Electrical networks and Lie theory

TL;DR

This paper introduces electrical Lie groups and algebras inspired by electrical network transformations, revealing their structure, isomorphisms, and parametrizations, and extends the concept to all Dynkin types.

Contribution

It defines electrical Lie groups and algebras, establishes their properties and isomorphisms, and generalizes the framework to all Dynkin types, connecting electrical network theory with Lie theory.

Findings

Electrical Lie groups are introduced and shown to relate to classical Lie groups.

The type A electrical Lie group is isomorphic to the symplectic group.

Decomposition and parametrization results parallel Lusztig's total nonnegativity theory.

Abstract

We introduce a new class of "electrical" Lie groups. These Lie groups, or more precisely their nonnegative parts, act on the space of planar electrical networks via combinatorial operations previously studied by Curtis-Ingerman-Morrow. The corresponding electrical Lie algebras are obtained by deforming the Serre relations of a semisimple Lie algebra in a way suggested by the star-triangle transformation of electrical networks. Rather surprisingly, we show that the type A electrical Lie group is isomorphic to the symplectic group. The nonnegative part (EL_{2n})_{\geq 0} of the electrical Lie group is a rather precise analogue of the totally nonnegative subsemigroup (U_{n})_{\geq 0} of the unipotent subgroup of SL_{n}. We establish decomposition and parametrization results for (EL_{2n})_{\geq 0}, paralleling Lusztig's work in total nonnegativity, and work of Curtis-Ingerman-Morrow and de Verdi\`{e}re-Gitler-Vertigan for networks. Finally, we suggest a generalization of electrical Lie algebras to all Dynkin types.