Electrical networks and Lie theory

Lam, Thomas; Pylyavskyy, Pavlo

doi:10.2140/ant.2015.9.1401

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Electrical networks and Lie theory

Thomas Lam and Pavlo Pylyavskyy

Vol. 9 (2015), No. 6, 1401–1418

DOI: 10.2140/ant.2015.9.1401

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 ${({EL}_{2 n})}_{\geq 0}$ of the electrical Lie group is an analogue of the totally nonnegative subsemigroup ${(U_{n})}_{\geq 0}$ of the unipotent subgroup of ${SL}_{n}$ . We establish decomposition and parametrization results for ${({EL}_{2 n})}_{\geq 0}$ , 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

Vol. 9, No. 6, 2015