论文标题
Tembyle-Lieb代数的同源
The homology of the Temperley-Lieb algebras
论文作者
论文摘要
本文研究了Temperley-Lieb代数TL_N（A）的同源性和共同体，并被解释为适当的TOR和EXT组。我们的主要结果适用于以下普遍假设：对于地面环中的某些单位V，A = V+V^{ - 1}，并指出同源性和共同体学已经消失至（包括程度（N-2））。为了实现这一目标，我们同时证明同源稳定性并计算稳定的同源性。我们表明，当n均匀时，消失的范围很清晰。 我们的方法灵感来自群体家庭的同源稳定性工具和技术。我们构建并利用了“平面式词语单词”的链复合物，该复合物类似于用于证明对称组稳定性的注射词的复合体。但是，在这种代数环境中，我们遇到了一个新颖的困难：对于m <n，tl_n（a）在tl_m（a）上并不平坦，因此不可用Shapiro的引理。我们通过构建相关模块的“归纳分辨率”来解决这一困难。 也可以从Jones-Wenzl投影仪的存在中获得temperley-Lieb代数的同源性和同源性的消失结果。通常，我们自己的消失结果远比这些结果要强得多，但是在有限的情况下，我们能够通过存在Jones-Wenzl投影仪获得其他消失的结果。 我们认为，这些结果以及第二作者在Iwahori-Hecke代数方面的工作，是同源稳定性技术首次应用于不是组代数的代数。
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.
