2013/06/15 by Tosiaki Kori, Tosiaki KORI
Mathematics · Physics and Astronomy · #Abelian group #Adjoint representation #Advanced Differential Geometry Research #Affine transformation #Algebra over a field #Dual (grammatical number) #Extension (predicate logic) #Geometric Analysis and Curvature Flows #Group (periodic table) #Homotopy and Cohomology in Algebraic Topology #Lie group #Representation of a Lie group #math.DG #msc:22E67 #msc:81R10
paper · pdf · doi:10.2969/jmsj/06630819
published as J.Math.Soc.Japan vol.66, No.3(2014) pp. 819-838
arxiv created 2013/06/15 · openalex publication_date 2014/07/01 · arxiv updated 2014/09/04 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 2π i \cal A3∗, where \cal A3∗ 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.