Extensions of Current Groups on S³ and the Adjoint Representations

الأصل · EN

Let Omega³(SU(n)) be the Lie group of based mappings from S³ to SU(n). We construct a Lie group extension of Omega³(SU(n)) for n>2 by the abelian group of the affine dual space of SU(n)-connections on S³. 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 Omega³(SU(2)) which seems not to be exact in Mickelsson's work, though his observation about the fact that the extension of Omega³(SU(2)) reduces to the extension by Z₂ is correct. Then we shall investigate the adjoint representation of the Lie group extension of Omega³(SU(n)) for n>2.