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2015 Electrical networks and Lie theory
Thomas Lam, Pavlo Pylyavskyy
Algebra Number Theory 9(6): 1401-1418 (2015). DOI: 10.2140/ant.2015.9.1401


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.


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Thomas Lam. Pavlo Pylyavskyy. "Electrical networks and Lie theory." Algebra Number Theory 9 (6) 1401 - 1418, 2015.


Received: 19 May 2014; Revised: 9 April 2015; Accepted: 11 June 2015; Published: 2015
First available in Project Euclid: 16 November 2017

zbMATH: 1325.05181
MathSciNet: MR3397406
Digital Object Identifier: 10.2140/ant.2015.9.1401

Primary: 05E15

Keywords: Electrical networks , Lie algebras , Serre relations

Rights: Copyright © 2015 Mathematical Sciences Publishers


Vol.9 • No. 6 • 2015
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