Electrical networks and Lie 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 and 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 electrically nonnegative part ${\left({EL}_{2n}\right)}_{\ge 0}$ of the electrical Lie group is an analogue of the totally nonnegative subsemigroup ${\left(U_n\right)}_{\ge 0}$ of the unipotent subgroup of ${SL}_n$. We establish decomposition and parametrization results for ${\left({EL}_{2n}\right)}_{\ge 0}$, paralleling Lusztig’s work in total nonnegativity, and work of Curtis, Ingerman and Morrow and of Colin de Verdière, Gitler and Vertigan for networks. Finally, we suggest a generalization of electrical Lie algebras to all Dynkin types.