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 $n\geq 3$.
Citation
Tosiaki KORI. "Extensions of current groups on $S^3$ and the adjoint representations." J. Math. Soc. Japan 66 (3) 819 - 838, July, 2014. https://doi.org/10.2969/jmsj/06630819
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