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The homology of the Temperley–Lieb algebras

Rachael Boyd and Richard Hepworth

Geometry & Topology 28 (2024) 1437–1499

We study 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 + v1 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.

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.

homological stability, Temperley–Lieb algebras
Mathematical Subject Classification
Primary: 20J06, 16E40
Secondary: 20F36
Received: 22 February 2022
Revised: 8 July 2022
Accepted: 17 August 2022
Published: 10 May 2024
Proposed: Nathalie Wahl
Seconded: Ulrike Tillmann, Ciprian Manolescu
Rachael Boyd
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
United Kingdom
School of Mathematics and Statistics
University of Glasgow
United Kingdom
Richard Hepworth
Institute of Mathematics
University of Aberdeen
United Kingdom

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