THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Generalized Weyl algebras and elliptic quantum groups

This thesis consists of four papers. In the first paper we present methods and explicit formulas for describing simple weight modules over twisted generalized Weyl algebras. Under certain conditions we obtain a classification of a class of locally finite simple weight modules from simple modules over tensor products of noncommutative tori. As an application we describe simple weight modules over the quantized Weyl algebra of rank two. In the second paper we derive necessary and sufficient conditions for an ambiskew polynomial ring to have a Hopf algebra structure of a certain type, generalizing many known Hopf algebras, for example U(sl2), Uq(sl2) and the enveloping algebra of the 3-dimensional Heisenberg Lie algebra. In a torsion-free case we describe the finite-dimensional simple modules, and prove a generalized Clebsch-Gordan theorem. We construct a Casimir type operator and prove that any finite-dimensional weight module is semisimple. In the third paper we define a notion of 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 ⊆ R. We prove that a weight module V is unitarizable iff it is isomorphic to its finitistic dual V . Using the classification of weight modules by Drozd, Guzner and Ovsienko, we prove necessary and sufficient conditions for an indecomposable weight module to be isomorphic to its finitistic dual, and thus to be unitarizable. Some examples are given, including Uq(sl2) for q a root of unity. In the fourth paper, using the language of h-Hopf algebroids, introduced by Etingof and Varchenko, we construct a dynamical quantum group, Fell(GL(n)), from Felder’s elliptic solution of the quantum dynamical Yang-Baxter equation with spectral parameter associated to the Lie algebra sln. We apply the generalized FRST construction and obtain a bialgebroidFell(M(n)) and study analogues of the exterior algebra and elliptic minors. We prove that the elliptic determinant it is grouplike and almost central. Localizing at this determinant and constructing an antipode we obtain the h-Hopf algebroid Fell(GL(n)).