标题： $\frak{g}$-拟Frobenius李代数 显示英文标题

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

摘要： 李代数版本的图雷夫的$\overline{G}$-Frobenius代数从二维同伦量子场论中被提出。这种李代数版本的基础是一种我们称之为\textit{$\frak{g}$-拟Frobenius 李代数}的结构，针对$\frak{g}$一个有限维李代数。 后者由一个拟Frobenius李代数$(\frak{q},\beta)$以及一个左$\frak{g}$-模结构组成，该结构通过导子作用于$\frak{q}$，并且对于$\beta$来说，它是$\frak{g}$-不变的。 从几何上看， $\frak{g}$-拟Frobenius李代数是与辛李群相关的李代数结构，该辛李群由一个李群$G$的作用，该作用通过辛李群自同构进行。 除了几何之外，$\frak{g}$-拟Frobenius李代数也可以从范畴论的角度得到动机。具体来说，$\frak{g}$-拟Frobenius李代数对应于\textit{拟Frobenius李对象}在$\mathbf{Rep}(\frak{g})$中。 如果$\frak{g}$现在配备了一个李双代数结构，那么在\cite{KP}中给出的$\overline{G}$-弗罗贝尼乌斯代数的范畴表述表明，$\overline{G}$-弗罗贝尼乌斯代数的李版本是$\mathbf{Rep}(D(\frak{g}))$中的一个拟弗罗贝尼乌斯李对象，其中$D(\frak{g})$是相关的（半经典）Drinfeld双代数。 我们证明如果$\frak{g}$是一个拟三角李双代数，那么每个$\frak{g}$-拟Frobenius 李代数都有一个诱导的$D(\frak{g})$-作用，使其具有$D(\frak{g})$-拟Frobenius 李代数的结构。 显示英文摘要

摘要： 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.