The homology of the Temperley–Lieb algebras

Boyd, Rachael¹ ; Hepworth, Richard²

¹ Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, United Kingdom, School of Mathematics and Statistics, University of Glasgow, Glasgow, United Kingdom
² Institute of Mathematics, University of Aberdeen, Aberdeen, United Kingdom

Geometry & topology, Tome 28 (2024) no. 3, pp. 1437-1499.

Résumé

We study the homology and cohomology of the Temperley–Lieb algebra ${\mathrm{TL}}_n(a)$, 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: ${\mathrm{TL}}_n(a)$ is not flat over ${\mathrm{TL}}_m(a)$ 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.

Vanishing results for the homology and cohomology of Temperley–Lieb algebras can also be obtained from the existence of the Jones–Wenzl projector. Our own vanishing results are in general far stronger than these, but in a restricted case we are able to obtain additional vanishing results via the existence of the Jones–Wenzl projector.

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.

Détail

DOI : 10.2140/gt.2024.28.1437

Keywords: homological stability, Temperley–Lieb algebras

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