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
electrical Lie group is isomorphic to the symplectic group. The electrically nonnegative
part
of the electrical Lie group is an analogue of the totally nonnegative subsemigroup
of the unipotent
subgroup of
.
We establish decomposition and parametrization results for
,
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.