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

TL;DR

This paper introduces $rak{g}$-quasi-Frobenius Lie algebras, a structure linking symplectic Lie groups, category theory, and quantum field theory, extending Frobenius algebra concepts to Lie algebra contexts.

Contribution

It defines $rak{g}$-quasi-Frobenius Lie algebras, explores their geometric and categorical interpretations, and connects them to Drinfeld doubles in the setting of Lie bialgebras.

Findings

$rak{g}$-quasi-Frobenius Lie algebras correspond to quasi Frobenius Lie objects in $ extbf{Rep}(rak{g})$

They induce $D(rak{g})$-actions when $rak{g}$ is quasitriangular

The structure generalizes Frobenius algebras in a Lie algebra framework

Abstract

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.