Jordanian Quantum (Super)Algebras $U_{h}(g)$ via Contraction Method and Mapping:Review

Abstract

Recently, a class of transformations of $R_q$-matrices was introduced such that the $q \to 1$ limit gives explicit nonstandard $R_h$-matrices. The transformation matrix is singular as $q \to 1$. For the transformed matrix, the singularities, however, cancel yielding a well-defined construction. We have shown that our method can be implemented systematically on $R_q$ matrices of all dimensions of $U_q(sl(N)), U_Q(osp(1|2))$ and $U_q(sl(2|1))$ algebras. Explicit constructions are presented for $U_q(sl(2)), U_q(sl(3)), U_q(osp(1|2))$ and $U_q(sl(2|1))$ algebras, while choosing $R_q$ matrix for (fund. rep.) \otimes (arbitrary irrep.). Our method yields nonstadard deformations along with a nonlinear map of the $h$-Borel subalgebra on the corresponding classical Borel subalgebra, which can be easily extended to the whole algebra. Following this approach we explicitly construct here the nonstandard Jordanian quantum (super)algebras $U_h(sl(2)), U_h(sl(3)), U_h(osp(1|2))$ and $U_h(sl(2|1))$. These Hopf (super)algebras are equipped with a remarkably simpler coalgebraic structure. Generalizing our results on $U_h(sl(3))$, we give the higher dimensional Jordanian (super)algebras $U_h(sl(N))$ for all $N$. The universal $R_h$ matrices are also given.