Algebra and Applications 1. Abdenacer Makhlouf

the product in V (Z, D), modifying only the action of the even part on the odd part and preserving the product of two even (respectively, two odd) elements. Denote with juxtaposition the product on V (Z, D). Let a, b ∈ Z. We define the new product ∙ by:

      In this way, we get another Jordan superalgebra V_1/2(Z, D) that is simple but not unital.

      It was proved in Zelmanov (2000) that:

      THEOREM 1.4.– Let J be a finite dimensional simple central non-unital Jordan superalgebra over a field F. Then J is isomorphic to one of the superalgebras on the list:

      1 i) the Kaplansky superalgebra K3 (example 1.12);

      2 ii) the field F has characteristic 3 and J is the degenerate Kac superalgebra (example 1.15);

      3 iii) a superalgebra V1/2(Z, D) (example 1.23).

      DEFINITION 1.14.– Let A be a Jordan superalgebra and let N be its radical, that is, the largest solvable ideal of A. The superalgebra A is said to be semisimple if N = (0).

      EXAMPLE 1.24.– Let B be a simple non-unital Jordan superalgebra and let H(B) = B + F1 be its unital hull. Then H(B) is a semisimple Jordan superalgebra that is not simple.

      THEOREM 1.5 (Zelmanov (2000)).– Let J be a finite dimensional Jordan superalgebra. Then J is semisimple if and only if

      where J₍₁₎,…, J_(t) are simple Jordan superalgebras and for every i = 1,…, s, the superalgebras Jij are simple non-unital Jordan superalgebras over the field extension Ki of F.

1.7. Finite dimensional representations

      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).

      1.7.1. Superalgebras of rank ≥ 3

      In this section, we consider Jordan bimodules over finite dimensional simple Jordan superalgebras of rank ≥ 3, that is, superalgebras Josp(n, 2m), n + m ≥ 3; Q(n)⁽⁺⁾, n ≥ 3; JP(n), n ≥ 3.

      In this case, the universal multiplicative enveloping superalgebra U(J) is finite dimensional and semisimple (Martin and Piard 1992). Hence every Jordan bimodule is completely reducible, as in the case of Jordan algebras.

      The superalgebras Josp(n, 2m) and JP(n) are of the type

      where A is a simple finite dimensional associative superalgebra and ∗ is an involution.

EXAMPLE 1.25.– An arbitrary right module over A is a one-sided module over H(A, ∗).

EXAMPLE 1.26.– The subspace K(A, ∗) = {k ∈ A | k^∗ = –k} with the action k ∙ a = ka + ak; k ∈ K(A, ∗), a ∈ H(A, ∗) is a unital H(A, ∗)-bimodule.

THEOREM 1.6 (Martin and Piard (1992)).– An arbitrary irreducible Jordan bimodule over Josp(n, 2m), n + m ≥ 3, or JP(n), n ≥ 3, is a bimodule of examples 1.25 and 1.26