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
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 or the regular bimodule.
The superalgebras
EXAMPLE 1.27.– Every right module over A gives rise to a one-sided Jordan bimodule over A(+).
Suppose now that the superalgebra A is equipped with an involution ∗.
EXAMPLE 1.28.– The subspaces H(A, ∗) and K(A, ∗) become Jordan A(+)- bimodules with the actions:
1 1) h ○ a = ha + a∗h;
2 2) h ○ a = ha∗ + ah;
3 3) k ○ a = ka + a∗k;
4 4) k ○ a = ka∗ + ak;
where h ∈ H(A, ∗), k ∈ K(A, ∗), a ∈ A.
The associative superalgebras Mm+n(F), where both m, n are odd, and Q(n), are not equipped with an involution, but they are equipped with a pseudoinvolution.
DEFINITION 1.20.– A graded linear map ∗ : A → A is called a pseudoinvolution if (ab)∗ = (–1)|a|∙|b|b∗a∗, (a∗)∗ = (–1)|a| a for arbitrary elements
EXAMPLE 1.29.– The mapping
EXAMPLE 1.30.– The mapping
Replacing the involution ∗ in example 1.28 with the pseudoinvolutions of examples 1.29 and 1.30, we get unital Jordan bimodules over Mm+n(F), where m, n are odd, and over Q(n)(+).
THEOREM 1.7 (see Martin and Piard (1992), Martínez et al. (2010)).– An arbitrary irreducible Jordan bimodule over
The exceptional Jordan superalgebra K10 has rank 3. Jordan bimodules over K10 have been classified by Shtern (1987).
THEOREM 1.8 (Shtern (1987)).– All Jordan bimodules over K10 are completely reducible. The only irreducible Jordan bimodules over K10 are the regular bimodule and its opposite.
1.7.2. Superalgebras of rank ≤ 2
If J is a Jordan superalgebra of rank ≤ 2, then, generally speaking, it is no longer true that its universal multiplicative algebra is finite dimensional and that any Jordan bimodule is completely reducible.
1.7.2(a) In the case J = Q(2)(+), however, it is true (see Martínez et al. (2010)). The universal multiplicative enveloping algebra U(Q(2)(+)) is finite dimensional and semisimple and the description of irreducible Jordan bimodules is similar to that of Q(n)(+), n ≥ 3.
1.7.2(b) Let us discuss bimodules over Kantor superalgebras. Recall that the Kantor superalgebras Kan(n) are Kantor doubles of the Grassmann superalgebras G(n), n ≥ 1, with respect
