%0 Journal Article
%A
%A
%T Real and complex asymptotic symmetries in quantum gravity, irreducible representations, polygons, polyhedra, and the *A, D, E* series
%D 1992
%R 10.1098/rsta.1992.0009
%J Philosophical Transactions of the Royal Society of London. Series A:
Physical and Engineering Sciences
%P 271-299
%V 338
%N 1650
%X The Bondi-Metzner-Sachs group B is 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 of B and 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 complexification CB of B itself are analysed. It is proved that all induced IRS of CB arise from IRS of compact 'little groups’. It follows that some IRS of CB are 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.
%U http://rsta.royalsocietypublishing.org/content/roypta/338/1650/271.full.pdf