论文标题
扭曲的广义Weyl代数的塔的盖环
Grothendieck rings of towers of twisted generalized Weyl algebras
论文作者
论文摘要
扭曲的广义Weyl代数（TGWAS）$ a（r，σ，t）$在基本环$ r $上定义为parameters $σ$和$ t $，其中$σ$是$ n $ n $ - $ n $ - $ t $ t $ a $ n $ n $ n $ - $ n $ - $ n $ of element $ r $ r $。我们表明，对于固定的$ r $和$σ$，有一个天然代数映射$ a（r，σ，tt'）\ to（r，σ，t）\ otimes_r a（r，σ，t'）$。这给出了模块上的张量产品操作，从而在$ a（r，σ，t）$的重量模块类别的Grothendieck组的直接总和（在所有$ t $上）诱发了环结构。当$ r = \ mathbb {c} [z] $时，我们以$ n = 1,2 $的形式提供了这些Grothendieck戒指的演示文稿。结果，对于$ n = 1 $，TGWA的任何不可分解的模块都可以写成通常的Weyl代数上的不可分解模块的张量产品。特别是，$ \ mathfrak {sl} _2 $上的任何有限维简单模块都是两个Weyl代数模块的张量产品。
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 $A(R,σ,tt')\to A(R,σ,t)\otimes_R A(R,σ,t')$. 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.
