Electrical Networks and Electrical Lie Theory of Classical Types.

[摘要] In this thesis we investigate the structure of electrical Lie algebras of finite Dynkin type. These Lie algebras were introduced by Lam-Pylyavskyy in the study of circular planar electrical networks. Among these electrical Lie algebras, the Lie group corresponding to type A electrical Lie algebra acts on such networks via some combinatorial operations studied by Curtis-Ingerman-Morrow and Colin de Verdi`ere-Gitler-Vertigan. Lam-Pylyavskyy studied the type A electrical Lie algebra of even rank in detail, and gave a conjecture for the dimension of electrical Lie algebras of finite Dynkin types. We prove this conjecture for all classical Dynkin types, that is, A, B, C, and D. Furthermore, we are able to explicitly describe the structure of some electrical Lie algebras of classical types as the semisimple product of the symplectic Lie algebra with its finite dimensional irreducible representations.We then introduce mirror symmetric circular planar electrical networks as the mirror symmetric subset of circular planar electrical networks studied by Curtis-Ingerman-Morrow [CIM] and Colin de Verdi`ere-Gitler-Vertigan [dVGV]. These mirror symmetric networks can be viewed as the type B generalization of circular planar electrical networks. We show that most of the properties of circular planar electricalviiinetworks are well inherited by these mirror symmetric electrical networks. In particular, the type B electrical Lie algebra has an infinitesimal action on such networks. Inspired by Lam [Lam], the space of mirror symmetric circular planar electrical net- works can be compactified using mirror symmetric cactus networks, which admit a stratification indexed by mirror symmetric matchings on [4n]. The partial order on the mirror symmetric matchings emerging from mirror symmetric electrical networks is dual to a subposet of affine Bruhat order of type C. We conjecture that this partial order is the closure partial order of the stratification of mirror symmetric cactus networks.

[发布机构] University of Michigan

[效力级别] Electrical Lie Algebra

[关键词] Electrical Networks;Electrical Lie Algebra;Finite Dynkin Type;Mathematics;Science;Mathematics