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The homology of the Temperley-Lieb algebras

Rachael Boyd, Richard Hepworth

Abstract

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.

The homology of the Temperley-Lieb algebras

Abstract

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.

Paper Structure

This paper contains 33 sections, 59 theorems, 163 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Temperley-Lieb algebras
  3. Homology of Temperley-Lieb algebras
  4. Jones-Wenzl projectors
  5. Planar injective words
  6. Spectral sequences and Shapiro's lemma
  7. Inductive resolutions
  8. Discussion: Homological stability for algebras
  9. Outline
  10. Acknowledgements
  11. Temperley-Lieb Algebras
  12. The Jones basis
  13. Induced modules of Temperley-Lieb Algebras
  14. Iwahori-Hecke algebras
  15. Inductive resolutions
  16. ...and 18 more sections

Key Result

Theorem A

Let $R$ be a commutative ring, and let $a$ be a unit in $R$. Then $\mathop{\mathrm{Tor}}\nolimits^{\mathrm{TL}_n(a)}_d(\mathbbm{1},\mathbbm{1})$ and $\mathop{\mathrm{Ext}}\nolimits_{\mathrm{TL}_n(a)}^d(\mathbbm{1},\mathbbm{1})$ both vanish for $d>0$.

Figures (10)

  • Figure 1: Diagrammatic relations in $\mathrm{TL}_n$.
  • Figure 2: Smoothings of $s_i$
  • Figure 3: The complex $C(m)$.
  • Figure 4: The complex $D(m)$.
  • Figure 5: The complex $\mathbbm{1}\otimes C(m)$
  • ...and 5 more figures

Theorems & Definitions (148)

  • Theorem A
  • Theorem B
  • Theorem C
  • Remark
  • Corollary
  • proof
  • Theorem D
  • Theorem E
  • Theorem F
  • Definition 2.1: The Temperley-Lieb algebra $\mathrm{TL}_n(a)$
  • ...and 138 more