Pseudo-unitarizable weight modules over generalized Weyl algebras

Details Pseudo-unitarizable weight modules over generalized Weyl algebras Pseudo-unitarizable weight modules over generalized Weyl algebras

2008-03-05

arxiv.org/abs/0803.0687v3

Mathematics

Rings and Algebras

38 pages

Scientific paper

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.

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