Generalized electrical Lie algebras

Abstract

We generalize the electrical Lie algebras originally introduced by Lam and Pylyavskyy in several ways. To each Kac-Moody Lie algebra $g$ we associate two types (vertex type and edge type) of the generalized electrical algebras. The electrical Lie algebras of vertex type are always subalgebras of $g$ and are flat deformations of the nilpotent Lie subalgebra of $g$. In many cases including $s{l}_n$, $s{o}_n$, and $s{p}_{2n}$ we find new (edge) models for our generalized electrical Lie algebras of vertex type. Finding an edge model in general is an interesting open problem.