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.

扭曲广义外尔代数 (TGWA) A ( R , σ , t )由参数σ和t在基环R上定义，其中σ是自同构的n元组，t是中心元素的n元组的R。我们证明，对于固定的R和σ，存在一个自然代数映射\(A(R,\sigma ,tt^{\prime })\to A(R,\sigma ,t)\otimes _{R} A (R,\sigma ,t^{\prime })\)。这给出了对模块的张量乘积运算，在直和上引入了一个环结构（在所有t ) A ( R , σ , t )的权重模块类别的 Grothendieck 群。当\(R=\mathbb {C}[z]\)时，我们给出了这些格罗腾迪克环对于n = 1,2 的演示。因此，对于n = 1，TGWA 的任何不可分解模块都可以写为不可分解模块在通常的 Weyl 代数上的张量积。特别是，\(\mathfrak {sl}_{2}\)上的任何有限维简单模块都是两个 Weyl 代数模块的张量积。