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We investigate the structure of electrical Lie algebras of finite Dynkintype. These Lie algebras were introduced by Lam-Pylyavskyy in the study of\textit{circular planar electrical networks}. The corresponding Lie group actson such networks via some combinatorial operations studied byCurtis-Ingerman-Morrow and Colin de Verdi\`{e}re-Gitler-Vertigan.Lam-Pylyavskyy studied the electrical Lie algebra of type $A$ of even rank indetail, and gave a conjecture for the dimension of electrical Lie algebras offinite Dynkin types. We prove this conjecture for all classical Dynkin types,that is, $A$, $B$, $C$, and $D$. Furthermore, we are able to explicitlydescribe the structure of the corresponding electrical Lie algebras as thesemisimple product of the symplectic Lie algebra with its finite dimensionalirreducible representations.

Electrical Lie Algebra of Classical Types