Finite dimensional semigroup quadratic algebras with the minimal number of relations.

A quadratic semigroup algebra is an algebra over a field given by the generators x, . . . , x and a finite set of quadratic relations each of which either has the shape x x = 0 or the shape x x = x x. 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 relations, where δ is the first integer greater than $${\frac{n^2+n}{4}}$$ . That is, the above Golod-Shafarevich-type estimate for semigroup algebras is sharp. [ABSTRACT FROM AUTHOR]