Finite dimensional semigroup quadratic algebras with the minimal number of relations

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Abstract

A quadratic semigroup algebra is an algebra over a field given by the generators x_1, . . . , x_n and a finite set of quadratic relations each of which either has the shape x_j x_k = 0 or the shape x_j x_k = x_l x_m . We prove that a quadratic semigroup algebra given by n generators and d=(n^2+n)/4 relations is always infinite dimensional. This strengthens the Golod–Shafarevich estimate for the above class of algebras. Our main result however is that for every n, there is a finite dimensional quadratic semigroup algebra with n generators and d_n relations, where d_n is the first integer greater than (n^2+n)/4 . That is, the above Golod–Shafarevich-type estimate for semigroup algebras is sharp.
Original languageEnglish
Pages (from-to)239-252
Number of pages14
JournalMonatshefte fur Mathematik
Volume168
Issue number2
Publication statusPublished - Nov 2012

ASJC Scopus subject areas

  • Mathematics(all)

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