1. Grothendieck Rings of Towers of Twisted Generalized Weyl Algebras
- Author
-
Daniele Rosso and Jonas T. Hartwig
- Subjects
16S35, 16D90 ,Pure mathematics ,Ring (mathematics) ,Weyl algebra ,Direct sum ,General Mathematics ,Center (category theory) ,Mathematics - Rings and Algebras ,Base (group theory) ,Tensor product ,Rings and Algebras (math.RA) ,FOS: Mathematics ,Representation Theory (math.RT) ,Indecomposable module ,Simple module ,Mathematics - Representation Theory ,Mathematics - Abstract
Twisted generalized Weyl algebras (TGWAs) $A(R,\sigma,t)$ are defined over a base ring $R$ by parameters $\sigma$ and $t$, where $\sigma$ 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 $\sigma$, there is a natural algebra map $A(R,\sigma,tt')\to A(R,\sigma,t)\otimes_R A(R,\sigma,t')$. 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 $A(R,\sigma,t)$. 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., Comment: 31 pages, 3 figures
- Published
- 2021