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-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.