Classification of factorial generalized down-up algebras

We determine when a generalized down-up algebra is a Noetherian unique factorisation domain or a Noetherian unique factorisation ring

A multiparameter family of irreducible representations of the quantum plane and of the quantum Weyl algebra

We construct a family of irreducible representations of the quantum plane and of the quantum Weyl algebra over an arbitrary field, assuming the deformation parameter is not a root of unity. We determine when two representations in this family are isomorphic, and when they are weight representations, in the sense of Bavula.Comment: 12 pages, Section 2 has been reorganized, new material added in a new Section

A Parametric Family of Subalgebras of the Weyl Algebra II. Irreducible Modules

An Ore extension over a polynomial algebra F[x] is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra A_h generated by elements x,y, which satisfy yx-xy = h, where h is in F[x]. When h is nonzero, these algebras are subalgebras of the Weyl algebra A_1 and can be viewed as differential operators with polynomial coefficients. In previous work, we studied the structure of A_h and determined its automorphism group and the subalgebra of invariants under the automorphism group. Here we determine the irreducible A_h-modules. In a sequel to this paper, we completely describe the derivations of A_h over any field.Comment: 30 pages, a few of the sections have been placed in a different order at the suggestion of the refere

Normally ordered forms of powers of differential operators and their combinatorics

We investigate the combinatorics of the general formulas for the powers of the operator h∂k, where h is a central element of a ring and ∂ is a differential operator. This generalizes previous work on the powers of operators h∂. New formulas for the generalized Stirling numbers are obtained.Ministerio de Economía y competitividad MTM2016-75024-PJunta de Andalucía P12-FQM-2696Junta de Andalucía FQM–33

Non-Noetherian generalized Heisenberg algebras

In this note, we classify the non-Noetherian generalized Heisenberg algebras H(f) introduced in [R. Lu and K. Zhao, Finite-dimensional simple modules over generalized Heisenberg algebras, Linear Algebra Appl. 475 (2015) 276-291]. In case deg f > 1, we determine all locally finite and also all locally nilpotent derivations of H(f) and describe the automorphism group of these algebras