Open Access
July, 2014 Extensions of current groups on $S^3$ and the adjoint representations
Tosiaki KORI
J. Math. Soc. Japan 66(3): 819-838 (July, 2014). DOI: 10.2969/jmsj/06630819


Let $\Omega^3(SU(n))$ be the Lie group of based mappings from $S^3$ to $SU(n)$. We construct a Lie group extension of $\Omega^3(SU(n))$ for $n\geq 3$ by the abelian group $\exp 2\pi i {\cal A}_3^{\ast}$, where ${\cal A}_3^{\ast}$ is the affine dual of the space of $SU(n)$-connections on $S^3$. J. Mickelsson in 1987 constructed a similar Lie group extension. In this article we give several improvement of his results, especially we give a precise description of the extension of those components that are not the identity component. We also correct several argument about the extension of $\Omega^3(SU(2))$ which seems not to be exact in Mickelsson's work, though his observation about the fact that the extension of $\Omega^3(SU(2))$ reduces to the extension by Z$_2$ is correct. Then we shall investigate the adjoint representation of the Lie group extension of $\Omega^3(SU(n))$ for $n\geq 3$.


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Tosiaki KORI. "Extensions of current groups on $S^3$ and the adjoint representations." J. Math. Soc. Japan 66 (3) 819 - 838, July, 2014.


Published: July, 2014
First available in Project Euclid: 24 July 2014

zbMATH: 1298.81107
MathSciNet: MR3238319
Digital Object Identifier: 10.2969/jmsj/06630819

Primary: 81R10
Secondary: 22E65 , 22E67

Keywords: adjoint representations , current groups , infinite dimensional Lie groups , Lie group extensions

Rights: Copyright © 2014 Mathematical Society of Japan

Vol.66 • No. 3 • July, 2014
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