 1994-2002): Yes, unless R ⊆ M21(F ).A detailed analysis shows: Mn(F ) with n = 1, 2, 3, 4, 5, 6, 8 mustreally be excluded.Here, R is simple or even prime, or satisfies some abstracttechnical conditions.
Matej Bresar Functional identities and their applications to graded algebras
![Page 126: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/126.jpg)
Examples of functional identities General FI Theory Applications
Lie homomorphisms of skew elements
Other Herstein’s questions on Lie homomorphismsR a ring with involution ∗,
K = {x ∈ R | x∗ = −x}
skew elements of R.Can a Lie isomorphism α : [K , K ]→ [K , K ] be extended to ahomomorphism?K. Beidar, M. Bresar, M. Chebotar, W. Martindale (series ofpapers 1994-2002): Yes, unless R ⊆ M21(F ).
A detailed analysis shows: Mn(F ) with n = 1, 2, 3, 4, 5, 6, 8 mustreally be excluded.Here, R is simple or even prime, or satisfies some abstracttechnical conditions.
Matej Bresar Functional identities and their applications to graded algebras
![Page 127: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/127.jpg)
Examples of functional identities General FI Theory Applications
Lie homomorphisms of skew elements
Other Herstein’s questions on Lie homomorphismsR a ring with involution ∗,
K = {x ∈ R | x∗ = −x}
skew elements of R.Can a Lie isomorphism α : [K , K ]→ [K , K ] be extended to ahomomorphism?K. Beidar, M. Bresar, M. Chebotar, W. Martindale (series ofpapers 1994-2002): Yes, unless R ⊆ M21(F ).A detailed analysis shows: Mn(F ) with n = 1, 2, 3, 4, 5, 6, 8 mustreally be excluded.
Here, R is simple or even prime, or satisfies some abstracttechnical conditions.
Matej Bresar Functional identities and their applications to graded algebras
![Page 128: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/128.jpg)
Examples of functional identities General FI Theory Applications
Lie homomorphisms of skew elements
Other Herstein’s questions on Lie homomorphismsR a ring with involution ∗,
K = {x ∈ R | x∗ = −x}
skew elements of R.Can a Lie isomorphism α : [K , K ]→ [K , K ] be extended to ahomomorphism?K. Beidar, M. Bresar, M. Chebotar, W. Martindale (series ofpapers 1994-2002): Yes, unless R ⊆ M21(F ).A detailed analysis shows: Mn(F ) with n = 1, 2, 3, 4, 5, 6, 8 mustreally be excluded.Here, R is simple
or even prime, or satisfies some abstracttechnical conditions.
Matej Bresar Functional identities and their applications to graded algebras
![Page 129: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/129.jpg)
Examples of functional identities General FI Theory Applications
Lie homomorphisms of skew elements
Other Herstein’s questions on Lie homomorphismsR a ring with involution ∗,
K = {x ∈ R | x∗ = −x}
skew elements of R.Can a Lie isomorphism α : [K , K ]→ [K , K ] be extended to ahomomorphism?K. Beidar, M. Bresar, M. Chebotar, W. Martindale (series ofpapers 1994-2002): Yes, unless R ⊆ M21(F ).A detailed analysis shows: Mn(F ) with n = 1, 2, 3, 4, 5, 6, 8 mustreally be excluded.Here, R is simple or even prime,
or satisfies some abstracttechnical conditions.
Matej Bresar Functional identities and their applications to graded algebras
![Page 130: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/130.jpg)
Examples of functional identities General FI Theory Applications
Lie homomorphisms of skew elements
Other Herstein’s questions on Lie homomorphismsR a ring with involution ∗,
K = {x ∈ R | x∗ = −x}
skew elements of R.Can a Lie isomorphism α : [K , K ]→ [K , K ] be extended to ahomomorphism?K. Beidar, M. Bresar, M. Chebotar, W. Martindale (series ofpapers 1994-2002): Yes, unless R ⊆ M21(F ).A detailed analysis shows: Mn(F ) with n = 1, 2, 3, 4, 5, 6, 8 mustreally be excluded.Here, R is simple or even prime, or satisfies some abstracttechnical conditions.
Matej Bresar Functional identities and their applications to graded algebras
![Page 131: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/131.jpg)
Examples of functional identities General FI Theory Applications
Graded algebras
G: abelian groupA (nonassociative) algebra R is G -graded if R = ∑g∈G Rg withRg Rh ⊆ Rgh.L : Lie algebra, J : Jordan algebra, A : associative algebraProblem: L ⊆ A−, L G -graded. Is then A G -graded,A = ∑g∈G Ag , and Lg = Ag ∩ L?Similar problem for J ⊆ A+.Example: A: algebra with involution, H: symmetric elements, K :skew elements, G = Z2.L = A− is G -graded with L0 = K ,L1 = H: [Li , Lj ] ⊆ Li+j . But A is not G -graded!The “right” problem: There may be two possibilities:Lg = Ag ∩ L or there exists t ∈ G with t2 = 1 such that:Lg = K (Ag , ∗) ∩ L⊕H(Atg , ∗) ∩ L.
Matej Bresar Functional identities and their applications to graded algebras
![Page 132: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/132.jpg)
Examples of functional identities General FI Theory Applications
Graded algebras
G: abelian group
A (nonassociative) algebra R is G -graded if R = ∑g∈G Rg withRg Rh ⊆ Rgh.L : Lie algebra, J : Jordan algebra, A : associative algebraProblem: L ⊆ A−, L G -graded. Is then A G -graded,A = ∑g∈G Ag , and Lg = Ag ∩ L?Similar problem for J ⊆ A+.Example: A: algebra with involution, H: symmetric elements, K :skew elements, G = Z2.L = A− is G -graded with L0 = K ,L1 = H: [Li , Lj ] ⊆ Li+j . But A is not G -graded!The “right” problem: There may be two possibilities:Lg = Ag ∩ L or there exists t ∈ G with t2 = 1 such that:Lg = K (Ag , ∗) ∩ L⊕H(Atg , ∗) ∩ L.
Matej Bresar Functional identities and their applications to graded algebras
![Page 133: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/133.jpg)
Examples of functional identities General FI Theory Applications
Graded algebras
G: abelian groupA (nonassociative) algebra R is G -graded if R = ∑g∈G Rg withRg Rh ⊆ Rgh.
L : Lie algebra, J : Jordan algebra, A : associative algebraProblem: L ⊆ A−, L G -graded. Is then A G -graded,A = ∑g∈G Ag , and Lg = Ag ∩ L?Similar problem for J ⊆ A+.Example: A: algebra with involution, H: symmetric elements, K :skew elements, G = Z2.L = A− is G -graded with L0 = K ,L1 = H: [Li , Lj ] ⊆ Li+j . But A is not G -graded!The “right” problem: There may be two possibilities:Lg = Ag ∩ L or there exists t ∈ G with t2 = 1 such that:Lg = K (Ag , ∗) ∩ L⊕H(Atg , ∗) ∩ L.
Matej Bresar Functional identities and their applications to graded algebras
![Page 134: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/134.jpg)
Examples of functional identities General FI Theory Applications
Graded algebras
G: abelian groupA (nonassociative) algebra R is G -graded if R = ∑g∈G Rg withRg Rh ⊆ Rgh.L : Lie algebra, J : Jordan algebra, A : associative algebra
Problem: L ⊆ A−, L G -graded. Is then A G -graded,A = ∑g∈G Ag , and Lg = Ag ∩ L?Similar problem for J ⊆ A+.Example: A: algebra with involution, H: symmetric elements, K :skew elements, G = Z2.L = A− is G -graded with L0 = K ,L1 = H: [Li , Lj ] ⊆ Li+j . But A is not G -graded!The “right” problem: There may be two possibilities:Lg = Ag ∩ L or there exists t ∈ G with t2 = 1 such that:Lg = K (Ag , ∗) ∩ L⊕H(Atg , ∗) ∩ L.
Matej Bresar Functional identities and their applications to graded algebras
![Page 135: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/135.jpg)
Examples of functional identities General FI Theory Applications
Graded algebras
G: abelian groupA (nonassociative) algebra R is G -graded if R = ∑g∈G Rg withRg Rh ⊆ Rgh.L : Lie algebra, J : Jordan algebra, A : associative algebraProblem: L ⊆ A−, L G -graded.
Is then A G -graded,A = ∑g∈G Ag , and Lg = Ag ∩ L?Similar problem for J ⊆ A+.Example: A: algebra with involution, H: symmetric elements, K :skew elements, G = Z2.L = A− is G -graded with L0 = K ,L1 = H: [Li , Lj ] ⊆ Li+j . But A is not G -graded!The “right” problem: There may be two possibilities:Lg = Ag ∩ L or there exists t ∈ G with t2 = 1 such that:Lg = K (Ag , ∗) ∩ L⊕H(Atg , ∗) ∩ L.
Matej Bresar Functional identities and their applications to graded algebras
![Page 136: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/136.jpg)
Examples of functional identities General FI Theory Applications
Graded algebras
G: abelian groupA (nonassociative) algebra R is G -graded if R = ∑g∈G Rg withRg Rh ⊆ Rgh.L : Lie algebra, J : Jordan algebra, A : associative algebraProblem: L ⊆ A−, L G -graded. Is then A G -graded,A = ∑g∈G Ag , and Lg = Ag ∩ L?
Similar problem for J ⊆ A+.Example: A: algebra with involution, H: symmetric elements, K :skew elements, G = Z2.L = A− is G -graded with L0 = K ,L1 = H: [Li , Lj ] ⊆ Li+j . But A is not G -graded!The “right” problem: There may be two possibilities:Lg = Ag ∩ L or there exists t ∈ G with t2 = 1 such that:Lg = K (Ag , ∗) ∩ L⊕H(Atg , ∗) ∩ L.
Matej Bresar Functional identities and their applications to graded algebras
![Page 137: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/137.jpg)
Examples of functional identities General FI Theory Applications
Graded algebras
G: abelian groupA (nonassociative) algebra R is G -graded if R = ∑g∈G Rg withRg Rh ⊆ Rgh.L : Lie algebra, J : Jordan algebra, A : associative algebraProblem: L ⊆ A−, L G -graded. Is then A G -graded,A = ∑g∈G Ag , and Lg = Ag ∩ L?Similar problem for J ⊆ A+.
Example: A: algebra with involution, H: symmetric elements, K :skew elements, G = Z2.L = A− is G -graded with L0 = K ,L1 = H: [Li , Lj ] ⊆ Li+j . But A is not G -graded!The “right” problem: There may be two possibilities:Lg = Ag ∩ L or there exists t ∈ G with t2 = 1 such that:Lg = K (Ag , ∗) ∩ L⊕H(Atg , ∗) ∩ L.
Matej Bresar Functional identities and their applications to graded algebras
![Page 138: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/138.jpg)
Examples of functional identities General FI Theory Applications
Graded algebras
G: abelian groupA (nonassociative) algebra R is G -graded if R = ∑g∈G Rg withRg Rh ⊆ Rgh.L : Lie algebra, J : Jordan algebra, A : associative algebraProblem: L ⊆ A−, L G -graded. Is then A G -graded,A = ∑g∈G Ag , and Lg = Ag ∩ L?Similar problem for J ⊆ A+.Example: A: algebra with involution, H: symmetric elements, K :skew elements, G = Z2.
L = A− is G -graded with L0 = K ,L1 = H: [Li , Lj ] ⊆ Li+j . But A is not G -graded!The “right” problem: There may be two possibilities:Lg = Ag ∩ L or there exists t ∈ G with t2 = 1 such that:Lg = K (Ag , ∗) ∩ L⊕H(Atg , ∗) ∩ L.
Matej Bresar Functional identities and their applications to graded algebras
![Page 139: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/139.jpg)
Examples of functional identities General FI Theory Applications
Graded algebras
G: abelian groupA (nonassociative) algebra R is G -graded if R = ∑g∈G Rg withRg Rh ⊆ Rgh.L : Lie algebra, J : Jordan algebra, A : associative algebraProblem: L ⊆ A−, L G -graded. Is then A G -graded,A = ∑g∈G Ag , and Lg = Ag ∩ L?Similar problem for J ⊆ A+.Example: A: algebra with involution, H: symmetric elements, K :skew elements, G = Z2.L = A− is G -graded with L0 = K ,L1 = H: [Li , Lj ] ⊆ Li+j .
But A is not G -graded!The “right” problem: There may be two possibilities:Lg = Ag ∩ L or there exists t ∈ G with t2 = 1 such that:Lg = K (Ag , ∗) ∩ L⊕H(Atg , ∗) ∩ L.
Matej Bresar Functional identities and their applications to graded algebras
![Page 140: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/140.jpg)
Examples of functional identities General FI Theory Applications
Graded algebras
G: abelian groupA (nonassociative) algebra R is G -graded if R = ∑g∈G Rg withRg Rh ⊆ Rgh.L : Lie algebra, J : Jordan algebra, A : associative algebraProblem: L ⊆ A−, L G -graded. Is then A G -graded,A = ∑g∈G Ag , and Lg = Ag ∩ L?Similar problem for J ⊆ A+.Example: A: algebra with involution, H: symmetric elements, K :skew elements, G = Z2.L = A− is G -graded with L0 = K ,L1 = H: [Li , Lj ] ⊆ Li+j . But A is not G -graded!
The “right” problem: There may be two possibilities:Lg = Ag ∩ L or there exists t ∈ G with t2 = 1 such that:Lg = K (Ag , ∗) ∩ L⊕H(Atg , ∗) ∩ L.
Matej Bresar Functional identities and their applications to graded algebras
![Page 141: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/141.jpg)
Examples of functional identities General FI Theory Applications
Graded algebras
G: abelian groupA (nonassociative) algebra R is G -graded if R = ∑g∈G Rg withRg Rh ⊆ Rgh.L : Lie algebra, J : Jordan algebra, A : associative algebraProblem: L ⊆ A−, L G -graded. Is then A G -graded,A = ∑g∈G Ag , and Lg = Ag ∩ L?Similar problem for J ⊆ A+.Example: A: algebra with involution, H: symmetric elements, K :skew elements, G = Z2.L = A− is G -graded with L0 = K ,L1 = H: [Li , Lj ] ⊆ Li+j . But A is not G -graded!The “right” problem: There may be two possibilities:Lg = Ag ∩ L or
there exists t ∈ G with t2 = 1 such that:Lg = K (Ag , ∗) ∩ L⊕H(Atg , ∗) ∩ L.
Matej Bresar Functional identities and their applications to graded algebras
![Page 142: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/142.jpg)
Examples of functional identities General FI Theory Applications
Graded algebras
G: abelian groupA (nonassociative) algebra R is G -graded if R = ∑g∈G Rg withRg Rh ⊆ Rgh.L : Lie algebra, J : Jordan algebra, A : associative algebraProblem: L ⊆ A−, L G -graded. Is then A G -graded,A = ∑g∈G Ag , and Lg = Ag ∩ L?Similar problem for J ⊆ A+.Example: A: algebra with involution, H: symmetric elements, K :skew elements, G = Z2.L = A− is G -graded with L0 = K ,L1 = H: [Li , Lj ] ⊆ Li+j . But A is not G -graded!The “right” problem: There may be two possibilities:Lg = Ag ∩ L or there exists t ∈ G with t2 = 1 such that:Lg = K (Ag , ∗) ∩ L⊕H(Atg , ∗) ∩ L.
Matej Bresar Functional identities and their applications to graded algebras
![Page 143: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/143.jpg)
Examples of functional identities General FI Theory Applications
Results
Series of papers by Bahturin, Kochetov, Montgomery, Shestakov,Zaicev...: classical finite dimensional Lie and Jordan algebras.New proofs and generalizations to infinite dimensional algebrasusing FI’s: Bahturin-Bresar (Lie algebras) andBahturin-Bresar-Shestakov (Jordan algebras).Result for a Lie ideal of skew elements K of a prime algebra A: onlythe first possibilty under very mild assumptions (e.g., dim A ≥ 441)Result for a prime algebra A: both possibilities, a technicalassumption also on GJordan case: similar results, but less restrictions
Matej Bresar Functional identities and their applications to graded algebras
![Page 144: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/144.jpg)
Examples of functional identities General FI Theory Applications
Results
Series of papers by Bahturin, Kochetov, Montgomery, Shestakov,Zaicev...: classical finite dimensional Lie and Jordan algebras.
New proofs and generalizations to infinite dimensional algebrasusing FI’s: Bahturin-Bresar (Lie algebras) andBahturin-Bresar-Shestakov (Jordan algebras).Result for a Lie ideal of skew elements K of a prime algebra A: onlythe first possibilty under very mild assumptions (e.g., dim A ≥ 441)Result for a prime algebra A: both possibilities, a technicalassumption also on GJordan case: similar results, but less restrictions
Matej Bresar Functional identities and their applications to graded algebras
![Page 145: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/145.jpg)
Examples of functional identities General FI Theory Applications
Results
Series of papers by Bahturin, Kochetov, Montgomery, Shestakov,Zaicev...: classical finite dimensional Lie and Jordan algebras.New proofs and generalizations to infinite dimensional algebrasusing FI’s:
Bahturin-Bresar (Lie algebras) andBahturin-Bresar-Shestakov (Jordan algebras).Result for a Lie ideal of skew elements K of a prime algebra A: onlythe first possibilty under very mild assumptions (e.g., dim A ≥ 441)Result for a prime algebra A: both possibilities, a technicalassumption also on GJordan case: similar results, but less restrictions
Matej Bresar Functional identities and their applications to graded algebras
![Page 146: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/146.jpg)
Examples of functional identities General FI Theory Applications
Results
Series of papers by Bahturin, Kochetov, Montgomery, Shestakov,Zaicev...: classical finite dimensional Lie and Jordan algebras.New proofs and generalizations to infinite dimensional algebrasusing FI’s: Bahturin-Bresar (Lie algebras) andBahturin-Bresar-Shestakov (Jordan algebras).
Result for a Lie ideal of skew elements K of a prime algebra A: onlythe first possibilty under very mild assumptions (e.g., dim A ≥ 441)Result for a prime algebra A: both possibilities, a technicalassumption also on GJordan case: similar results, but less restrictions
Matej Bresar Functional identities and their applications to graded algebras
![Page 147: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/147.jpg)
Examples of functional identities General FI Theory Applications
Results
Series of papers by Bahturin, Kochetov, Montgomery, Shestakov,Zaicev...: classical finite dimensional Lie and Jordan algebras.New proofs and generalizations to infinite dimensional algebrasusing FI’s: Bahturin-Bresar (Lie algebras) andBahturin-Bresar-Shestakov (Jordan algebras).Result for a Lie ideal of skew elements K of a prime algebra A: onlythe first possibilty under very mild assumptions (e.g., dim A ≥ 441)
Result for a prime algebra A: both possibilities, a technicalassumption also on GJordan case: similar results, but less restrictions
Matej Bresar Functional identities and their applications to graded algebras
![Page 148: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/148.jpg)
Examples of functional identities General FI Theory Applications
Results
Series of papers by Bahturin, Kochetov, Montgomery, Shestakov,Zaicev...: classical finite dimensional Lie and Jordan algebras.New proofs and generalizations to infinite dimensional algebrasusing FI’s: Bahturin-Bresar (Lie algebras) andBahturin-Bresar-Shestakov (Jordan algebras).Result for a Lie ideal of skew elements K of a prime algebra A: onlythe first possibilty under very mild assumptions (e.g., dim A ≥ 441)Result for a prime algebra A: both possibilities, a technicalassumption also on G
Jordan case: similar results, but less restrictions
Matej Bresar Functional identities and their applications to graded algebras
![Page 149: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/149.jpg)
Examples of functional identities General FI Theory Applications
Results
Series of papers by Bahturin, Kochetov, Montgomery, Shestakov,Zaicev...: classical finite dimensional Lie and Jordan algebras.New proofs and generalizations to infinite dimensional algebrasusing FI’s: Bahturin-Bresar (Lie algebras) andBahturin-Bresar-Shestakov (Jordan algebras).Result for a Lie ideal of skew elements K of a prime algebra A: onlythe first possibilty under very mild assumptions (e.g., dim A ≥ 441)Result for a prime algebra A: both possibilities, a technicalassumption also on GJordan case: similar results, but less restrictions
Matej Bresar Functional identities and their applications to graded algebras
![Page 150: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/150.jpg)
Examples of functional identities General FI Theory Applications
Idea of proof
A algebra over F , L ⊆ A−, L G -graded,H = FG group algebraρ : L⊗H → L⊗H
ρ(ag ⊗ h) = ag ⊗ gh
is a Lie isomorphism of L⊗H ⊆ A⊗H.A⊗H is not prime etc., use deeper results.
Matej Bresar Functional identities and their applications to graded algebras
![Page 151: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/151.jpg)
Examples of functional identities General FI Theory Applications
Idea of proof
A algebra over F , L ⊆ A−, L G -graded,
H = FG group algebraρ : L⊗H → L⊗H
ρ(ag ⊗ h) = ag ⊗ gh
is a Lie isomorphism of L⊗H ⊆ A⊗H.A⊗H is not prime etc., use deeper results.
Matej Bresar Functional identities and their applications to graded algebras
![Page 152: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/152.jpg)
Examples of functional identities General FI Theory Applications
Idea of proof
A algebra over F , L ⊆ A−, L G -graded,H = FG group algebraρ : L⊗H → L⊗H
ρ(ag ⊗ h) = ag ⊗ gh
is a Lie isomorphism of L⊗H ⊆ A⊗H.A⊗H is not prime etc., use deeper results.
Matej Bresar Functional identities and their applications to graded algebras
![Page 153: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/153.jpg)
Examples of functional identities General FI Theory Applications
Idea of proof
A algebra over F , L ⊆ A−, L G -graded,H = FG group algebraρ : L⊗H → L⊗H
ρ(ag ⊗ h) = ag ⊗ gh
is a Lie isomorphism of L⊗H ⊆ A⊗H.A⊗H is not prime etc., use deeper results.
Matej Bresar Functional identities and their applications to graded algebras
![Page 154: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/154.jpg)
Examples of functional identities General FI Theory Applications
Idea of proof
A algebra over F , L ⊆ A−, L G -graded,H = FG group algebraρ : L⊗H → L⊗H
ρ(ag ⊗ h) = ag ⊗ gh
is a Lie isomorphism of L⊗H ⊆ A⊗H.
A⊗H is not prime etc., use deeper results.
Matej Bresar Functional identities and their applications to graded algebras
![Page 155: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/155.jpg)
Examples of functional identities General FI Theory Applications
Idea of proof
A algebra over F , L ⊆ A−, L G -graded,H = FG group algebraρ : L⊗H → L⊗H
ρ(ag ⊗ h) = ag ⊗ gh
is a Lie isomorphism of L⊗H ⊆ A⊗H.A⊗H is not prime etc.,
use deeper results.
Matej Bresar Functional identities and their applications to graded algebras
![Page 156: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/156.jpg)
Examples of functional identities General FI Theory Applications
Idea of proof
A algebra over F , L ⊆ A−, L G -graded,H = FG group algebraρ : L⊗H → L⊗H
ρ(ag ⊗ h) = ag ⊗ gh
is a Lie isomorphism of L⊗H ⊆ A⊗H.A⊗H is not prime etc., use deeper results.
Matej Bresar Functional identities and their applications to graded algebras
![Page 157: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/157.jpg)
Examples of functional identities General FI Theory Applications
Lie superhomomorphisms
A = A0 ⊕ A1: associative superalgebra
[a, b]s = ab− (−1)|a||b|ba
(|a| = i if a ∈ Ai ), A becomes a Lie superalgebra.Lie superhomomorphism: preserves [a, b]s .Problem: Describe it!Bahturin-Bresar: extending a Lie superhomomorphism to theGrassman envelope makes it possible to use FI’s.
Matej Bresar Functional identities and their applications to graded algebras
![Page 158: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/158.jpg)
Examples of functional identities General FI Theory Applications
Lie superhomomorphisms
A = A0 ⊕ A1: associative superalgebra
[a, b]s = ab− (−1)|a||b|ba
(|a| = i if a ∈ Ai ), A becomes a Lie superalgebra.Lie superhomomorphism: preserves [a, b]s .Problem: Describe it!Bahturin-Bresar: extending a Lie superhomomorphism to theGrassman envelope makes it possible to use FI’s.
Matej Bresar Functional identities and their applications to graded algebras
![Page 159: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/159.jpg)
Examples of functional identities General FI Theory Applications
Lie superhomomorphisms
A = A0 ⊕ A1: associative superalgebra
[a, b]s = ab− (−1)|a||b|ba
(|a| = i if a ∈ Ai ), A becomes a Lie superalgebra.
Lie superhomomorphism: preserves [a, b]s .Problem: Describe it!Bahturin-Bresar: extending a Lie superhomomorphism to theGrassman envelope makes it possible to use FI’s.
Matej Bresar Functional identities and their applications to graded algebras
![Page 160: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/160.jpg)
Examples of functional identities General FI Theory Applications
Lie superhomomorphisms
A = A0 ⊕ A1: associative superalgebra
[a, b]s = ab− (−1)|a||b|ba
(|a| = i if a ∈ Ai ), A becomes a Lie superalgebra.Lie superhomomorphism: preserves [a, b]s .
Problem: Describe it!Bahturin-Bresar: extending a Lie superhomomorphism to theGrassman envelope makes it possible to use FI’s.
Matej Bresar Functional identities and their applications to graded algebras
![Page 161: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/161.jpg)
Examples of functional identities General FI Theory Applications
Lie superhomomorphisms
A = A0 ⊕ A1: associative superalgebra
[a, b]s = ab− (−1)|a||b|ba
(|a| = i if a ∈ Ai ), A becomes a Lie superalgebra.Lie superhomomorphism: preserves [a, b]s .Problem: Describe it!
Bahturin-Bresar: extending a Lie superhomomorphism to theGrassman envelope makes it possible to use FI’s.
Matej Bresar Functional identities and their applications to graded algebras
![Page 162: Functional identities and their applications to graded ...liejor/2009/Conference_Manaus/Bresar.pdf · Matej Bre sar Functional identities and their applications to graded algebras](https://reader033.vdocument.in/reader033/viewer/2022050415/5f8b67dc542e3b178b64531c/html5/thumbnails/162.jpg)
Examples of functional identities General FI Theory Applications
Lie superhomomorphisms
A = A0 ⊕ A1: associative superalgebra
[a, b]s = ab− (−1)|a||b|ba
(|a| = i if a ∈ Ai ), A becomes a Lie superalgebra.Lie superhomomorphism: preserves [a, b]s .Problem: Describe it!Bahturin-Bresar: extending a Lie superhomomorphism to theGrassman envelope makes it possible to use FI’s.
Matej Bresar Functional identities and their applications to graded algebras