General Construction of Nonstandard $R_h$-matrices as Contraction Limits of $R_{q}$-matrices

Details General Construction of Nonstandard $R_h$-matrices as Contraction Limits of $R_{q}$-matrices General Construction of Nonstandard $R_h$-matrices as Contraction Limits of $R_{q}$-matrices

1997-06-28

arxiv.org/abs/q-alg/9706033v1

Mathematics

Quantum Algebra

11 pages, no figures

Scientific paper

10.1142/S021773239800084X

A class of transformations of $R_q$-matrices is introduced such that the $q\to 1$ limit gives explicit nonstandard $R_{h}$-matrices. The transformation matrix is singular itself at $q\to 1$ limit. For the transformed matrix, the singularities, however, cancel yielding a well-defined construction. Our method can be implemented systematically for R-matrices of all dimensions and not only for $sl(2)$ but also for algebras of higher dimensions. Explicit constructions are presented starting with ${\cal U}_q(sl(2))$ and ${\cal U}_q(sl(3))$, while choosing $R_q$ for (fund. rep.)$\otimes$(arbitrary irrep.). The treatment for the general case and various perspectives are indicated. Our method yields nonstandard deformations along with a nonlinear map of the $h$-Borel subalgebra on the corresponding classical Borel subalgebra. For ${\cal U}_h(sl(2))$ this map is extended to the whole algebra and compared with another one proposed by us previously.

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