The Distributivity Of A Class Of Finite Dimensional Algebras

Koszul algebras play a very important role in representation theory of algebras and other related mathematical branches. There is a close relationship between the Koszulity of algebras and the distributivity of algebras.Let Q be a finite quiver. This paper is devoted to finding some conditions for the graded algebra A=TkQokQ1/R1to be distributive. By Gerasimov’s theorem, the graded algebra TkkQ1/R’ is distributive if dimk R’=1. In the paper we show that if Q is a quiver with finitely many vertices and, for any positive integer n, there exists at most one path of length n between any two vertices in Q, then all the quotient al-gebras of the path algebra kQ are distributive, that is, for any admissible ideal R’ of kQ, A=TkQokQ1/R’ is distributive. Therefore, Gerasimov’s theorem is general-ized to the case of non-connected graded algebras.