On a class of algebras associated to directed graphs

Hilbert series of algebras associated to directed graphs

Few changes. We compute the Hilbert series of some algebras associated to directed graphs and related to factorizations of noncommutative polynomials.

On a class of Koszul algebras associated to directed graphs

In math.QA/0506507 I. Gelfand and the authors introduced and studied 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 (i.e. ranked) graphs.

Factorizations of Polynomials over Noncommutative Algebras and Sufficient Sets of Edges in Directed Graphs

To directed graphs with unique sink and source we associate a noncommutative associative alsgebra and a polynomial over this algebra. Edges of the graph correspond to pseudo-roots of the polynomial. We give a sufficient condition when coefficients of the polynomial can be rationally expressed via elements of a given set of pseudo-roots (edges). Our results are based on a new theorem for directed graphs also proved in this paper.

有向图的关系态射

Relation morphisms of directed graphs

本文提出关系图类RG，通过定义可接受图关系，构建一个同时泛化协变与逆变构造的子范畴，解决统一两种构造的难题。

The Minimal Non-Koszul A(Gamma)

The algebras $A(Γ)$, where $Γ$ is a directed layered graph, were first constructed by I. Gelfand, S. Serconek, V. Retakh and R. Wilson. These algebras are generalizations of the algebras $Q_n$, which are related to factorizations of non-commutative polynomials. It was conjectured that these algebras were Koszul. In 2008, T.Cassidy and B.Shelton found a counterexample to this claim, a non-Koszul $A(Γ)$ corresponding to a graph $Γ$ with 18 edges and 11 vertices. We produce an example of a directed layered graph $Γ$ with 13 edges and 9 vertices which produces a non-Koszul $A(Γ)$. We also show thi…

Connection matrices and Lie algebra weight systems for multiloop chord diagrams

We give necessary and sufficient conditions for a weight system on multiloop chord diagrams to be obtainable from a metrized Lie algebra representation, in terms of a bound on the ranks of associated connection matrices. Here a multiloop chord diagram is a graph with directed and undirected edges so that at each vertex precisely one directed edge is entering and precisely one directed edge is leaving, and each vertex is incident with precisely one undirected edge. Weight systems on multiloop chord diagrams yield the Vassiliev invariants for knots and links. The $k$-th connection matrix of a fu…