Symmetry groups and non-abelian cohomology

by R. F. Streater, Communications in Mathematical Phys., 132, 201-215, 1990.

ABSTRACT

We consider the implementation of symmetry groups of automorphisms of an algebra of observables in a reducible representation whose multipliers in general are non-commuting operators in the commutant of the representation. The multipliers obey a non-abelian cocycle relation which generalizes the 2-cohomology of the group. Examples are given from the theory of spin algebras and continuous tensor products. For type I representations we show that the multiplier can be chosen to lie in the centre, giving an isomorphism with abelian theory.

REMARKS

Go to my paper with Gajdziniski for a surprising application of the theory.

This paper was written in response to a request by A. Jaffe, the editor of Communications in Mathematical Physics, for a contribution to the issue celebrating the work of R. Jost and A. S. Wightman. I had just moved to Virginia Tech., and Jaffe's letter was sent to King's College London, who did not forward it until there was only two weeks to go before the invited papers were due. I had been working on the theory of chemical cotransport but this was not ready for publication, and might not be considered suitable for CMP (it did not mention von Neumann factors of type III). But I could not let this chance to honour my main mentors go by. I then resolved to resurrect my previous attempts to redo the Wigner theory of cocycles of group representations when the representation is reducible. Naturally, I chose to set the problem in the context of a factor representation of an algebra of observables, in the spirit of Haag. It is the condition, that the product of two symmetry automorphisms is another such, that leads to the Baer formula for "factor sets". I then found a theorem and an example, so the paper was finished in two weeks.

This idea has been extended to covariant systems of positive-operator-valued measures in a nice paper by Thomas Breuer.