The Quantum Symplectic Cayley-Klein Groups

The contraction method applied to the construction of the nonsemisimple quantum symplectic Cayley-Klein groups $Fun(Sp_q(n;j))$. This groups has been realised as Hopf algebra of the noncommutative functions over the algebra with nilpotent generators. The dual quantum algebras $sp_q(n;j)$ are constructed.Comment: 6 pages, LaTeX, submitted to Proceedings of ' II International Workshop on Classical and Quantum Integrible Systems' (Dubna, 8-12 July,1996), to be published in Int.J.Mod.Phy

Contractions of Integrable Equations

The contraction is applied to obtaining of integrable systems associated with nonsemisimple algebras. The effect of contraction is splitting off some components from initial system without loss of integrability.Comment: 6 pages, LaTeX, submitted to Proceedings of ' II International Workshop on Classical and Quantum Integrible Systems' (Dubna, 8-12 July,1996), to be published in Int.J.Mod.Phy

Non-commutative low dimension spaces and superspaces associated with contracted quantum groups and supergroups

Quantum planes which correspond to all one parameter solutions of QYBE for the two-dimensional case of GL-groups are summarized and their geometrical interpretations are given. It is shown that the quantum dual plane is associated with an exotic solution of QYBE and the well-known quantum $h$-plane may be regarded as the quantum analog of the flag (or fiber) plane. Contractions of the quantum supergroup $GL_q(1|2)$ and corresponding quantum superspace $C_q(1|2)$ are considered in Cartesian basis. The contracted quantum superspace $C_h(1|2;\iota)$ is interpreted as the non-commutative analog of the superspace with the fiber odd part.Comment: Talk given at the XIII Int. Coll. on Integrable Systems and Quantum Groups, June 17-19, 2004, Prague, Czech Republic. Submitted in Czech. J. of Physic