Abstract
Twisted generalized Weyl algebras (TGWAs) A(R,σ,t) are defined over a base ring R by parameters σ and t, where σ is an n-tuple of automorphisms, and t is an n-tuple of elements in the center of R. We show that, for fixed R and σ, there is a natural algebra map \(A(R,\sigma ,tt^{\prime })\to A(R,\sigma ,t)\otimes _{R} A(R,\sigma ,t^{\prime })\). This gives a tensor product operation on modules, inducing a ring structure on the direct sum (over all t) of the Grothendieck groups of the categories of weight modules for A(R,σ,t). We give presentations of these Grothendieck rings for n = 1,2, when \(R=\mathbb {C}[z]\). As a consequence, for n = 1, any indecomposable module for a TGWA can be written as a tensor product of indecomposable modules over the usual Weyl algebra. In particular, any finite-dimensional simple module over \(\mathfrak {sl}_{2}\) is a tensor product of two Weyl algebra modules.
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Presented by: Pramod Achar
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The first author gratefully acknowledges support from Simons Collaboration Grant for Mathematicians, award number 637600. The second author was supported in this research by a Grant-In-Aid of Research and a Summer Faculty Fellowship from Indiana University Northwest.
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Hartwig, J.T., Rosso, D. Grothendieck Rings of Towers of Twisted Generalized Weyl Algebras. Algebr Represent Theor 25, 1345–1377 (2022). https://doi.org/10.1007/s10468-021-10070-w
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DOI: https://doi.org/10.1007/s10468-021-10070-w