Abstract
This paper studies the homology and cohomology of the Temperley-Lieb algebra TLn(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: TLn(a) is not flat over TLm(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.
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: TLn(a) is not flat over TLm(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.
| Original language | English |
|---|---|
| Journal | Geometry & Topology |
| Publication status | Accepted/In press - 17 Aug 2022 |