The Temperley-Lieb Tower and the Weyl Algebra

Abstract

We define a monoidal category $W$ and a closely related 2-category $2Weyl$ using diagrammatic methods. We show that $2Weyl$ acts on the category $\mathbf{TL}:={⨁}_n{\mathrm{TL}}_n-\mathrm{mod}$ of modules over Temperley-Lieb algebras, with its generating 1-morphisms acting by induction and restriction. The Grothendieck groups of $W$ and a third category we define $W^{\infty }$ are closely related to the Weyl algebra. We formulate a sense in which $K_0(W^{\infty })$ acts asymptotically on $K_0(\mathbf{TL})$.

arXiv comment39 pages, many figures. Comments particularly encouraged!