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![Page 1: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/1.jpg)
A characterization of the Kostrikin radical of aLie algebra.
International Workshop: ”Infinite-Dimensional Lie Algebras”
M. Gomez Lozano
Universidad de Malaga
Newfoundland, July 18-23 (2010)
(joint work with E. Garcıa)
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 2: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/2.jpg)
History
TheoremLet R associative. Then
semiprime and d.c.c. on left ideals.R ∼=
⊕ki=1Mn1(∆ni )
Semiprime, I / R, I2 = 0 =⇒ I = 0d.c.c. on left ideals,
I1 ⊂ I2 ⊂ · · · ⊂ In ⊂ · · ·
∃m | Im = Im+s for all s ∈ N
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 3: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/3.jpg)
History
TheoremLet R associative. Then
semiprime and d.c.c. on left ideals.R ∼=
⊕ki=1Mn1(∆ni )
Semiprime, I / R, I2 = 0 =⇒ I = 0
d.c.c. on left ideals,
I1 ⊂ I2 ⊂ · · · ⊂ In ⊂ · · ·
∃m | Im = Im+s for all s ∈ N
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 4: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/4.jpg)
History
TheoremLet R associative. Then
semiprime and d.c.c. on left ideals.R ∼=
⊕ki=1Mn1(∆ni )
Semiprime, I / R, I2 = 0 =⇒ I = 0d.c.c. on left ideals,
I1 ⊂ I2 ⊂ · · · ⊂ In ⊂ · · ·
∃m | Im = Im+s for all s ∈ N
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 5: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/5.jpg)
History
TheoremLet R associative. Then
semiprime and d.c.c. on left ideals.R ∼=
⊕ki=1Mn1(∆ni )
Semiprime, I / R, I2 = 0 =⇒ I = 0.
if a ∈ R | aRa = 0 then a = 0
L Lie, semiprime: 0 6= I / L, [I, I] 6= 0L Lie, nondegenerate: if a ∈ L | [[a, L], a] = 0 then a = 0
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 6: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/6.jpg)
History
TheoremLet R associative. Then
semiprime and d.c.c. on left ideals.R ∼=
⊕ki=1Mn1(∆ni )
Semiprime, I / R, I2 = 0 =⇒ I = 0.if a ∈ R | aRa = 0 then a = 0
L Lie, semiprime: 0 6= I / L, [I, I] 6= 0L Lie, nondegenerate: if a ∈ L | [[a, L], a] = 0 then a = 0
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 7: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/7.jpg)
History
TheoremLet R associative. Then
semiprime and d.c.c. on left ideals.R ∼=
⊕ki=1Mn1(∆ni )
Semiprime, I / R, I2 = 0 =⇒ I = 0.if a ∈ R | aRa = 0 then a = 0
L Lie, semiprime: 0 6= I / L, [I, I] 6= 0
L Lie, nondegenerate: if a ∈ L | [[a, L], a] = 0 then a = 0
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 8: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/8.jpg)
History
TheoremLet R associative. Then
semiprime and d.c.c. on left ideals.R ∼=
⊕ki=1Mn1(∆ni )
Semiprime, I / R, I2 = 0 =⇒ I = 0.if a ∈ R | aRa = 0 then a = 0
L Lie, semiprime: 0 6= I / L, [I, I] 6= 0L Lie, nondegenerate: if a ∈ L | [[a, L], a] = 0 then a = 0
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 9: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/9.jpg)
History
TheoremLet R associative. Then
semiprime and d.c.c. on left ideals.R ∼=
⊕ki=1Mn1(∆ni )
d.c.c. on left ideals,
I1 ⊂ I2 ⊂ · · · ⊂ In ⊂ · · ·
∃m | Im = Im+s for all s ∈ N
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 10: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/10.jpg)
History
TheoremLet R associative. Then
semiprime and d.c.c. on left ideals.R ∼=
⊕ki=1Mn1(∆ni ).
semiprime and d.c.c. on right ideals.semiprime and d.c.c. on inner ideals.
I inner: if IRI ⊂ I.
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 11: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/11.jpg)
History
TheoremLet R associative. Then
semiprime and d.c.c. on left ideals.R ∼=
⊕ki=1Mn1(∆ni ).
semiprime and d.c.c. on right ideals.
semiprime and d.c.c. on inner ideals.
I inner: if IRI ⊂ I.
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 12: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/12.jpg)
History
TheoremLet R associative. Then
semiprime and d.c.c. on left ideals.R ∼=
⊕ki=1Mn1(∆ni ).
semiprime and d.c.c. on right ideals.semiprime and d.c.c. on inner ideals.
I inner: if IRI ⊂ I.
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 13: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/13.jpg)
History
TheoremLet R associative. Then
semiprime and d.c.c. on left ideals.R ∼=
⊕ki=1Mn1(∆ni ).
semiprime and d.c.c. on right ideals.semiprime and d.c.c. on inner ideals.
I inner: if IRI ⊂ I.
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 14: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/14.jpg)
History
TheoremLet L be a simple Lie algebra over a field F of characteristic 0or greater than 7. Then L is Artinian and nondegenerate if andonly if it is one of the following:
1 A division Lie algebra.2 A (finite dimensional over its centroid) simple exceptional
Lie algebra.3 [R, R]/[R, R] ∩ Z (R), where R is a simple Artinian
associative algebra.4 [K , K ]/[K , K ] ∩ Z (R), where K = Skew(R, ∗) and R is a
simple associative algebra with involution ∗ whichcoincides with its socle, such that...
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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History
TheoremLet L nondegenerate. Soc(L) :=
∑Iα, Iα minimal inner ideal of
LSoc(L) = ⊕Mi with Mi simple nondegenerate coincidingwith its socle.
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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History
TheoremL Lie simple, nondegenerate, char = 0 or > 7 with I minimal,then:
1 A (finite dimensional over its centroid) simple exceptionalLie algebra with Jordan elements.
2 L ∼= [R, R]/Z (R) ∩ [R, R], R simple associative with soclenot a division algebra.
3 L ∼= [K , K ]/Z (R) ∩ [K , K ] for K = Skew(R, ∗), R simpleassociative algebra with isotropic involution ∗, with socleand where either Z (R) = 0 or the dimension of R overZ (R) is greater than 16.
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 17: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/17.jpg)
Objectives
L Lie, Nondegenerate ⇐⇒ S.P. Strongly Prime?
L Lie algebra
Anti-commutativity, [x , x ] = 0 for every x ∈ L.Jacobi identity, [x , [y , z]] = [[x , y ], z] + [y , [x , z]] for allx , y , z ∈ L.
L non-degenerate, [x , [x , L]] = 0 implies x = 0.L strongly prime, prime and non-degenerate.L is a sub-direct product of strongly prime if
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 18: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/18.jpg)
Objectives
L Lie, Nondegenerate ⇐⇒ S.P. Strongly Prime?L Lie algebra
Anti-commutativity, [x , x ] = 0 for every x ∈ L.Jacobi identity, [x , [y , z]] = [[x , y ], z] + [y , [x , z]] for allx , y , z ∈ L.
L non-degenerate, [x , [x , L]] = 0 implies x = 0.L strongly prime, prime and non-degenerate.L is a sub-direct product of strongly prime if
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 19: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/19.jpg)
Objectives
L Lie, Nondegenerate ⇐⇒ S.P. Strongly Prime?L Lie algebra
Anti-commutativity, [x , x ] = 0 for every x ∈ L.Jacobi identity, [x , [y , z]] = [[x , y ], z] + [y , [x , z]] for allx , y , z ∈ L.
L non-degenerate, [x , [x , L]] = 0 implies x = 0.
L strongly prime, prime and non-degenerate.L is a sub-direct product of strongly prime if
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 20: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/20.jpg)
Objectives
L Lie, Nondegenerate ⇐⇒ S.P. Strongly Prime?L Lie algebra
Anti-commutativity, [x , x ] = 0 for every x ∈ L.Jacobi identity, [x , [y , z]] = [[x , y ], z] + [y , [x , z]] for allx , y , z ∈ L.
L non-degenerate, [x , [x , L]] = 0 implies x = 0.L strongly prime, prime and non-degenerate.
L is a sub-direct product of strongly prime if
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 21: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/21.jpg)
Objectives
L Lie, Nondegenerate ⇐⇒ S.P. Strongly Prime?L Lie algebra
Anti-commutativity, [x , x ] = 0 for every x ∈ L.Jacobi identity, [x , [y , z]] = [[x , y ], z] + [y , [x , z]] for allx , y , z ∈ L.
L non-degenerate, [x , [x , L]] = 0 implies x = 0.L strongly prime, prime and non-degenerate.L is a sub-direct product of strongly prime if
L ↪→∏
α∈∆ Lα
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 22: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/22.jpg)
Objectives
L Lie, Nondegenerate ⇐⇒ S.P. Strongly Prime?L Lie algebra
Anti-commutativity, [x , x ] = 0 for every x ∈ L.Jacobi identity, [x , [y , z]] = [[x , y ], z] + [y , [x , z]] for allx , y , z ∈ L.
L non-degenerate, [x , [x , L]] = 0 implies x = 0.L strongly prime, prime and non-degenerate.L is a sub-direct product of strongly prime if
L ↪→∏
α∈∆ Lα → Lα onto for all α
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 23: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/23.jpg)
Objectives
L Lie, Nondegenerate ⇐⇒ S.P. Strongly Prime?L Lie algebra
Anti-commutativity, [x , x ] = 0 for every x ∈ L.Jacobi identity, [x , [y , z]] = [[x , y ], z] + [y , [x , z]] for allx , y , z ∈ L.
L non-degenerate, [x , [x , L]] = 0 implies x = 0.L strongly prime, prime and non-degenerate.L is a sub-direct product of strongly prime if
L ↪→∏
α∈∆ Lα → Lα onto for all α
∃{Iα}α∈∆ ideals of L such that L/Iα is s.p and⋂
α∈∆ Iα = 0
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 24: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/24.jpg)
R associative
B(R) Baer radical (prime) de R.R/B(R) semiprime.
I / R semiprime (prime), if R/I is semiprime (prime).
B(R) =⋂
Iα/primeR Iα.
R semiprime ⇐⇒ R sub-direct product of prime ones.{xn}n∈N, xn+1 := xnanxn,
x1 ∈ B(R) ⇐⇒ ∃k |xk = 0
• Semiprime, I / R, I2 = 0 =⇒ I = 0
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 25: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/25.jpg)
R associative
B(R) Baer radical (prime) de R.R/B(R) semiprime.
I / R semiprime (prime), if R/I is semiprime (prime).
B(R) =⋂
Iα/primeR Iα.
R semiprime ⇐⇒ R sub-direct product of prime ones.{xn}n∈N, xn+1 := xnanxn,
x1 ∈ B(R) ⇐⇒ ∃k |xk = 0
• Semiprime, I / R, I2 = 0 =⇒ I = 0• Prime, I, J / R, IJ = 0 =⇒ I = 0 or J = 0
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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R associative
B(R) Baer radical (prime) de R.R/B(R) semiprime.
I / R semiprime (prime), if R/I is semiprime (prime).
B(R) =⋂
Iα/primeR Iα.
R semiprime ⇐⇒ R sub-direct product of prime ones.{xn}n∈N, xn+1 := xnanxn,
x1 ∈ B(R) ⇐⇒ ∃k |xk = 0
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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R associative
B(R) Baer radical (prime) de R.R/B(R) semiprime.
I / R semiprime (prime), if R/I is semiprime (prime).
B(R) =⋂
Iα/primeR Iα.
R semiprime ⇐⇒ R sub-direct product of prime ones.
{xn}n∈N, xn+1 := xnanxn,
x1 ∈ B(R) ⇐⇒ ∃k |xk = 0
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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R associative
B(R) Baer radical (prime) de R.R/B(R) semiprime.
I / R semiprime (prime), if R/I is semiprime (prime).
B(R) =⋂
Iα/primeR Iα.
R semiprime ⇐⇒ R sub-direct product of prime ones.{xn}n∈N, xn+1 := xnanxn,
x1 ∈ B(R) ⇐⇒ ∃k |xk = 0
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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R non associative
B(R) Baer radical (prime) de R.R/B(R) semiprime.
I / R semiprime (prime), si R/I is semiprime (prime).
B(R) =⋂
Iα/primeR Iα.
R semiprime ⇐⇒ R sub-direct product of prime ones.{xn}n∈N, xn+1 := xn(anxn),
x1 ∈ B(R) ⇐⇒ ∃k |xk = 0
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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Kostrikin radical
x ∈ L absolute zero divisor, [x , [x , L]] = 0.L non degenerate si [x , [x , L]] = 0 =⇒ x = 0.L strongly prime if is prime and non-degenerate.
K (L) =⋃
α Kα(L) donde Kα(L): (Kostrikin radical)K1(L) / L ideal generate by A.Z.D L.Kα(L) =
⋃β<α Kβ(L) with α ordinal limit, and
Kα(L)/Kα−1(L) = K1(L/Kα−1(L)) other case.
I / L non degenerate (s.p.) if L/I is non-degenerate (s.p.).
1 L non-degenerate ⇐⇒ L is a S.P of strongly prime?2 {xn}n∈N, xn+1 := [xn, [an, xn]]. If x1 ∈ K (L), finite
m-sequence?
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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Kostrikin radical
x ∈ L absolute zero divisor, [x , [x , L]] = 0.L non degenerate si [x , [x , L]] = 0 =⇒ x = 0.L strongly prime if is prime and non-degenerate.K (L) =
⋃α Kα(L) donde Kα(L): (Kostrikin radical)
K1(L) / L ideal generate by A.Z.D L.Kα(L) =
⋃β<α Kβ(L) with α ordinal limit, and
Kα(L)/Kα−1(L) = K1(L/Kα−1(L)) other case.
I / L non degenerate (s.p.) if L/I is non-degenerate (s.p.).
1 L non-degenerate ⇐⇒ L is a S.P of strongly prime?2 {xn}n∈N, xn+1 := [xn, [an, xn]]. If x1 ∈ K (L), finite
m-sequence?
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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Kostrikin radical
x ∈ L absolute zero divisor, [x , [x , L]] = 0.L non degenerate si [x , [x , L]] = 0 =⇒ x = 0.L strongly prime if is prime and non-degenerate.K (L) =
⋃α Kα(L) donde Kα(L): (Kostrikin radical)
K1(L) / L ideal generate by A.Z.D L.Kα(L) =
⋃β<α Kβ(L) with α ordinal limit, and
Kα(L)/Kα−1(L) = K1(L/Kα−1(L)) other case.
I / L non degenerate (s.p.) if L/I is non-degenerate (s.p.).
1 L non-degenerate ⇐⇒ L is a S.P of strongly prime?2 {xn}n∈N, xn+1 := [xn, [an, xn]]. If x1 ∈ K (L), finite
m-sequence?
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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Associative-Lie
TheoremR associative, Φ with no 2-torsion, K (R−)
1 R prime, R−/Z (R) strongly prime.2 K (R−) is the intersection of all s.p. ideals of R−.3 K (R−) = π−1(Z (R/r(R))), r(R) Baer radical of R and
π : R → R/r(R) the canonical projection.4 K (R−) = {x ∈ R | m-sequence in x has finite length}.
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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Associative-Lie
Theorem(R, ∗) associative with involution, Φ with no 2-torsion,L := Skew(R, ∗) and K (L) Kostrikin radical.
1 R prime, L/(Z (R) ∩ Skew(R, ∗)) strongly prime??.2 K (L) is the intersection of all s.p. ideals of L.3 K (L) = π−1(Z (L/(r(R) ∩ L))), r(R) Baer radical of R and
π : R → R/r(R) the canonical projection.4 K (L) = {x ∈ R | m-sequence in x has finite length}.
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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J Jordan algebra
Mc(J) McCrimmon radical.J/Mc(J) nondegenerate.I / J nondegenerate if J/I nondegenerate.I / J strongly prime if J/I prime and nondegenerate.Mc(J) =
⋂Iα/f.p.J Iα.
1 J nondegenerate ⇐⇒ J subdirect product of s.p.2 m-sequences: {xn}n∈N, xn+1 := Uxnan,
x1 ∈ Mc(J) ⇐⇒ ∃k |xk = 0
J nondegenerate: UxJ := 0 =⇒ x = 0
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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Lie-Jordan 12 , 1
3 ∈ φ
x ∈ L Jordan element, if ad3x = 0.
Jordan algebra L in x :
L with product a • b := 12 [[a, x ], b]
ker(x) := {z ∈ L | [x , [x , z]] = 0} / (L, •)Lx := (L/ ker(x), •).
Uab =18
ad2a ad2
x b, for all a, b ∈ L, and
{a, b, c} = −14
[a, [ad2x b, c]] for all a, b, c ∈ L
Theorem: L , x ∈ L Jordan element
Mc(Lx) ⊂ {a ∈ Lx | [x , [x , a]] ⊂ K (L)}.
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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Lie-Jordan 12 , 1
3 ∈ φ
x ∈ L Jordan element, if ad3x = 0.
Jordan algebra L in x :
L with product a • b := 12 [[a, x ], b]
ker(x) := {z ∈ L | [x , [x , z]] = 0} / (L, •)Lx := (L/ ker(x), •).
Uab =18
ad2a ad2
x b, for all a, b ∈ L, and
{a, b, c} = −14
[a, [ad2x b, c]] for all a, b, c ∈ L
Theorem: L , x ∈ L Jordan element
Mc(Lx) ⊂ {a ∈ Lx | [x , [x , a]] ⊂ K (L)}.
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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Lie-Jordan 12 , 1
3 ∈ φ
x ∈ L Jordan element, if ad3x = 0.
Jordan algebra L in x :L with product a • b := 1
2 [[a, x ], b]
ker(x) := {z ∈ L | [x , [x , z]] = 0} / (L, •)Lx := (L/ ker(x), •).
Uab =18
ad2a ad2
x b, for all a, b ∈ L, and
{a, b, c} = −14
[a, [ad2x b, c]] for all a, b, c ∈ L
Theorem: L , x ∈ L Jordan element
Mc(Lx) ⊂ {a ∈ Lx | [x , [x , a]] ⊂ K (L)}.
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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Lie-Jordan 12 , 1
3 ∈ φ
x ∈ L Jordan element, if ad3x = 0.
Jordan algebra L in x :L with product a • b := 1
2 [[a, x ], b]ker(x) := {z ∈ L | [x , [x , z]] = 0} / (L, •)
Lx := (L/ ker(x), •).
Uab =18
ad2a ad2
x b, for all a, b ∈ L, and
{a, b, c} = −14
[a, [ad2x b, c]] for all a, b, c ∈ L
Theorem: L , x ∈ L Jordan element
Mc(Lx) ⊂ {a ∈ Lx | [x , [x , a]] ⊂ K (L)}.
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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Lie-Jordan 12 , 1
3 ∈ φ
x ∈ L Jordan element, if ad3x = 0.
Jordan algebra L in x :L with product a • b := 1
2 [[a, x ], b]ker(x) := {z ∈ L | [x , [x , z]] = 0} / (L, •)Lx := (L/ ker(x), •).
Uab =18
ad2a ad2
x b, for all a, b ∈ L, and
{a, b, c} = −14
[a, [ad2x b, c]] for all a, b, c ∈ L
Theorem: L , x ∈ L Jordan element
Mc(Lx) ⊂ {a ∈ Lx | [x , [x , a]] ⊂ K (L)}.
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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Lie-Jordan 12 , 1
3 ∈ φ
x ∈ L Jordan element, if ad3x = 0.
Jordan algebra L in x :L with product a • b := 1
2 [[a, x ], b]ker(x) := {z ∈ L | [x , [x , z]] = 0} / (L, •)Lx := (L/ ker(x), •).
Uab =18
ad2a ad2
x b, for all a, b ∈ L, and
{a, b, c} = −14
[a, [ad2x b, c]] for all a, b, c ∈ L
Theorem: L , x ∈ L Jordan element
Mc(Lx) ⊂ {a ∈ Lx | [x , [x , a]] ⊂ K (L)}.
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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Technic property Char(φ) = p
Definitionα an ordinal. L satisfies Hα if β ≤ α not ordinal limit everysubmodule Lβ := Kβ(L)/Kβ−1(L) invariant by innerautomorphisms of Lβ is an ideal of Lβ.L satisfies H if satisfies Hα, ∀ α.
Examples
L over a field φ of zero characteristic.L Generate by ad-nilpotent elements index ≤ k . p > 2k − 2L =
⊕ni=1 Li , Z-graded L0 =
∑ni=1[Li , L−i ]. p > 4n
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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Technic property Char(φ) = p
Definitionα an ordinal. L satisfies Hα if β ≤ α not ordinal limit everysubmodule Lβ := Kβ(L)/Kβ−1(L) invariant by innerautomorphisms of Lβ is an ideal of Lβ.L satisfies H if satisfies Hα, ∀ α.
ExamplesL over a field φ of zero characteristic.
L Generate by ad-nilpotent elements index ≤ k . p > 2k − 2L =
⊕ni=1 Li , Z-graded L0 =
∑ni=1[Li , L−i ]. p > 4n
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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Technic property Char(φ) = p
Definitionα an ordinal. L satisfies Hα if β ≤ α not ordinal limit everysubmodule Lβ := Kβ(L)/Kβ−1(L) invariant by innerautomorphisms of Lβ is an ideal of Lβ.L satisfies H if satisfies Hα, ∀ α.
ExamplesL over a field φ of zero characteristic.L Generate by ad-nilpotent elements index ≤ k . p > 2k − 2
L =⊕n
i=1 Li , Z-graded L0 =∑n
i=1[Li , L−i ]. p > 4n
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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Technic property Char(φ) = p
Definitionα an ordinal. L satisfies Hα if β ≤ α not ordinal limit everysubmodule Lβ := Kβ(L)/Kβ−1(L) invariant by innerautomorphisms of Lβ is an ideal of Lβ.L satisfies H if satisfies Hα, ∀ α.
ExamplesL over a field φ of zero characteristic.L Generate by ad-nilpotent elements index ≤ k . p > 2k − 2L =
⊕ni=1 Li , Z-graded L0 =
∑ni=1[Li , L−i ]. p > 4n
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 46: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/46.jpg)
L Lie algebra with H
TheoremLet x ∈ L Jordan element. ThenMc(Lx) = {a ∈ Lx | [x , [x , a]] ∈ K (L)}.
CorollaryLet x ∈ L Jordan element. Then any m-sequence on x hasfinite length . In particular, Mc(Lx) = Lx .
PropositionLet M an m-sequence of Jordan elements.If P / L maximalM ∩ P = ∅, Then P is s.p. ideal of L.
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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L Lie algebra with H
TheoremLet x ∈ L Jordan element. ThenMc(Lx) = {a ∈ Lx | [x , [x , a]] ∈ K (L)}.
CorollaryLet x ∈ L Jordan element. Then any m-sequence on x hasfinite length . In particular, Mc(Lx) = Lx .
PropositionLet M an m-sequence of Jordan elements.If P / L maximalM ∩ P = ∅, Then P is s.p. ideal of L.
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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L Lie algebra with H
TheoremLet x ∈ L Jordan element. ThenMc(Lx) = {a ∈ Lx | [x , [x , a]] ∈ K (L)}.
CorollaryLet x ∈ L Jordan element. Then any m-sequence on x hasfinite length . In particular, Mc(Lx) = Lx .
PropositionLet M an m-sequence of Jordan elements.If P / L maximalM ∩ P = ∅, Then P is s.p. ideal of L.
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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L Lie with H Char(φ) = p
TheoremIf L nondegenerate and every nonzero ideal of L contains anonzero Jordan elements. Then the intersection of all stronglyprime ideal of L is zero.
Examples
L Lie over φ of characteristic zero such that every idealcontains a nonzero Jordan elements.L generate by ad-nilpotent elements of index ≤ k , withp > 2k − 2 such that every ideal contains a nonzeroJordan elements.L =
⊕ni=1 Li , Z-graded with L0 =
∑ni=1[Li , L−i ], p > 4n
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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L Lie with H Char(φ) = p
TheoremIf L nondegenerate and every nonzero ideal of L contains anonzero Jordan elements. Then the intersection of all stronglyprime ideal of L is zero.
ExamplesL Lie over φ of characteristic zero such that every idealcontains a nonzero Jordan elements.
L generate by ad-nilpotent elements of index ≤ k , withp > 2k − 2 such that every ideal contains a nonzeroJordan elements.L =
⊕ni=1 Li , Z-graded with L0 =
∑ni=1[Li , L−i ], p > 4n
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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L Lie with H Char(φ) = p
TheoremIf L nondegenerate and every nonzero ideal of L contains anonzero Jordan elements. Then the intersection of all stronglyprime ideal of L is zero.
ExamplesL Lie over φ of characteristic zero such that every idealcontains a nonzero Jordan elements.L generate by ad-nilpotent elements of index ≤ k , withp > 2k − 2 such that every ideal contains a nonzeroJordan elements.
L =⊕n
i=1 Li , Z-graded with L0 =∑n
i=1[Li , L−i ], p > 4n
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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L Lie with H Char(φ) = p
TheoremIf L nondegenerate and every nonzero ideal of L contains anonzero Jordan elements. Then the intersection of all stronglyprime ideal of L is zero.
ExamplesL Lie over φ of characteristic zero such that every idealcontains a nonzero Jordan elements.L generate by ad-nilpotent elements of index ≤ k , withp > 2k − 2 such that every ideal contains a nonzeroJordan elements.L =
⊕ni=1 Li , Z-graded with L0 =
∑ni=1[Li , L−i ], p > 4n
M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
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Papers
1 A construction of gradings of Lie Algebras. Int. Math. Res.Not. (2007)
2 The Jordan Algebras of a Lie Algebra. J. Algebra (2007)3 The socle of a nondegenerate Lie algebras. J. Algebra
(2008)4 An Artinian theory for Lie algebras. J. Algebra (2008)5 Inner ideal structure of nearly Artinian Lie algebras. Proc.
A.M.S (2008-9)6 A characterization of the Kostrikin radical of a Lie algebra.
preprint (???)
Authors: E. Neher, A. Fernandez Lopez, E. Garcia, —M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 54: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/54.jpg)
Papers
1 A construction of gradings of Lie Algebras. Int. Math. Res.Not. (2007)
2 The Jordan Algebras of a Lie Algebra. J. Algebra (2007)
3 The socle of a nondegenerate Lie algebras. J. Algebra(2008)
4 An Artinian theory for Lie algebras. J. Algebra (2008)5 Inner ideal structure of nearly Artinian Lie algebras. Proc.
A.M.S (2008-9)6 A characterization of the Kostrikin radical of a Lie algebra.
preprint (???)
Authors: A. Fernandez Lopez, E. Garcia, —M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 55: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/55.jpg)
Papers
1 A construction of gradings of Lie Algebras. Int. Math. Res.Not. (2007)
2 The Jordan Algebras of a Lie Algebra. J. Algebra (2007)3 The socle of a nondegenerate Lie algebras. J. Algebra
(2008)
4 An Artinian theory for Lie algebras. J. Algebra (2008)5 Inner ideal structure of nearly Artinian Lie algebras. Proc.
A.M.S (2008-9)6 A characterization of the Kostrikin radical of a Lie algebra.
preprint (???)
Authors: . Draper, A. Fernandez Lopez, E. Garcia, —M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 56: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/56.jpg)
Papers
1 A construction of gradings of Lie Algebras. Int. Math. Res.Not. (2007)
2 The Jordan Algebras of a Lie Algebra. J. Algebra (2007)3 The socle of a nondegenerate Lie algebras. J. Algebra
(2008)4 An Artinian theory for Lie algebras. J. Algebra (2008)
5 Inner ideal structure of nearly Artinian Lie algebras. Proc.A.M.S (2008-9)
6 A characterization of the Kostrikin radical of a Lie algebra.preprint (???)
Authors: A. Fernandez Lopez, E. Garcia, —M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 57: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/57.jpg)
Papers
1 A construction of gradings of Lie Algebras. Int. Math. Res.Not. (2007)
2 The Jordan Algebras of a Lie Algebra. J. Algebra (2007)3 The socle of a nondegenerate Lie algebras. J. Algebra
(2008)4 An Artinian theory for Lie algebras. J. Algebra (2008)5 Inner ideal structure of nearly Artinian Lie algebras. Proc.
A.M.S (2008-9)
6 A characterization of the Kostrikin radical of a Lie algebra.preprint (???)
Authors: A. Fernandez Lopez, E. Garcia, —M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.
![Page 58: A characterization of the Kostrikin radical of a Lie ... · PDF fileA characterization of the Kostrikin radical of a Lie algebra. International Workshop: ”Infinite-Dimensional Lie](https://reader037.vdocument.in/reader037/viewer/2022110111/5a95d3547f8b9a451b8c8dd3/html5/thumbnails/58.jpg)
Papers
1 A construction of gradings of Lie Algebras. Int. Math. Res.Not. (2007)
2 The Jordan Algebras of a Lie Algebra. J. Algebra (2007)3 The socle of a nondegenerate Lie algebras. J. Algebra
(2008)4 An Artinian theory for Lie algebras. J. Algebra (2008)5 Inner ideal structure of nearly Artinian Lie algebras. Proc.
A.M.S (2008-9)6 A characterization of the Kostrikin radical of a Lie algebra.
preprint (???)
Authors: A. Fernandez Lopez, E. Garcia, —M. Gomez Lozano Universidad de Malaga
A characterization of the Kostrikin radical of a Lie algebra.