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

Source abstract

An Ore extension over a polynomial algebra F [ x ] \mathbb {F}[x] is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra A h \mathsf {A}_h generated by elements x , y x,y , which satisfy y x − x y = h yx-xy = h , where h ∈ F [ x ] h\in \mathbb {F}[x] . When h ≠ 0 h \neq 0 , the algebras A h \mathsf {A}_h are subalgebras of the Weyl algebra A 1 \mathsf {A}_1 and can be viewed as differential operators with polynomial coefficients. In previous work, we studied the structure of A h \mathsf {A}_h and determined its automorphism group A u t F ( A h ) \mathsf {Aut_\mathbb {F}}(\mathsf {A}_h) and the subalgebra of invariants under A u t F ( A h ) \mathsf {Aut_\mathbb {F}}(\mathsf {A}_h) . Here we determine the irreducible A h \mathsf {A}_h -modules. In a sequel to this paper, we completely describe the derivations of A h \mathsf {A}_h over any field.