Representations of modular and quantum Weyl algebras and of generalized Heisenberg algebras

The first half of this thesis deals with the representation theory of the nth Weyl algebra Wn over a field of prime characteristic. One of the main ideas introduced is the use of modular Lie algebra techniques to produce modules for Wn, which is a quotient of the universal enveloping algebra of the Heisenberg Lie algebra Hn by the ideal generated by z - 1, where z is the central element of Hn.; The irreducible representations of Wn had been constructed previously by Peters-Shi on spaces of truncated polynomials weighted by exponentials. We show that they have an alternate description coming from induced modules for the Heisenberg algebra Hn, a restricted Lie algebra, by inducing from a one-dimensional module for a "Borel" subalgebra of Hn. We derive a criterion for when such an induced Hn-module is irreducible (precisely when the eigenvalue lambda0 of the central element is nonzero). When lambda0 = 0, we prove that the resulting module is uniserial and indecomposable, but not irreducible. We also apply this approach to produce new indecomposable representations of Hn. Lastly in this part, we prove that the tensor product of two irreducible p-dimensional modules Vlambda and Vmu for the Heisenberg algebra H1 is either indecomposable or splits into a direct sum of p isomorphic irreducible modules Vlambda+mu.; In the second half, we broaden the scope of the investigation by establishing connections with the work on generalized Weyl algebras and their weight modules, as developed by Drozd and Bavula - Van Oystaeyen. We also consider certain generalizations of the Heisenberg algebra defined by Bergen and show how these algebras can be described via the generalized Weyl algebra construction. This enables us to describe the center and classify all the irreducible weight modules for generalized Heisenberg algebras.; Finally, we consider certain q-deformations of the Heisenberg algebra and Weyl algebra, introduced by Hayashi, which have a natural description in terms of derivation operators and multiplication operators acting on the skew polynomial ring. We describe their center and determine all their irreducible weight modules after realizing these q-deformations as generalized Weyl algebras.