CLIFFORD AND WEYL SUPERALGEBRAS AND SPINOR REPRESENTATIONS
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Springer Science+Business Media New York (2019)
CLIFFORD AND WEYL SUPERALGEBRAS AND SPINOR REPRESENTATIONS JONAS T. HARTWIG
VERA SERGANOVA
Department of Mathematics Iowa State University Ames, IA-50011, USA [email protected]
Department of Mathematics University of California Berkeley, CA-94720, USA [email protected]
Abstract. We construct a family of twisted generalized Weyl algebras which includes Weyl–Clifford superalgebras and quotients of the enveloping algebras of gl(m|n) and osp(m|2n). We give a condition for when a canonical representation by differential operators is faithful. Lastly, we give a description of the graded support of these algebras in terms of pattern-avoiding vector compositions.
1. Introduction Twisted generalized Weyl algebras (TGWAs) were introduced by Mazorchuk and Turowska in [20], [21] in an attempt to include a wider range of examples than Bavula’s generalized Weyl algebras (GWAs) [1]. Their structure and representations have been studied in [20], [21], [19], [24], [12], [13], [14], [15], [10]. Known examples of TGWAs include multiparameter quantized Weyl algebras [21], [12], [10], the Mickelsson–Zhelobenko step algebras associated to (gln+1 , gln ⊕ gl1 ) [19] and some primitive quotients of enveloping algebras [16]. In this paper we take a step further by proving that supersymmetric analogs of some classical algebras are also examples of TGWAs. Specifically, we show that Weyl–Clifford superalgebras and some quotients of the enveloping algebras of gl(m|n) and osp(m|2n) can be realized as twisted generalized Weyl (TGW) algebras. This suggests that much of the general representation theory from [21], [19], [12] could be applied to the study of certain families of superalgebras. In addition our new algebras provide a large supply of consistent but non-regular TGW algebras (i.e., certain elements ti are zero-divisors). This motivates future development of the theory to include such algebras. It is also worth mentioning that, as a special case, we show that Clifford algebras can be presented as TGW algebras. This shows that TGW algebras can be finitedimensional. To summarize the contents of the present paper, in Section 2 we recall the definition of TGW algebras from [21] which includes certain scalars µij that in our case will be ±1. Some known results that will be used are also stated. In Section 3 DOI: 10.1007/S00031-019-09542-7 Received July 11, 2018. Accepted December 23, 2018. Corresponding Author: Jonas T. Hartwig, e-mail: [email protected]
JONAS T. HARTWIG, VERA SERGANOVA
we prove that the Weyl–Clifford superalgebra from [23] can be realized as a TGW algebra. The main object of the paper is introduced in Section 4, in which we define a family of TGW algebras A(γ)± which depend on a certain matrix γ with integer entries. These algebras naturally come with an algebra homomorphism ϕ from A(γ)± to a Clifford-Weyl algebra. This is a generalization of the construction in [16]. A sufficient condition for ϕγ to be injective is given in Section 4.2. This condition is related
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