On a class of Koszul algebras associated to directed graphs

Abstract

In [I. Gelfand, V. Retakh, S. Serconek, R.L. Wilson, On a class of algebras associated to directed graphs, Selecta Math. (N.S.) 11 (2005), math.QA/0506507] I. Gelfand and the authors of this paper introduced a new class of algebras associated to directed graphs. In this paper we show that these algebras are Koszul for a large class of layered graphs.

Hilbert series of algebras associated to directed graphs

“…In [2] we constructed a linear basis in A(Γ ). In [4] we showed that algebras A(Γ ) are defined by quadratic relations for a large class of directed graphs and proved that in this case they are Koszul algebras. It follows immediately that the dual algebras to A(Γ ) are also Koszul and that their Hilbert series are related.…”

Representations of Aut(A(Γ)) acting on homogeneous components of A

“…A layered graph is uniform if for every v ∈ V j , j 2, every pair of vertices u, w in S 1 (v) satisfies S 1 (u) ∩ S 1 (w) = ∅ ("diamond condition"). [11,Proposition 3.5].) Let Γ be a uniform layered graph.…”

Koszulity of splitting algebras associated with cell complexes☆☆Some of our results were later generalized by T. Cassidy, C. Phan, and B. Shelton in their paper “Noncommutative Koszul algebras from combinatorial topology” in arXiv:0811.3450.

“…For certain graphs Γ (i.e., uniform graphs as defined in [8]) these relations are consequences of a family of quadratic relations and so the splitting algebra A(Γ ) possesses a quadratic dual algebra A(Γ ) ! .…”

“…. In [8] we claimed that any algebra A(Γ ) defined by a uniform layered graph is Koszul. Later, T. Cassidy and B. Shelton constructed a counter-example to this statement and this example forced us to rethink the situation.…”

“…The analysis of [8] does provide a sufficient condition for A(Γ ) to be Koszul. We discuss this in Section 6 and use this sufficient condition to show that if Γ is either a complete layered graph or the graph associated to an abstract simplicial complex then A(Γ ) is Koszul.…”