Factoriality of representations of the group of paths on SU(n) (Q794784)

The Sobolev-Lie group of mappings from a manifold I into a compact semi- simple Lie group G has been studied in several papers. When the manifold I is one-dimensional, a representation of the group of mappings can be realized by left translations on sample paths of Brownian motion with values in G: \[ (u^ L(\psi)f)(\eta)\equiv((d\mu_ 0(\psi^{- 1}\eta)/d\mu_ 0(\eta))^{1/2}f(\psi^{-1}\eta) \] where \(\psi \in G^ I\), the group of \(C^{\infty}\) mappings from I into G, \(\eta \in C(I,G)\), the set of continuous mappings from I to G equipped with the Brownian motion measure \(\mu_ 0\), and \(f\in {\mathcal H}=L^ 2(C(I,G),\mu_ 0).\) (The representation realised by right translations \(U^ R\) is unitarily equivalent to \(U^ L).\) The present work is an extension of the results of a previous paper by the same authors and \textit{A. Vershik} [J. Funct. Anal. 51, 115-131 (1983; Zbl 0522.22013)], in which they showed the factoriality of the representation \(U^ L\) in the case \(G=SU(2)\). The extension of these results to the case of SU(n) is achieved by proving a crucial lemma on the algebra of SU(n).