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

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>2 by the abelian group of the affine dual space of SU(n)-connections on S^3. In this article we give several improvement of J. Mickelsson's results in 1987, 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>2.