Real and complex asymptotic symmetries in quantum gravity, irreducible representations, polygons, polyhedra, and theA, D, Eseries

Abstract

The Bondi-Metzner-Sachs groupBis the common asymptotic group of all asymptotically flat (lorentzian) space-times, and is the best candidate for the universal symmetry group of general relativity. However, in quantum gravity, complexified or euclidean versions of general relativity are frequently considered, and the question arises: Are there similar symmetry groups for these versions of the theory? In this paper it is shown that there are such analogues ofBand a variety of further ones, either real in any signature, or complex. The relationships between these various groups are described. Irreducible unitary representations (IRS) of the complexificationCBofBitself are analysed. It is proved that all induced IRS ofCBarise from IRS ofcompact'little groups’. It follows that some IRS ofCBare controlled by the IRS of the ‘A,D,E' series of finite symmetry groups of regular polygons and polyhedra in ordinary euclidean 3-space. Possible applications to quantum gravity are indicated.