A duality of a twisted group algebra of the hyperoctahedral group and the queer Lie superalgebra (Q2741045)

The author establishes a duality relation (Theorem 4.2) between one of the twisted group algebras of the hyperoctahedral group \(H_k\) (or the Weyl group of type \(B_k\)) and a Lie superalgebra \(q(n_0)\oplus q(n_1)\) for any integers \(k>4\) and \(n_0,n_1>1\). Here \(q(n_0)\) and \(q(n_1)\) denote the ``queer'' Lie superalgebras. The twisted group algebra \({\mathcal B}_k'\) in the focus in this paper belongs to a cocycle different from the one for \({\mathcal B}_k\) used by A. N. Sergeev and by the present author in their previous separate works. This \({\mathcal B}_k\) contains the twisted group algebra \({\mathcal A}_k\) of the symmetric group \({\mathfrak S}_k\) in a straightforward manner (cf. \S 1.1.1), and has a structure similar to the semidirect product of \({\mathcal A}_k\) and \(\mathbb{C}[(\mathbb{Z}/2\mathbb{Z})^k]\). In \S 2, he constructs the \(\mathbb{Z}_2\)-graded simple \({\mathcal B}_k'\)-modules using an analogue of the little group method. These simple \({\mathcal B}_k'\)-modules are slightly different from the non-graded simple \({\mathcal B}_k'\)-modules constructed by Stembridge before because of the difference between \(\mathbb{Z}_2\)-graded and non-graded theories, but they can easily be translated into each other. The construction of the simple \({\mathcal B}_k'\)-modules leads to a construction of the simple \({\mathcal C}_k\otimes{\mathcal B}_k'\)-modules in \S 3, where \({\mathcal C}_k\) is the \(2^k\)-dimensional Clifford algebra and \(\otimes\) denotes the \(\mathbb{Z}_2\)-graded tensor product. In \S 4, he defines a representation of \({\mathcal C}_k\otimes{\mathcal B}_k'\) in the \(k\)-fold tensor product \(W=V^{\otimes k}\) of \(V={\mathcal C}^{n_0+n_1}\oplus{\mathcal C}^{n_0+n_1}\), the space of the natural representation of the Lie superalgebra \(q(n_0+n_1)\). This representation of \({\mathcal C}_k\otimes{\mathcal B}_k'\) depends on \(n_0\) and \(n_1\), not just on \(n_0+n_1\). He shows that the centralizer of \({\mathcal C}_k\otimes{\mathcal B}_k'\) in \(\text{End}(W)\) is generated by the action of the Lie superalgebra \(q(n_0+n_1)\) (Theorem 4.1). Moreover he shows that \({\mathcal B}_k'\) and \(q(n_0+n_1)\) act on a subspace \(W^\varepsilon\) of \(W\) ``as mutual centralizers of each other'' (Theorem 4.2). Note that \({\mathcal A}_k\) and \(q(n)\) act on the same space \(W^\varepsilon\) ``as mutual centralizers of each other'' (cf. Theorem B).NEWLINENEWLINEFor the entire collection see [Zbl 0963.00024].