Subalgebras of the Fomin–Kirillov algebra

Abstract

The Fomin---Kirillov algebra $${\mathcal {E}}_n$$En is a noncommutative quadratic algebra with a generator for every edge of the complete graph on n vertices. For any graph G on n vertices, we define $${{\mathcal {E}}_G}$$EG to be the subalgebra of $${\mathcal {E}}_n$$En generated by the edges of G. We show that these algebras have many parallels with Coxeter groups and their nil-Coxeter algebras: for instance, $${\mathcal {E}}_G$$EG is a free $${\mathcal {E}}_H$$EH-module for any $$H\subseteq G$$H⊆G, and if $${\mathcal {E}}_G$$EG is finite-dimensional, then its Hilbert series has symmetric coefficients. We determine explicit monomial bases and Hilbert series for $${\mathcal {E}}_G$$EG when G is a simply laced finite Dynkin diagram or a cycle, in particular showing that $${\mathcal {E}}_G$$EG is finite-dimensional in these cases. We also present conjectures for the Hilbert series of $${\mathcal {E}}_{\tilde{D}_n}$$ED~n, $${\mathcal {E}}_{\tilde{E}_6}$$EE~6, and $${\mathcal {E}}_{\tilde{E}_7}$$EE~7, as well as the graphs G on six vertices for which $$\mathcal {E}_G$$EG is finite-dimensional.