Finite group

An algebraic description of basic discrete symmetries (space reversal P , time reversal T and their combination P T) is studied. Discrete subgroups of orthogonal groups of multidimensional spaces over the fields of real and complex numbers are considered in terms of fundamental automorphisms of Clifford algebras. In accordance with a division ring structure, a complete classification of automorphism groups is established for the Clifford algebras over the field of real numbers. The correspondence between eight double coverings (Dabrowski groups) of the orthogonal group and eight types of the real Clifford algebras is defined with the use of isomorphisms between the automorphism groups and finite groups. Over the field of complex numbers there is a correspondence between two nonisomorphic double coverings of the complex orthogonal group and two types of complex Clifford algebras. It is shown that these correspondences associate with a well-known Atiyah-Bott-Shapiro periodicity. Generalized Brauer-Wall groups are introduced on the extended sets of the Clifford algebras. The structure of the inequality between the two Clifford-Lipschitz groups with mutually opposite signatures is elucidated. The physically important case of the two different double coverings of the Lorentz groups is considered in details.

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 P T , CP , CT , CP T) is given. Discrete subgroups {1, P, T, P T } 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 (CP T-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 CP T-group. An extension of universal coverings (Clifford-Lipschitz groups) of the orthogonal groups is given in terms of CP T-structures which include well-known Shirokov-Dabrowski P Tstructures 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. An equivalence of the multiplication tables of the groups {1, P, T, P T } and Aut(Cℓ) = {Id, ⋆, , ⋆} proves this isomorphism (in virtue of the commutativity (A ⋆) = (A) ⋆ and the involution conditions (⋆) 2 = () 2