A Duality of a Twisted Group Algebra of the Hyperoctahedral Group and the Queer Lie Superalgebra

We establish a duality relation (Theorem 4.2) between one of the twisted group algebras, of the hyperoctahedral group Hk (or the Weyl group of type Bk) and a Lie superalgebra q(n0)⊕ q(n1) for any integers k ≥ 4 and n0, n1 ≥ 1. Here q(n0) and q(n1) denote the “queer” Liesuperalgebras as called by some authors. The twisted group algebra B′ k in focus in this paper belongs to a different cocycle from the one Bk used by A. N. Sergeev in his work [8] on a duality with q(n) and by the present author in a previous work [11]. This B′ k contains the twisted group algebra Ak of the symmetric group Sk in a straightforward manner (§1. 1. 1), and has a structure similar to the semidirect product of Ak and C[(Z/2Z)]. (B′ k and Bk were denoted by C[−1,+1,+1]Wk and C[+1,+1,−1]Wk respectively by J. R. Stembridge in [10].) In §2, we construct the Z2-graded simple B′ k-modules (where Z2 = Z/2Z) using an analogue of the little group method. These simple B′ k modules are slightly different from the non-graded simple B′ k-modules constructed by Stembridge in [10] because of the difference between Z2-graded and non-graded theories, but they can easily be translated into each other. We will use the algebra Ck . ⊗B′ k, where Ck is the 2-dimensional Clifford algebra (cf. (3.2)) and . ⊗ denotes the Z2-graded tensor product (cf. [1], [2], [11, §1]), as an intermediary for establishing our duality, as we explain below. The construction of the simple B′ k-modules leads to a construction of the simple Ck . ⊗ B′ k-modules in §3. In §4, we define a representation of Ck . ⊗ B′ k in the k-fold tensor product W = V ⊗k of V = C01 ⊕ C01 , the space of the natural representation of the Lie superalgebra q(n0 + n1). This representation of Ck . ⊗ B′ k depends on n0 and n1, not just n0 + n1. Note that Bk can be regarded as a subalgebra of Ck . ⊗ B′ k, since Bk is isomorphic to Ck . ⊗ Ak by our previous result (cf. (3.3) of [11]). Under this embedding, our representation of Ck . ⊗ B′ k restricts to the representation of Bk in