Electrical networks, electrical Lie algebras and Lie groups of finite Dynkin type
نویسنده
چکیده
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 EL2n whose positive part (EL2n)≥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 el2n 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 eb2n . The nonnegative part (EB2n)≥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. Résumé Curtis, Ingerman et Morrow ont étudié l’espace des réseaux électriques circulaires plans et ont classifié toutes les matrices de réponses possibles pour ces réseaux. Lam et Pylyavskyy ont trouvé un groupe de Lie EL2n dont la partie positive (EL2n)≥0 agit naturellement sur le réseau électrique circulaire plan par une description combinatoire, où l’action est inspirée par la transformation étoile vers triangle des réseaux électriques. L’algèbre de Lie el2n est semi-simple et isomorphe à l’algèbre symplectique. A la fin de leur article, ils proposent une généralisation des algèbres de Lie électriques pour tous les types de Dynkin finis. Nous donnons la structure de l’algèbre de Lie électrique eb2n du type B. La partie positive (EB2n)≥0 du groupe de Lie correspondant agit conjecturalement sur une famille de ”miroirs réseaux électriques circulaires symétriques plans”. Cette famille de réseaux a des propriétés combinatoires intéressantes. Nous donnons enfin quelques résultats partiels de l’algèbres de Lie électrique du type C et du type D, où une étude analogue doit être développée.
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