Contraction of Algebraical Structures and Different Couplings of Cayley-Klein and Hopf Structures

Abstract

Contractions (and graded contractions) of Lie algebra, Lie bialgebra and Hopf algebra are discussed. It is noticed the fundamental role of E.In{ö}n{ü} and E.P.Wigner idea of degenerate transformations. A constructive algorithm for description of contractions of quantum Cayley-Klein algebras $ so_{z}(n+1; {\bf j}) $ with different choice of the set of primitive operators is suggested. For nonsemisimple quantum algebras it gives nonisomorphic Hopf algebras. From physical point of view this algorithm gives the different physical interpretations of primitive operators for mathematically the same nonsemisimple quantum algebra. The case of $ so_{z}(3; {\bf j}) $ is regarded in detail.