href="#ulink_0374180c-3866-58b3-a768-b200629b2f4b">section 1.8) in characteristic 0.
Let us denote
THEOREM 1.3 (Martínez and Zelmanov (2010)).– Let
1.6.4. Non-unital simple Jordan superalgebras
Finally, let us consider non-unital simple Jordan superalgebras. As we have seen, K3 the three-dimensional Kaplansky superalgebra and K9 the nine-dimensional degenerate Kac superalgebra are examples of such superalgebras.
EXAMPLE 1.23.– Let Z be a unital associative commutative algebra, D : Z → Z a derivation. Assume that Z is D-simple and that the only constants are elements α1, α ∈ F.
Let us consider in Z the bracket { , } given by:
The above bracket is a Jordan bracket, so the Kantor double V(Z, D) = Z + Zv = KJ(Z, { , }) is a simple unital Jordan superalgebra.
Now we will change 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 V1/2(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
Then Vop 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 EndF 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 EndF V(1/2) generated by all linear transformations RV (1/2)(a) : V(1/2)
