Journal of the Mathematical Society of Japan

Extensions of current groups on $S^3$ and the adjoint representations

Tosiaki KORI

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Abstract

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 $n\geq 3$.

Article information

Source
J. Math. Soc. Japan, Volume 66, Number 3 (2014), 819-838.

Dates
First available in Project Euclid: 24 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1406206974

Digital Object Identifier
doi:10.2969/jmsj/06630819

Mathematical Reviews number (MathSciNet)
MR3238319

Zentralblatt MATH identifier
1298.81107

Subjects
Primary: 81R10: Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $W$-algebras and other current algebras and their representations [See also 17B65, 17B67, 22E65, 22E67, 22E70]
Secondary: 22E67: Loop groups and related constructions, group-theoretic treatment [See also 58D05] 22E65: Infinite-dimensional Lie groups and their Lie algebras: general properties [See also 17B65, 58B25, 58H05]

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

Citation

KORI, Tosiaki. Extensions of current groups on $S^3$ and the adjoint representations. J. Math. Soc. Japan 66 (2014), no. 3, 819--838. doi:10.2969/jmsj/06630819. https://projecteuclid.org/euclid.jmsj/1406206974


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