|
|||||
|
|
|
|
|
|
|
|
|
|
|
|
|
The
homology of the Temperley–Lieb algebras
Rachael Boyd and Richard Hepworth |
|
Geometry & Topology 28 (2024) 1437–1499
|
 |
Abstract |
|
We study the homology and cohomology of the Temperley–Lieb algebra , interpreted as appropriate Tor and Ext groups. Our main result applies under the common assumption that for some unit in the ground ring, and states that the homology and cohomology vanish up to and including degree . To achieve this we simultaneously prove homological stability and compute the stable homology. We show that our vanishing range is sharp when 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: is not flat over for , 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. |
Keywords
homological stability, Temperley–Lieb algebras
|
Mathematical Subject Classification
Primary: 20J06, 16E40
Secondary: 20F36
|
References |
Publication
Received: 22 February 2022
Revised: 8 July 2022
Accepted: 17 August 2022
Published: 10 May 2024
Proposed: Nathalie Wahl
Seconded: Ulrike Tillmann, Ciprian Manolescu
|
Authors |
| Rachael Boyd | |
| Department of Pure Mathematics and
Mathematical Statistics University of Cambridge Cambridge United Kingdom |
School of Mathematics and
Statistics University of Glasgow Glasgow United Kingdom |
| https://www.maths.gla.ac.uk/~rboyd/ | |
| Richard Hepworth | |
| Institute of Mathematics University of Aberdeen Aberdeen United Kingdom |
|
| http://homepages.abdn.ac.uk/r.hepworth/pages/ | |
| © 2024 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
Open Access made possible by participating institutions via Subscribe to Open.
