Idempotents and homology of diagram algebras

IF 1.3 2区 数学 Q1 MATHEMATICS Mathematische Annalen Pub Date : 2024-08-28 DOI:10.1007/s00208-024-02960-3
Guy Boyde
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Abstract

This paper provides a systematization of some recent results in homology of algebras. Our main theorem gives criteria under which the homology of a diagram algebra is isomorphic to the homology of the subalgebra on diagrams having the maximum number of left-to-right connections. From this theorem, we deduce the ‘invertible-parameter’ cases of the Temperley–Lieb and Brauer results of Boyd–Hepworth and Boyd–Hepworth–Patzt. We are also able to give a new proof of Sroka’s theorem that the homology of an odd-strand Temperley–Lieb algebra vanishes, as well as an analogous result for Brauer algebras and an interpretation of both results in the even-strand case. Our proofs are relatively elementary: in particular, no auxiliary chain complexes or spectral sequences are required. We briefly discuss the relationship to cellular algebras in the sense of Graham–Lehrer.

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图代数的幂等性和同源性
本文系统阐述了图代数同调学的一些最新成果。我们的主要定理给出了图代数的同调与具有最大左右连接数的图上子代数的同调同构的标准。根据这个定理,我们推导出了博伊德-赫普沃思和博伊德-赫普沃思-帕茨特的 Temperley-Lieb 和 Brauer 结果的 "可逆参数 "情形。我们还给出了斯洛卡关于奇股滕伯里-李布代数的同调消失定理的新证明,以及布劳尔代数的类似结果和这两个结果在偶股情况下的解释。我们的证明相对简单:特别是不需要辅助链复数或谱序列。我们简要讨论了与格雷厄姆-莱勒意义上的细胞代数的关系。
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来源期刊
Mathematische Annalen
Mathematische Annalen 数学-数学
CiteScore
2.90
自引率
7.10%
发文量
181
审稿时长
4-8 weeks
期刊介绍: Begründet 1868 durch Alfred Clebsch und Carl Neumann. Fortgeführt durch Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück und Nigel Hitchin. The journal Mathematische Annalen was founded in 1868 by Alfred Clebsch and Carl Neumann. It was continued by Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguigon, Wolfgang Lück and Nigel Hitchin. Since 1868 the name Mathematische Annalen stands for a long tradition and high quality in the publication of mathematical research articles. Mathematische Annalen is designed not as a specialized journal but covers a wide spectrum of modern mathematics.
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