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 $\mathfrak{g}$ 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 $\mathfrak{g}$ and are flat deformations of the nilpotent Lie subalgebra of $\mathfrak{g}$. 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 an open problem.

AmsLaTeX, 19 pages, few more misprints corrected, to appear in Advances