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We consider the multiplication and differentiation operators x and d/dx, which generate the Weyl algebra. Fix a nonzero polynomial h=h(x) and let y be the operator h.d/dx, so that x and y satisfy the commutation relation [y, x]=h. The algebra generated by x and y is denoted by A_h and is a subalgebra of the Weyl algebra. Noteworthy algebras in this family are the Weyl algebra A_1, the enveloping algebra of the two-dimensional non-abelian Lie algebra A_x, and the Jordan plane A_x^2. We will discuss the representations of the algebras A_h over a field of arbitrary characteristic, including all the irreducible representations. When the base field has prime characteristic some interesting combinatorics emerge, which we will discuss and phrase in the language of partitions. This is joint work in progress with Georgia Benkart and Matt Ondrus.
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