Vol. 9, No. 6, 2015

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Electrical networks and Lie theory

Thomas Lam and Pavlo Pylyavskyy

Vol. 9 (2015), No. 6, 1401–1418
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 (EL2n)0 of the electrical Lie group is an analogue of the totally nonnegative subsemigroup (Un)0 of the unipotent subgroup of SLn. We establish decomposition and parametrization results for (EL2n)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.

Keywords
Lie algebras, Serre relations, electrical networks
Mathematical Subject Classification 2010
Primary: 05E15
Milestones
Received: 19 May 2014
Revised: 9 April 2015
Accepted: 11 June 2015
Published: 7 September 2015
Authors
Thomas Lam
Department of Mathematics
University of Michigan
530 Church St.
Ann Arbor, MI 48109
United States
Pavlo Pylyavskyy
Department of Mathematics
University of Minnesota
206 Church St. SE
Minneapolis, MN 55455
United States