The Algebra $P_n$ is Koszul

Abstract

The algebras $Q_n$ describe the relationship between the roots and coefficients of a non-commutative polynomial. I.Gelfand, S.Gelfand, and V. Retakh have defined quotients of these algebras corresponding to graphs. In this work we find the Hilbert series of the class of algebras corresponding to the $n$-vertex path, $P_n$. We also show this algebra is Koszul. We do this by first looking at class of quadratic algebras we call Partially Generator Commuting. We then find a sufficient condition for a PGC-Algebra to be Koszul and use this to show a similar class of PGC algebras, which we call ch$P_n$, is Koszul. Then we show it is possible to extend what we did to the algebras $P_n$ although they are not PGC. Finally we examine the Hilbert Series of the algebras $P_n$