Extensions of current groups on S³ and the adjoint representations

抄録

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.