Finite dimensional semigroup quadratic algebras with the minimal number of relations

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 ${d\leq \frac{n^2+n}{4}}$ 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 δ _n relations, where δ _n is the first integer greater than ${\frac{n^2+n}{4}}$ . That is, the above Golod–Shafarevich-type estimate for semigroup algebras is sharp.