Algebra and Applications 1. Abdenacer Makhlouf

, Q(n)⁽⁺⁾ are of the type A⁽⁺⁾, where A is a simple finite dimensional associative superalgebra.

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 is a pseudoinvolution on Q(n).

EXAMPLE 1.30.– The mapping is a pseudoinvolution on Mm+n(F).

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 , m + n ≥ 3 or Q(n)⁽⁺⁾, n ≥ 3, is one of examples 1.27 and 1.28 with an involution or a pseudoinvolution, or a regular bimodule.

      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 to the Poisson bracket

      Let Kan(n) = G(n) + G(n)v. The Grassmann superalgebra G(n) is embeddable in the associative commutative superalgebra A = F [t, ξ₁,…, ξn] = F [t] ⊗_F G(n).

      For an arbitrary scalar α ∈ F, the Poisson bracket [ , ] extends to the Jordan bracket on A defined by [t, ξi] = 0, [ξi, ξj] = –δij, [ξi, 1] = 0, [t, 1] = αt. The Kantor double Kan(n) = G(n) + G(n)v embeds in the Kantor double Kan(A, [ , ]) = A + Av. The subspace V (α) = tG(n) + tG(n)v is an irreducible unital Jordan bimodule over K(n). The square of the multiplication operator by the element v acts on V (α) as the scalar multiplication by α.

      The simple superalgebras Kan(n), n ≥ 2 are exceptional (see Martínez et al. (2001)). Therefore, they do not have non-zero one-sided Jordan bimodules.

      THEOREM 1.9 (Stern (1995), Martínez and Zelmanov (2009), Solarte and Shestakov (2016)).– Every finite dimensional irreducible Jordan bimodule over Kan(n), n ≥ 2 is isomorphic to V (α) or V (α)^op, α ∈ F.

      In Solarte and Shestakov (2016), the theorem above was proved for algebras over a field of characteristic p > 2.

      1.7.2(c) Jordan superalgebras of a superform. Let be a ℤ/2ℤ-graded vector space with a non-degenerate supersymmetric bilinear form. Assume , and choose a basis e₁,…, em in with 〈ei, ej〉 = δij and a basis v₁, w₁,…, vn, wn in such that

      Let Cl(m) be the Clifford algebra of the restriction of the form 〈 , 〉 to , and let

      be the simple Weyl algebra.

      Then the tensor product S = Cl(m) ⊗F Wn is the universal associative enveloping superalgebra of the Jordan superalgebra J = V + F ∙ 1.

      Since the algebra Wn, n ≥ 1 is infinite dimensional, it follows that the superalgebra J does not have non-zero finite dimensional one-sided Jordan bimodules unless n = 0.

      Consider in the algebra