Categories of bistochastic measures, and representations of some infinite-dimensional groups

The following groups are considered: the automorphism group of a Lebesgue measure space (with finite or $\sigma$-finite measure), groups of measurable functions with values in a Lie group, and diffeomorphism groups of manifolds. It turns out that the theory of representations of all these groups is closely related to the theory of representations of some category, which will be called the category of $G$-polymorphisms. Objects of this category are measure spaces, and a morphism from $M$ to $N$ is a probability measure on $M\times N\times G$, where $G$ is a fixed Lie group. For some of the above-mentioned infinite-dimensional groups $\mathfrak{G}$ it is shown that any representation of $\mathfrak{G}$ extends canonically to a representation of some category of $G$-polymorphisms. For automorphism groups of measure spaces this makes it possible to obtain a classification of all unitary representations. Also “new” examples of representations of groups of area-preserving diffeomorphisms of two-dimensional manifolds are constructed.