正交群的泛覆盖
Universal Coverings of the Orthogonal Groups
摘要：
利用Clifford-Lipschitz群研究了正交群及其扩张的泛覆盖。给出了基本离散对称性（空间反演$P$、时间反演$T$、电荷共轭$C$及其组合$PT$、$CP$、$CT$、$CPT$）的代数描述。利用Clifford代数的基本自同构，研究了实数域和复数域上多维空间正交群的离散子群$\{1, P, T, PT\}$.基本自同构形成一个4阶有限群。电荷共轭由Clifford代数的复共轭伪自同构表示。这样的描述允许扩展自同构群。证明了一个扩展的自同构群（$CPT$ -群）构成一个8阶有限群.研究了扩张自同构群与有限群的群结构和同构。证明了$CPT$ -群存在64种不同的实现。给出了正交群的泛覆盖（Clifford-Lipschitz群）在$CPT$ -结构下的一个推广，其中包括著名的Shirokov-Dcabrowski $PT$ -结构作为特例.引入了商Clifford-Lipschitz群和商表示。结果表明，商群的完全分类依赖于扩展自同构群的各种子群的结构。
英文摘要：
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-Dcabrowski $PT$-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.
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