TH`ESE A NEW INVARIANT OF QUADRATIC LIE ALGEBRAS AND ...

Chapter 4
Pseudo-Euclidean Jordan algebras
4.1 Preliminaries
Definition 4.1.1. A (non-associative) algebra J over C is called a (commutative) Jordan algebra
if its product is commutative and satisfies the following identity (Jordan identity):
(xy)x 2 = x(yx 2 ), ∀ x,y,z ∈ J. (I)
For instance, any commutative algebra with an associative product is a Jordan algebra. A
trivial case is when the product xy = 0 for all x,y ∈ J. In this case, we say that J is Abelian. A
Jordan algebra J is called nilpotent if there is an integer k ∈ N such that J k = {0}. The smallest
k for which this condition satisfied is called the nilindex of J and we say that J is k − 1-step
nilpotent. If J is non-Abelian and it has only two ideals {0} and J then we say J simple.
Given an algebra A, the commutator [x,y] = xy − yx, for all x,y ∈ A measures the commutativity
of A. Similarly the associator defined by
(x,y,z) = (xy)z − x(yz), ∀ x,y,z ∈ A.
measures the associativity of A. In term of associators, the Jordan identity in a Jordan algebra J
becomes
(x,y,x 2 ) = 0, ∀ x,y,z ∈ J. (II)
An algebra A is called a power-associative algebra if the subalgebra generated by any element
x ∈ A is associative (see [Sch66] for more details). A Jordan algebra is an example of
a power-associative algebra. A power-associative algebra A is called trace-admissible if there
exists a bilinear form τ on A that satisfies:
(1) τ(x,y) = τ(y,x),
(2) τ(xy,z) = τ(x,yz),
(3) τ(e,e) = 0 for any idempotent e of A,
(4) τ(x,y) = 0 if xy is nilpotent or xy = 0.
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