Curtis-Ingerman-Morrow studied the space of circular planar electrical networks, and classified all possible response matrices for such networks. Lam and Pylyavskyy found a Lie group $EL_{2n}$ whose positive part $(EL_{2n})_{\geq 0}$ naturally acts on the circular planar electrical network via some combinatorial description, where the action is inspired by the star-triangle transformation of the electrical networks. The Lie algebra $el_{2n}$ is semisimple and isomorphic to the symplectic algebra. In the end of their paper, they suggest a generalization of electrical Lie algebras to all finite Dynkin types. We give the structure of the type $B$ electrical Lie algebra $e_{b_{2n}}$. The nonnegative part $(E_{B_{2n}})_{\geq 0}$ of the corresponding Lie group conjecturally acts on a class of "mirror symmetric circular planar electrical networks". This class of networks has interesting combinatorial properties. Finally, we mention some partial results for type $C$ and $D$ electrical Lie algebras, where an analogous story needs to be developed.

Source : oai:HAL:hal-01207599v1

Volume: DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014)

Section: Proceedings

Published on: January 1, 2014

Submitted on: November 21, 2016

Keywords: network,electrical Lie algebra,finite Dynkin type,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]

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