| Title:
|
$\mathfrak{g}$-quasi-Frobenius Lie algebras (English) |
| Author:
|
Pham, David N. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
52 |
| Issue:
|
4 |
| Year:
|
2016 |
| Pages:
|
233-262 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
symplectic Lie groups |
| Keyword:
|
quasi-Frobenius Lie algebras |
| Keyword:
|
Lie bialgebras |
| Keyword:
|
Drinfeld double |
| Keyword:
|
group actions |
| MSC:
|
18A05 |
| MSC:
|
18E05 |
| MSC:
|
22E60 |
| MSC:
|
22Exx |
| MSC:
|
53D05 |
| idZBL:
|
Zbl 06674902 |
| idMR:
|
MR3610652 |
| DOI:
|
10.5817/AM2016-4-233 |
| . |
| Date available:
|
2016-12-20T21:51:01Z |
| Last updated:
|
2018-01-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145931 |
| . |
| Reference:
|
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