Abdenacer Makhlouf - Algebra and Applications 1

DEFINITION 1.1.– A (linear ) Jordan algebra is a vector space J with a linear binary operation ( x , y ) ↦ xy satisfying the following identities :

(J1) xy = yx (commutativity );

(J2) ( x 2 y ) x = x 2( yx ) ∀ x, y ∈ J (Jordan identity ).

Instead of (J2 ) we can consider the corresponding linearized identity :

( J’2 ) ( xy )( zu ) + (xz )( yu ) + (xu )( yz ) = (( xy ) z ) u + (( xu ) z ) y + (( yu ) z ) x ∀ x, y, z, u ∈ J .

REMARK 1.1.– A Lie algebra L is a vector space with a linear binary operation ( x, y ) ↦ [ x, y ] satisfying the following identities:

(L1) [ x, y ] = – [y, x ] (anticommutativity);

(L2) [[ x, y ], z ] + [[ y, z ], x ] + [[ z, x ], y ] = 0 for arbitrary elements x , y , z ∈ J (Jacobi identity).

EXAMPLE 1.1.– If A is an associative algebra, then ( A (+), ∙), where a ∙ b = ab + ba is a Jordan algebra, and ( A (–), [, ]), where [ a, b ] = ab – ba is a Lie algebra. Both A (+)and A (–)have the same underlying vector space as A .

DEFINITION 1.2.– A superalgebra A is an algebra with a ℤ /2 ℤ-grading. So is a direct sum of two vector spaces and

Elements of are called homogeneous elements. The parity of a homogeneous element a, denoted | a |, is defined by | a | = 0 if and | a | = 1 if .

Elements in are called even and elements in are called odd .

Note that is a subalgebra of A, but is not, instead it can be seen as a bimodule over .

EXAMPLE 1.2.– If V is a vector space of countable dimension, then G = G(V ) denotes the Grassmann (or exterior) algebra over V , that is, the quotient of the tensor algebra over the ideal generated by the symmetric tensors v ⊗ w + w ⊗ v , v , w ∈ V . This algebra G(V ) is ℤ/ 2 ℤ -graded. Indeed, , where the “even part” is the linear span of all tensors of even length and the “odd part” is the linear span of all tensors of odd length.

G(V ) is an example of a superalgebra.

DEFINITION 1.3.– Consider a variety of algebras V defined by homogeneous identities (see Jacobson (1968 ) or Zhevlakov et al. ( 1982 )). We say that a superalgebra is a V-superalgebra if the even part of A ⊗ F G(V ) lies in the variety, that is

DEFINITION 1.4.– The algebra is called the Grassmann envelope of the superalgebra A and will be denoted as G ( A ).

Let us consider V the variety of associative, commutative, anticommutative, Jordan or Lie algebras, respectively. Then we get:

EXAMPLE 1.3.– A superalgebra is an associative superalgebra if and only if it is a ℤ/2ℤ-graded associative algebra.

EXAMPLE 1.4.– A superalgebra is a commutative superalgebra if it satisfies:

for any x , y homogeneous elements of A .

EXAMPLE 1.5.– A superalgebra A is an anticommutative superalgebra if

for every x , y homogeneous elements of A .

EXAMPLE 1.6.– A Jordan superalgebra is a superalgebra that is commutative and satisfies the graded identity:

for every homogeneous elements .

EXAMPLE 1.7.– An anticommutative superalgebra A is a Lie superalgebra if it satisfies:

for every .

DEFINITION 1.5.– If is a Jordan superalgebra and , then their triple product is defined by :