The homology of the Temperley-Lieb algebras
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866911882506403840 |
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| author | Boyd, Rachael Hepworth, Richard |
| author_facet | Boyd, Rachael Hepworth, Richard |
| contents | 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.
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 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2006_04256 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The homology of the Temperley-Lieb algebras Boyd, Rachael Hepworth, Richard Algebraic Topology Geometric Topology Representation Theory 20J06, 16E40 (primary), 20F36 (secondary) 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. 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 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. |
| title | The homology of the Temperley-Lieb algebras |
| topic | Algebraic Topology Geometric Topology Representation Theory 20J06, 16E40 (primary), 20F36 (secondary) |
| url | https://arxiv.org/abs/2006.04256 |