jordan algebras - george washington universityspwm/jordan algebras.pdf · 2011-08-30 · jordan...

Basics Background Information Formally Real Special and Exceptional Summary

Jordan Algebras

Katie Sember, Lindsey Scoppetta, Amanda Clemm

George Washington University Summer Program for Women in Mathematics

July 7, 2011

K. Sember, L. Scoppetta, A. Clemm Jordan Algebras

DefinitionA ring is a commutative group under addition equipped with asecond binary operation, namely multiplication, that satisfies:

ClosureAssociativityDistribution over addition

DefinitionA field is a commutative ring whose nonzero elements form agroup under multiplication. Thus every element x of a field hasan inverse x−1 that is also in the field.

An Algebra: Review

Let (V,+, ·) be a vector space over a field. Then V is analgebra when it has a binary operation × : V × V → V , calledmultiplication, and an identity element referred to as the unit,such that the following properties hold:

(u + v)× w = (u× w) + (v × w)

u× (v + w) = (u× v) + (u× w)

(au)× (bv) = (ab)(u× v)

There exists an element 1 ∈ V such that 1× u = u× 1 = u

Note: Algebras are not assumed to be associative undermultiplication.

An Algebra: Review

(u + v)× w = (u× w) + (v × w)

u× (v + w) = (u× v) + (u× w)

(au)× (bv) = (ab)(u× v)

An Algebra: Review

(u + v)× w = (u× w) + (v × w)

u× (v + w) = (u× v) + (u× w)

(au)× (bv) = (ab)(u× v)

An Algebra: Review

(u + v)× w = (u× w) + (v × w)

u× (v + w) = (u× v) + (u× w)

(au)× (bv) = (ab)(u× v)

Jordan Algebras!

DefinitionA Jordan algebra is a non-associative algebra A over a field Fthat satisfies the identities:

a× b = b× a

a2 × (b× a) = (a2 × b)× a for all a, b ∈ A

Note: non-associative refers to for all a, b, c ∈ A,(a× b)× c 6= a× (b× c).

DefinitionA matrix is self-adjoint if it is equal to its conjugate transpose.

Jordan Algebras!

a× b = b× a

Jordan Algebras!

a× b = b× a

Jordan Algebras!

a× b = b× a

Jordan Algebras!

a× b = b× a

Important Mathematicians

Professor Pascual Jordan - 1933

applications of Jordan algebra:Quantum MechanicsNumber TheoryExtremal Black Holes

Born: October 18th, 1902 - Hanover, Germanyquantum mechanics and field theoryMay of 1933 - Nazi Party

considered "politically unreliable"

1954 - Nobel Prize in Physics given to Max BornAbraham Adrian Albert - Albert algebra

Question

Why was Jordan led away from associative algebras?

Answer: Quantum Mechanics

Quantum Mechanics

DefinitionPhysicist study observablesA system observable is a property of the system statethat can be determined by some sequence of physicaloperations.

ExampleSubmitting the system to electromagnetic fields and reading avalue off of a gauge.

Quantum Mechanics

Multiplication of Observables

In quantum mechanics, observables are closed under additionand scalar multiplication, but multiplying observables is morecomplicated.

Commutative Product of Observables

x ◦ y = 12((x + y)2 − x2 − y2) = 1

2(xy + yx)

Physicists like this definition because it considered partialordering.

Power Associative

DefinitionA sub-algebra is a subset of an algebra, closed under all itsoperations, and carrying the induced operations.

DefinitionAn algebra is said to be power associative if the sub-algebragenerated by any element is associative.

Power Associative

DefinitionA sub-algebra is a subset of an algebra, closed under all itsoperations, and carrying the induced operations.

DefinitionAn algebra is said to be power associative if the sub-algebragenerated by any element is associative.

Formally Real Jordan Algebras

DefinitionFormally real Jordan algebra: commutative,power-associative algebra J satisfyingx21 + · · ·+ x2n = 0⇒ x1 = · · · = xn = 0 ∀n

Infinite Families of Formally Real Jordan Algebras

The Jordan algebra of n× n self-adjoint real matrices.The Jordan algebra of n× n self-adjoint complex matrices.The Jordan algebra of n× n self-adjoint quaternionicmatrices.The Jordan algebra freely generated by Rn with therelations where the right-hand side is defined using theusual inner product on Rn. This is sometimes called a spinfactor or a Jordan algebra of Clifford type.the exceptional Jordan algebra

The Jordan algebra of 3× 3 self-adjoint octonionic matrices.

Associative Algebra

Let U be an associative algebra over a field F that does nothave characteristic two. Aside from the associativemultiplication of the elements of U , defined a · b, we will definetwo new operations:

[a, b] = a · b− b · aab = 1

2(a · b + b · a)

Associative Algebra

Let U be an associative algebra over a field F that does nothave characteristic two. Aside from the associativemultiplication of the elements of U , defined a · b, we will definetwo new operations:

[a, b] = a · b− b · aab = 1

2(a · b + b · a)

Two New Algebras: U−, U+

If we keep the vector space structure of this associative algebraU that we have defined, but replace its multiplication by [a, b],the result is a new algebra, U−.

The multiplication implemented by U−, [a, b], is called thecommutator of elements in U .

U− with the Commutator

Upon investigation it can be seen that U− satisfies the JacobiIdentity,

[A, [B,C]] + [B, [C,A]] + [C, [A,B]] = 0 where A,B,C ∈ U ,

and is a Lie algebra.

Poincare−Birkhoff −Witt Theorem

The Poincare−Birkhoff −Witt theorem states that everyLie algebra is isomorphic to a sub-algebra of some algebra U−.

RecallA sub-algebra is a subset of an algebra, closed under all itsoperations, and carrying the induced operations.

Multiplication of U+

If we take the vector space structure of U and replace itsassociative multiplication with ab, we attain another newalgebra, U+.

Recall

ab = 12(a · b + b · a)

Now let’s consider U+ with the multiplication ab. We can seethat this multiplication is commutative and satisfies theidentities we have defined for a Jordan algebra.

Thus U+ is a Jordan algebra.

Poincare−Birkhoff −Witt Theorem

The analogue of the Poincare−Birkhoff −Witt theorem isnot true for Jordan algebras.

However, if a Jordan Algebra W is isomorphic to a sub-algebraof U+, then W is called a special Jordan algebra.

All other Jordan algebras, those that are not special, are calledexceptional Jordan algebras.

Examples of Special and Exceptional Jordan Algebras

DefinitionA self-adjoint matrix B is a matrix such that B is equal to itsconjugate transpose, (B = B∗).

Some Examples:The set of self-adjoint real, complex, or quaternionicmatrices with multiplication

(xy+yx)2

The set of 3x3 self-adjoint matrices over thenon-associative octonions, again with multiplication

(xy+yx)2 ,

"The" exceptional Jordan algebra.K. Sember, L. Scoppetta, A. Clemm Jordan Algebras

(xy+yx)2

(xy+yx)2 ,

(xy+yx)2

(xy+yx)2 ,

(xy+yx)2

(xy+yx)2 ,

Review of Key Concepts

A Jordan algebra was created by Pascual Jordan forquantum mechanics and is a power-associate algebra.By investigating the algebra U+, we can observe twodifferent categories of Jordan algebras, special andexceptional.A specific type of Jordan algebras known as the Formallyreal Jordan algebras can be classified as real, complex,quaternionic, and octonionic matrices.

jordan algebras - george washington universityspwm/jordan algebras.pdf · 2011-08-30 · jordan...

Documents