African Diaspora Journal of Mathematics

Automorphisms of Cotangent Bundles of Lie Groups

A. Diatta and B. Manga

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Abstract

Let $G$ be a Lie group, ${\mathcal G}$ its Lie algebra and $T^*G$ its cotangent bundle. On $T^*G,$ we consider the Lie group structure obtained by performing a left trivialization and endowing the resulting trivial bundle $G\times {\mathcal G}^*$ with the semi-direct product, using the co-adjoint action of $G$ on the dual space ${\mathcal G}^*$ of ${\mathcal G}$. We investigate the group of automorphisms of the Lie algebra ${\mathcal D}:=T^*{\mathcal G}$ of $T^*G.$ More precisely, we fully characterize the Lie algebra of all derivations of ${\mathcal D},$ exhibiting a finer decomposition into components made of well known spaces. Further, we specialize to the cases where $G$ has a bi-invariant Riemannian or pseudo-Riemannian metric, with the semi-simple and compact cases investigated as particular cases.

Article information

Source
Afr. Diaspora J. Math. (N.S.), Volume 17, Number 2 (2014), 20-46.

Dates
First available in Project Euclid: 11 August 2015

Permanent link to this document
https://projecteuclid.org/euclid.adjm/1439297830

Mathematical Reviews number (MathSciNet)
MR3366685

Zentralblatt MATH identifier
1330.17021

Subjects
Primary: 22C05: Compact groups 22E60: Lie algebras of Lie groups {For the algebraic theory of Lie algebras, see 17Bxx} 22E15: General properties and structure of real Lie groups 22E10: General properties and structure of complex Lie groups [See also 32M05]

Keywords
Cotangent bundle automorphism dérivation Lie group Lie algebra bi-invariant metric bi-invariant tensor supersymmetric Lie group Lie superalgebra Lie supergroup

Citation

Diatta, A.; Manga, B. Automorphisms of Cotangent Bundles of Lie Groups. Afr. Diaspora J. Math. (N.S.) 17 (2014), no. 2, 20--46. https://projecteuclid.org/euclid.adjm/1439297830


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