Generalized electrical Lie algebras

TL;DR

This paper introduces generalized electrical Lie algebras associated with Kac-Moody Lie algebras, exploring their structures, deformations, and models, extending prior work on electrical Lie algebras.

Contribution

It generalizes electrical Lie algebras to all Kac-Moody Lie algebras, defines vertex and edge types, and identifies new models for classical cases like sl_n, so_n, and sp_{2n}.

Findings

Vertex type electrical Lie algebras are subalgebras of the associated Lie algebra.

Vertex type electrical Lie algebras are flat deformations of nilpotent subalgebras.

New models for electrical Lie algebras are found in classical Lie algebra cases.

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 $sl_n$, $so_n$, and $sp_{2n}$ 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.