Universal coverings of 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 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.

Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups

“…When N is odd, a simultaneous reflection along all axes commutes with SO(N ) and we can further simplify O(N ) ∼ = SO(N ) × Z 2 . The analogous relation for the groups P in ± (N ) is [41]…”

“…In this section we review aspects of these Chern-Simons theories. A discussion of the relevant group theory may be found in [38][39][40][41][42], while aspects of their bundles are described in [38,39].…”

Constitutive relations in optics in terms of geometric algebra

“…In GA formulation no additional spacetime symmetry considerations are needed. As shown in the paper [11] the identity, inversion, reversion and Clifford conjugation operations in GA are isomorphic to group of four, which consists of identity operation, space P and time T reversals, and the combination PT. Thus in geometric algebra the symmetry operations P T PT 1, , , { } are automatically satisfied.…”

“…In GA the objects are multivectors of different grades that have geometric interpretation and can be simply manipulated including their transformations between same grade as well as different grade subspaces. The multivectors of GA satisfy a number of outermorphisms (or involutions) which automatically ensure symmetry properties of the spacetime, namely, P and T transformations of the space and time, and their combination PT [11]. Of all 64 irreducible Clifford algebras represented by 8-periodicity table [12], in optics and electrodynamics the two are the most important, namely, Cl 3,0 algebra which describes Euclidean 3D space and Cl 1,3 algebra which describes Minkowskian relativistic 4D spacetime.…”

Spinor Structure and Matter Spectrum

“…In our case, pure states of the form (3) correspond to charged states. At this point, the sign of charge is changed under action of the pseudoautomorphism A → A of the complex spinor structure (for more details see [24,25,26]). Following to analogy with the Lagrangian formalism, where neutral particles are described by real fields, we introduce vector states of the form…”