Temperley-Lieb 代数的同调
The homology of the Temperley-Lieb algebras
AI总结：
本文借鉴群族同调稳定性方法，构造平面内射词链复形与归纳分解，证明 TL_n(a) 的同调和上同调在次数 n-2 以内消失，并确定偶数 n 时范围最佳。
摘要：
本文研究 Temperley-Lieb 代数 TL_n(a) 的同调与上同调，并将其解释为相应的 Tor 群和 Ext 群。我们的主要结果在通常假设下成立，即对于基环中的某个单位元 v，有 a=v+v^{-1}；该结果表明，其同调与上同调在直至并包括次数 (n-2) 的范围内均为零。为实现这一点，我们同时证明了同调稳定性并计算了稳定同调。我们还表明，当 n 为偶数时，这一消失范围是最佳的。我们的方法受到群族同调稳定性所用工具和技术的启发。我们构造并利用了一个“平面内射词”链复形，它类似于用于证明对称群稳定性的内射词复形。然而，在这一代数环境中，我们遇到了一个新的困难：对于 m<n，TL_n(a) 在 TL_m(a) 上不是平坦的，因此无法使用 Shapiro 引理。我们通过构造所谓的相关模的“归纳分解”来解决这一困难。Temperley-Lieb 代数同调与上同调的消失结果也可以由 Jones-Wenzl 投影元的存在得到。一般而言，我们自己的消失结果远比这些结果强，但在一种受限情形下，我们能够借助 Jones-Wenzl 投影元的存在获得额外的消失结果。我们相信，这些结果连同第二作者关于 Iwahori-Hecke 代数的工作，首次将同调稳定性技术应用于非群代数的代数。
英文摘要：
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.
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