On a class of algebras associated to directed graphs

Abstract

To any directed graph we associate an algebra with edges of the graph as generators and with relations defined by all pairs of directed paths with the same origin and terminus. Such algebras are related to factorizations of polynomials over noncommutative algebras. We also construct a basis for our algebras asssociated to layered graphs.

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

“…Certain algebras, denoted A(Γ ), associated to directed graphs were first defined by Gelfand, Retakh, Serconek, and Wilson [7,8,10]. We recall the definitions of A(Γ ) and gr A(Γ ) following the development found in [12, §2].…”

“…The construction of this algebra is described in [8] (see Section 1). In brief (using the definition given in Proposition 1.2), the generators are the vertices and the relations are that two paths which have the same starting and ending vertices are equivalent.…”

“…However, the elements on the right-hand side are all of lower degree than those on the left-hand side [8,Lemma 2.2]. Note that the elements on the left-hand side have degree k + l and are in (T (E)/R) [(k+l)|v|−(k+l)(k+l+1) /2] .…”

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 v ∈ V define S − (v) to be the set of all vertices w ∈ V covered by v, i.e. such that there exists an edge with the tail v and the head w. In [4] for each layered graph Γ = (V , E) we constructed an associative algebra generated by the edges of Γ . Let T (E) denote the free associative algebra on E over a field K .…”

Weakly Cohen–Macaulay posets and a class of finite-dimensional graded quadratic algebras

“…The algebra R Γ has a relatively complex history -we give a brief summary. In [4], Gelfand, Retakh, Serconek and Wilson associate to Γ a connected graded F-algebra A Γ , which is called the splitting algebra of Γ; splitting algebras are related to the problem of factoring non-commuting polynomials. Retakh, Serconek and Wilson later showed that if Γ satisfies a combinatorial condition called uniform, then an associated graded algebra of the splitting algebra is quadratic, and it follows that A Γ is quadratic (c.f.…”