Homological stability for Temperley-Lieb algebras

Rachael Boyd, Richard Hepworth

Published 2020-06-07Version 1

This paper studies the homology and cohomology of the Temperley-Lieb algebra TL_n(a), interpreted as appropriate Tor and Ext groups. Our main result applies under the common assumption that a=v+v^{-1} for some unit v in the ground ring, and states that the homology and cohomology vanish up to and including degree (n-2). To achieve this we simultaneously prove homological stability and compute the stable homology. We show that our vanishing range is sharp when n is even. Our methods are inspired by the tools and techniques of homological stability for families of groups. We construct and exploit a chain complex of 'planar injective words' that is analogous to the complex of injective words used to prove stability for the symmetric groups. However, in this algebraic setting we encounter a novel difficulty: TL_n(a) is not flat over TL_m(a) for m<n, so that Shapiro's lemma is unavailable. We resolve this difficulty by constructing what we call 'inductive resolutions' of the relevant modules. We believe that these results, together with the second author's work on Iwahori-Hecke algebras, are the first time the techniques of homological stability have been applied to algebras that are not group algebras.