FRT Quantization Theory for the Nonsemisimple Cayley-Klein Groups

Details FRT Quantization Theory for the Nonsemisimple Cayley-Klein Groups FRT Quantization Theory for the Nonsemisimple Cayley-Klein Groups

1997-11-26

arxiv.org/abs/q-alg/9711024v1

Mathematics

Quantum Algebra

15 pages, Latex file. Submitted to Proc. Int. Conf. "Quantum Groups, Deformations and Contractions", Istanbul, Turkey, 17-24 S

Scientific paper

The quantization theory of the simple Lie groups and algebras was developed by Faddeev-Reshetikhin-Takhtadjan (FRT). In group theory there is a remarkable set of groups, namely the motion groups of n-dimensional spaces of constant curvature or the orthogonal Cayley-Klein (CK) groups. In some sense the CK groups are in the nearest neighborhood with the simple ones. The well known groups of physical interest such as Euclidean E(n), Poincare P(n), Galileian G(n) and other nonsemisimple groups are in the set of CK groups. But many standart algebraical costructions are not suitable for the nonsemisimple groups and algebras, in particular Killing form is degenerate, Cartan matrix do not exist. Nevertheless it is possible to describe and to quantize all CK groups and algebras, as it was made for the simple ones. The principal proposal is to consider CK groups as the groups over an associative algebra $D$ with nilpotent commutative generators and the corresponding quantum CK groups as the algebra of noncommutative functions over $D$.

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