On contractions of quantum orthogonal groups

Abstract

The standard Faddeev quantization of the simple groups is modified in such a way that the quantum analogs of the nonsemisimple groups are obtained by contractions. The contracted quantum groups are regarded as the algebras of noncommutative functions generated by elements $J_{ik}t_{ik},$ where $J_{ik}$ are some products of generators of the algebra ${\bf D}(\iota)$ and $t_{ik}$ are the noncommutative generators of guantum group. Possible contractions of quantum orthogonal groups essentially depend on the choice of primitive elements of the Hopf algebra. All such choices are considered for quantum group $SO_{q}(N;C)$ and all allowed contractions in Cayley--Klein scheme are described. The quantum deformations of the complex kinematical groups have been investigated as a contractions of $SO_q(5;C).$ The quantum Euclead $E_q(4;C)$ and Newton $N_q(4;C)$ groups with unchanged deformation parameter as well as Newton group $N_v(4;C)$ with transformed deformation parameter are obtained. But there is no quantum analog of the (complex) Galilei group $G(1,3).$ According to correspondence principle a new physical theory must include an old one as a particular case. For space-time symmetries this principle is realized as the chain of contractions of the kinematical groups: $$ S^{\pm}(1,3)\stackrel{K \to 0}{\longrightarrow} P(1,3)\stackrel{c \to \infty}{\longrightarrow}G(1,3). $$ As it was mentioned above there is no quantum deformation of the complex Galilei group in the standard Cayley--Klein scheme, therefore it is not possible to construct the quantum analog of the full chain of contractions of the (1+3) kinematical groups even at the level of a complex groups.