Álgebras associadas a grafos orientados em níveis e a propriedade da Koszulidade

Abstract

In this work we study classes of algebras associated with layered directed graphs. Let 􀀀 be a layered directed graph. We determine the algebra Ap􀀀q; generated by the edges of the graph, satisfying a set of quadratic relations R; and the dual algebra Ap􀀀q!, associated with grpAp􀀀qq. For each P Autp􀀀q we determine: the algebra Ap􀀀 q; where 􀀀 is the subgraph of 􀀀 whose vertices are xed by ; the graded trace generating functions Tr pAp􀀀q; tq and Tr pAp􀀀q!; tq: We also determine the multiplicities of the irreducible representations of AutpAp􀀀qq acting on Ap􀀀q and Ap􀀀q!: We show that for a layered directed graph 􀀀, satisfying some hypotheses, AutpAp􀀀qq K Autp􀀀q. Finally, we verify the property Tr pAp􀀀q; tq Tr pAp􀀀q!; tq 1 for all P Autp􀀀q, called koszulity property. We consider two classes of algebras, the algebra associated to the Hasse graph of the partially ordered set of faces of a star polygon, Ap􀀀 q; and the algebra associated with the Hasse graph of the lattice of subespaces of a nite dimensional vector space over Fq; ApLpn; qqq: