Читать книгу Algebra and Applications 1 - Abdenacer Makhlouf - Страница 20

Jacobson (1968) developed the theory of bimodules over semisimple finite dimensional Jordan algebras.

In this section, we discuss representations (bimodules) of finite dimensional Jordan superalgebras.

DEFINITION 1.15.– The rank of a Jordan superalgebra J is the maximal number of pairwise orthogonal idempotents in the even part.

Unless otherwise stated we will assume char F = 0.

DEFINITION 1.16.– Let V be a ℤ/2ℤ-graded vector space with bilinear mappings V × J → V, J × V → V. We call V a Jordan bimodule if the split null extension V + J is a Jordan superalgebra.

Recall that in the split null extension the multiplication extends the multiplication on J, products V ∙ J and J ∙ V are defined via the bilinear mappings above and V ∙ V = (0).

Let be a Jordan bimodule over J. Consider the vector space , where is a copy of with different parity. Define the action of J on V^op via

Then V^op is also a Jordan bimodule over J. We call it the opposite module of V.

Let V be the free Jordan J-bimodule on one free generator.

DEFINITION 1.17.– The associative subsuperalgebra U(J) of End_F V generated by all linear transformations RV(a) : V → V, v ↦ va, a ∈ J, is called the universal multiplicative enveloping superalgebra of J.

Every Jordan bimodule over J is a right module over U(J).

DEFINITION 1.18.– A bimodule V over J is called a one-sided bimodule if {J, V, J} = (0).

Let V(1/2) be the free one-sided Jordan J-bimodule on one free generator.

DEFINITION 1.19.– The associative subsuperalgebra S(J) of End_F V(1/2) generated by all linear transformations RV (1/2)(a) : V(1/2) → V(1/2), v ↦ va, a ∈ J, is called the universal associative enveloping algebra of J.

Every one-sided Jordan J-bimodule is a right module over S(J).

Finally, let J be a unital Jordan superalgebra with the identity element e. Let V (1) denote the free unital J-bimodule on one free generator. The associative subsuperalgebra U₁(J) of EndFV(1) generated by {RV_(1/2)(a)}_{a ∈ J} is called the universal unital enveloping algebra of J.

For an arbitrary Jordan bimodule V, the Peirce decomposition

is a decomposition of V into a direct sum of unital and one-sided bimodules. Hence U(J) ≅ U₁(J) ⊕ S(J).