Extensions of current groups on S~3 and the adjoint representations

摘要

Let Ω~3(SU(n)) be the Lie group of based mappings from S~3 to SU(n). We construct a Lie group extension of Ω~3(SU(n)) for n ≥ 3 by the abelian group exp2πiA_3~*, where A_3~* 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 Ω~3(SU(2)) which seems not to be exact in Mickelsson's work, though his observation about the fact that the extension of Ω~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 Ω~3(SU(n)) for n ≥ 3.