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

摘要

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 ∈ F[x]. When h ≠ 0, the algebras A_h 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 Aut_F(A_h) and the subalgebra of invariants under Aut_F(A_h). 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.