1 Koszul Algebras

The algebras Qn 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, Pn. 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 chPn, is Koszul. Then we show it is possible to extend what we did to the algebras Pn although they are not PGC. Finally we examine the Hilbert Series of the algebras Pn 1 Koszul Algebras There are a number of equivalent definitions of Koszul algebras including this lattice definition from Ufnarovskij [7]. Definition 1. A quadratic algebra A = {V, R} (where V is the span of the generators and R the span of the generating relations in V ⊗ V) is Koszul if the collection of n − 1 subspaces {V ⊗i−1 ⊗ R ⊗ V ⊗n−i−1 }i generates a distributive lattice in V ⊗n for any n. The characterization of Koszulity we will need arises from this definition and is based on the diamond lemma. Suppose that A is a quadratic algebra with relations R in V ⊗ V and a monomial ordering exists so that every overlap ambiguity of degree three resolves. Then A is a PBW-algebra (see chapter one of [5]). The following result is due to S. Priddy and found in [6]. Theorem 1. Any quadratic PBW-algebra is Koszul. We will also need the following theorem from [7]. Theorem 2. A quadratic algebra A is Koszul iff its dual algebra A * is Koszul. In the situation where they are both Koszul the Hilbert series of A is given by 1 h(−x) where h(x) is the Hilbert series of A. Let P (x) = x n − an−1x n−1 + an−2x n−2 − · · · + (−1) n a0 be a polynomial over a division algebra. I. Gelfand and V. Retakh [3] studied relationships between the coefficients ai and a generic set {x1, · · · , xn} of solutions of P (x) = 0. For any ordering (i1, · · · , in) of {1, …