Grothendieck Rings of Towers of Twisted Generalized Weyl Algebras

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 , σ , t t ′ ) → A ( R , σ , t ) ⊗ R A ( R , σ , t ′ ) $A(R,\sigma ,tt^{\prime })\to A(R,\sigma ,t)\otimes _{R} A(R,\sigma ,t^{\prime })$ . 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 = ℂ [ z ] $R=\mathbb {C}[z]$ . 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 s l 2 $\mathfrak {sl}_{2}$ is a tensor product of two Weyl algebra modules.