Abstract
Let Ω3(SU(n)) be the Lie group of based mappings from S3 to SU(n). We construct a Lie group extension of Ω3(SU(n)) for n ≥ 3 by the abelian group exp 2πi 𝒜*3, where 𝒜*3 is the affine dual of the space of SU(n)-connections on S3. 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 Ω3(SU(2)) which seems not to be exact in Mickelsson's work, though his observation about the fact that the extension of Ω3(SU(2)) reduces to the extension by Z2 is correct. Then we shall investigate the adjoint representation of the Lie group extension of Ω3(SU(n)) for n ≥ 3.
