Finite dimensional semigroup quadratic algebras with minimal number of relations

Details Finite dimensional semigroup quadratic algebras with minimal number of relations Finite dimensional semigroup quadratic algebras with minimal number of relations

2011-04-11

arxiv.org/abs/1104.2029v3

Mathematics

Rings and Algebras

V3: corrected typos and stylistic changes, accepted for publication in Monatshefte fuer Mathematik

Scientific paper

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 $x_jx_k=0$ or the shape $x_jx_k=x_lx_m$. We prove that a quadratic semigroup algebra given by $n$ generators and $d\leq \frac{n^2+n}{4}$ 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 $\delta_n$ relations, where $\delta_n$ is the first integer greater than $\frac{n^2+n}{4}$. This shows that the above Golod-Shafarevich type estimate for semigroup algebras is sharp.

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