Cassidy, T., Phan, C. and Shelton, B. (2010) Noncommutative Koszul algebras from combinatorial topology. Journal für die reine und angewandte Mathematik (Crelles Journal), 2010(646), pp. 45-63. (doi: 10.1515/CRELLE.2010.065)
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Abstract
Associated to any uniform finite layered graph Γ there is a noncommutative graded quadratic algebra A(Γ) given by a construction due to Gelfand, Retakh, Serconek and Wilson. It is natural to ask when these algebras are Koszul. Unfortunately, a mistake in the literature states that all such algebras are Koszul. That is not the case and the theorem was recently retracted. We analyze the Koszul property of these algebras for two large classes of graphs associated to finite regular CW-complexes, X. Our methods are primarily topological. We solve the Koszul problem by introducing new cohomology groups HX(n, k), generalizing the usual cohomology groups Hn(X). Along with several other results, our methods give a new and primarily topological proof of the main result of [Serconek and Wilson, J. Algebra 278: 473–493, 2004] and [Piontkovski, J. Alg. Comput. 15, 643–648, 2005].
| Item Type: | Articles |
|---|---|
| Status: | Published |
| Refereed: | Yes |
| Glasgow Author(s) Enlighten ID: | Phan, Dr Christopher |
| Authors: | Cassidy, T., Phan, C., and Shelton, B. |
| Subjects: | Q Science > QA Mathematics |
| College/School: | College of Science and Engineering > School of Mathematics and Statistics > Mathematics |
| Journal Name: | Journal für die reine und angewandte Mathematik (Crelles Journal) |
| Journal Abbr.: | J. Reine Angew. Math. |
| ISSN: | 0075-4102 |
| ISSN (Online): | 1435-5345 |
| Published Online: | 17 September 2010 |
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