$\frak{g}$-quasi-Frobenius Lie algebras

A Lie version of Turaev's $\overline{G}$-Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a \textit{$\frak{g}$-quasi-Frobenius Lie algebra} for $\frak{g}$ a finite dimensional Lie algebra. The latter consists of a quasi-Frobenius Lie algebra $(\frak{q},\beta)$ together with a left $\frak{g}$-module structure which acts on $\frak{q}$ via derivations and for which $\beta$ is $\frak{g}$-invariant. Geometrically, $\frak{g}$-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, $\frak{g}$-quasi-Frobenius Lie algebras can also be motivated from the point of view of category theory. Specifically, $\frak{g}$-quasi Frobenius Lie algebras correspond to \textit{quasi Frobenius Lie objects} in $\mathbf{Rep}(\frak{g})$. If $\frak{g}$ is now equipped with a Lie bialgebra structure, then the categorical formulation of $\overline{G}$-Frobenius algebras given in \cite{KP} suggests that the Lie version of a $\overline{G}$-Frobenius algebra is a quasi-Frobenius Lie object in $\mathbf{Rep}(D(\frak{g}))$, where $D(\frak{g})$ is the associated (semiclassical) Drinfeld double. We show that if $\frak{g}$ is a quasitriangular Lie bialgebra, then every $\frak{g}$-quasi-Frobenius Lie algebra has an induced $D(\frak{g})$-action which gives it the structure of a $D(\frak{g})$-quasi-Frobenius Lie algebra.