$𝔤$-quasi-Frobenius Lie algebras

Pham, David N.. "$\mathfrak {g}$-quasi-Frobenius Lie algebras." Archivum Mathematicum 052.4 (2016): 233-262. <http://eudml.org/doc/287569>.

@article{Pham2016,
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 $\mathfrak \{g\}$-quasi-Frobenius Lie algebra for $\mathfrak \{g\}$ a finite dimensional Lie algebra. The latter consists of a quasi-Frobenius Lie algebra $(\mathfrak \{q\},\beta )$ together with a left $\mathfrak \{g\}$-module structure which acts on $\mathfrak \{q\}$ via derivations and for which $\beta $ is $\mathfrak \{g\}$-invariant. Geometrically, $\mathfrak \{g\}$-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, $\mathfrak \{g\}$-quasi-Frobenius Lie algebras can also be motivated from the point of view of category theory. Specifically, $\mathfrak \{g\}$-quasi Frobenius Lie algebras correspond to quasi Frobenius Lie objects in $\mathbf \{Rep\}(\mathfrak \{g\})$. If $\mathfrak \{g\}$ is now equipped with a Lie bialgebra structure, then the categorical formulation of $\overline\{G\}$-Frobenius algebras given in [16] suggests that the Lie version of a $\overline\{G\}$-Frobenius algebra is a quasi-Frobenius Lie object in $\mathbf \{Rep\}(D(\mathfrak \{g\}))$, where $D(\mathfrak \{g\})$ is the associated (semiclassical) Drinfeld double. We show that if $\mathfrak \{g\}$ is a quasitriangular Lie bialgebra, then every $\mathfrak \{g\}$-quasi-Frobenius Lie algebra has an induced $D(\mathfrak \{g\})$-action which gives it the structure of a $D(\mathfrak \{g\})$-quasi-Frobenius Lie algebra.},
author = {Pham, David N.},
journal = {Archivum Mathematicum},
keywords = {symplectic Lie groups; quasi-Frobenius Lie algebras; Lie bialgebras; Drinfeld double; group actions},
language = {eng},
number = {4},
pages = {233-262},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {$\mathfrak \{g\}$-quasi-Frobenius Lie algebras},
url = {http://eudml.org/doc/287569},
volume = {052},
year = {2016},
}

TY - JOUR
AU - Pham, David N.
TI - $\mathfrak {g}$-quasi-Frobenius Lie algebras
JO - Archivum Mathematicum
PY - 2016
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 052
IS - 4
SP - 233
EP - 262
AB - 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 $\mathfrak {g}$-quasi-Frobenius Lie algebra for $\mathfrak {g}$ a finite dimensional Lie algebra. The latter consists of a quasi-Frobenius Lie algebra $(\mathfrak {q},\beta )$ together with a left $\mathfrak {g}$-module structure which acts on $\mathfrak {q}$ via derivations and for which $\beta $ is $\mathfrak {g}$-invariant. Geometrically, $\mathfrak {g}$-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, $\mathfrak {g}$-quasi-Frobenius Lie algebras can also be motivated from the point of view of category theory. Specifically, $\mathfrak {g}$-quasi Frobenius Lie algebras correspond to quasi Frobenius Lie objects in $\mathbf {Rep}(\mathfrak {g})$. If $\mathfrak {g}$ is now equipped with a Lie bialgebra structure, then the categorical formulation of $\overline{G}$-Frobenius algebras given in [16] suggests that the Lie version of a $\overline{G}$-Frobenius algebra is a quasi-Frobenius Lie object in $\mathbf {Rep}(D(\mathfrak {g}))$, where $D(\mathfrak {g})$ is the associated (semiclassical) Drinfeld double. We show that if $\mathfrak {g}$ is a quasitriangular Lie bialgebra, then every $\mathfrak {g}$-quasi-Frobenius Lie algebra has an induced $D(\mathfrak {g})$-action which gives it the structure of a $D(\mathfrak {g})$-quasi-Frobenius Lie algebra.
LA - eng
KW - symplectic Lie groups; quasi-Frobenius Lie algebras; Lie bialgebras; Drinfeld double; group actions
UR - http://eudml.org/doc/287569
ER -