Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: $\frak{g}$-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.

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The categories of this paper are: hep-lat
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The categories of this paper are: q-alg
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The categories of this paper are: quant-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: stat
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The categories of this paper are: physics
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The categories of this paper are: adap-org
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The categories of this paper are: supr-con
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The categories of this paper are: dg-ga
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The categories of this paper are: bayes-an
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The categories of this paper are: funct-an
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The categories of this paper are: hep-ex
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The categories of this paper are: q-bio
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The categories of this paper are: math-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-th