Pseudo-unitarizable weight modules over generalized Weyl algebras

TL;DR

This paper introduces the concept of pseudo-unitarizability for weight modules over generalized Weyl algebras, establishing criteria for when such modules are isomorphic to their duals, with applications to quantum groups.

Contribution

It defines pseudo-unitarizability in this context and provides necessary and sufficient conditions for modules to be isomorphic to their duals, extending classification results.

Findings

Characterization of pseudo-unitarizable modules

Criteria for modules to be isomorphic to their duals

Examples including quantum groups at roots of unity

Abstract

We define a notion of pseudo-unitarizability for weight modules over a generalized Weyl algebra (of rank one, with commutative coeffiecient ring $R$), which is assumed to carry an involution of the form $X^*=Y$, $R^*\subseteq R$. We prove that a weight module $V$ is pseudo-unitarizable iff it is isomorphic to its finitistic dual $V^\sharp$. Using the classification of weight modules by Drozd, Guzner and Ovsienko, we obtain necessary and sufficient conditions for an indecomposable weight module to be isomorphic to its finitistic dual, and thus to be pseudo-unitarizable. Some examples are given, including $U_q(\mathfrak{sl}_2)$ for $q$ a root of unity.