NilCoxeter algebras categorify the Weyl algebra

In this paper we present an example of elaborate categorical structures hidden in very simple algebraic objects. We look at the algebra of polynomial differential operators in one variable x, also known as the Weyl algebra, and its irreducible representation in the ring of polynomials Q[x]. We construct an abelian category C whose Grothendieck group can be naturally identified with the ring of polynomials and define exact functors FX : C → C and FD : C → C such that (a) on the Grothendieck group K(C) of the category C functors FX and FD act as the multiplication by x and differentiation, respectively, (b) there is a functor isomorphism FDFX ∼= FXFD ⊕ Id, which lifts the defining relation ∂x = x∂ + 1 of the Weyl algebra, (c) functors FX and FD have nice adjointness properties: FX is left adjoint to FD and right adjoint to FD, twisted by an automorphism of C. The category C is the direct sum of categories Cn over all n ≥ 0, where Cn is the category of finite dimensional representations of the nilCoxeter algebra An, which is generated by Yi, 1 ≤ i ≤ n − 1, subject to relations Y 2 i = 0, YiYj = YjYi for |i − j| > 1 and YiYi+1Yi = Yi+1YiYi+1. The nilCoxeter algebra is the algebra of divided difference operators (see Macdonald [M], Fomin and Stanley [FS])