g-QUASI-FROBENIUS LIE ALGEBRAS

A Lie version of Turaev’s G-Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a g-quasi-Frobenius Lie algebra for g a finite dimensional Lie algebra. The latter consists of a quasi-Frobenius Lie algebra (q, β) together with a left g-module structure which acts on q via derivations and for which β is g-invariant. Geometrically, g-quasi-Frobenius Lie algebras are the Lie algebra structures associated to symplectic Lie groups with an action by a Lie group G which acts via symplectic Lie group automorphisms. In addition to geometry, g-quasi-Frobenius Lie algebras can also be motivated from the point of view of category theory. Specifically, g-quasi Frobenius Lie algebras correspond to quasi Frobenius Lie objects in Rep(g). If g is now equipped with a Lie bialgebra structure, then the categorical formulation of G-Frobenius algebras given in [16] suggests that the Lie version of a G-Frobenius algebra is a quasi-Frobenius Lie object in Rep(D(g)), where D(g) is the associated (semiclassical) Drinfeld double. We show that if g is a quasitriangular Lie bialgebra, then every g-quasi-Frobenius Lie algebra has an induced D(g)-action which gives it the structure of a D(g)-quasi-Frobenius Lie algebra.