136 research outputs found
Noncommutative Koszul Algebras from Combinatorial Topology
Associated to any uniform finite layered graph Gamma there is a
noncommutative graded quadratic algebra A(Gamma) 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 H_X(n,k), generalizing the usual cohomology groups H^n(X).
Along with several other results, our methods give a new and primarily
topological proof of a result of Serconek and Wilson and of Piontkovski.Comment: 22 pages, 1 figur
Hilbert series of quadratic algebras associated with pseudo-roots of noncommutative polynomials
The quadratic algebras Q_n are associated with pseudo-roots of noncommutative
polynomials. We compute the Hilbert series of the algebras Q_n and of the dual
quadratic algebras Q_n^!Comment: Amstex, 24 page
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