Universal Coverings of the Orthogonal Groups

Abstract

Universal coverings of the orthogonal groups and their extensions are studied in terms of Clifford-Lipschitz groups. An algebraic description of basic discrete symmetries (space inversion $P$, time reversal $T$, charge conjugation $C$ and their combinations $PT$, $CP$, $CT$, $CPT$) is given. Discrete subgroups $\{1, P, T, PT\}$ of orthogonal groups of multidimensional spaces over the fields of real and complex numbers are considered in terms of fundamental automorphisms of Clifford algebras. The fundamental automorphisms form a finite group of order 4. The charge conjugation is represented by a complex conjugation pseudoautomorphism of the Clifford algebra. Such a description allows one to extend the automorphism group. It is shown that an extended automorphism group ($CPT$-group) forms a finite group of order 8. The group structure and isomorphisms between the extended automorphism groups and finite groups are studied in detail. It is proved that there exist 64 different realizations of $CPT$-group. An extension of universal coverings (Clifford-Lipschitz groups) of the orthogonal groups is given in terms of $CPT$-structures which include well-known Shirokov-D\c{a}browski $PT$-structures as a particular case. Quotient Clifford-Lipschitz groups and quotient representations are introduced. It is shown that a complete classification of the quotient groups depends on the structure of various subgroups of the extended automorphism group.