Electrical Lie Algebra of Classical Types

Abstract

We investigate the structure of electrical Lie algebras of finite Dynkin type. These Lie algebras were introduced by Lam-Pylyavskyy in the study of \textit{circular planar electrical networks}. The corresponding Lie group acts on such networks via some combinatorial operations studied by Curtis-Ingerman-Morrow and Colin de Verdi\`{e}re-Gitler-Vertigan. Lam-Pylyavskyy studied the electrical Lie algebra of type $A$ of even rank in detail, and gave a conjecture for the dimension of electrical Lie algebras of finite Dynkin types. We prove this conjecture for all classical Dynkin types, that is, $A$, $B$, $C$, and $D$. Furthermore, we are able to explicitly describe the structure of the corresponding electrical Lie algebras as the semisimple product of the symplectic Lie algebra with its finite dimensional irreducible representations.