Semidirect products of groups of loops and groups of diffeomorphisms of real, complex and quaternion manifolds

Abstract

This article is devoted to the investigation of semidirect products of groups of loops and groups of diffeomorphisms of finite and infinte dimensional real, complex and quaternion manifolds. Necessary statements about quaternion manifolds with quaternion holomorphic transition mappings between charts of atlases are proved. Unitary representations of these groups including irreducible are constructed with the help of stochastic processes and quasi-invariant transition measures on groups $G$ relative to dense subgroups $G'$. It is proved, that this procedure provides a family of the cardinality $card ({\bf R})$ of pairwise nonequivalent irreducible unitary representations. A differentiability of such representations is studied.