The 2-plectic structures induced by the Lie bialgebras

Mohammad Shafiee

doi:10.3934/jgm.2017003

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2017, Volume 9, Issue 1: 83-90. Doi: 10.3934/jgm.2017003

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The 2-plectic structures induced by the Lie bialgebras

Mohammad Shafiee

Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran

Received: June 2016

Revised: January 2017

Published: May 2017

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Abstract

In this paper we show that if the Lie algebra $\mathfrak{g}$ admits a Lie bialgebra structure and $\mathcal{D}$ is a Lie group with Lie algebra $\mathfrak{d}$, the double of $\mathfrak{g}$, then $\mathcal{D}$ or its quotient by a suitable Lie subgroup admits a $2$-plectic structure. In particular it is shown that the imaginary part of the Killing form on $\mathfrak{sl}(n, \mathbb{C})$ (as a real Lie algebra) induces a $2$-plectic structure on $SL(n, \mathbb{C})$.

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Mathematics Subject Classification: 53DXX, 17B37, 17B62.

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