2004/05/25 by B. Abdesselam, Abdesselam, B., R. Chakrabarti +7
Engineering · Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Optical Network Technologies #math.QA
paper · pdf · doi:10.48550/arxiv.math/0405477
Talk given by M.B. Zahaf to the Seventh Constantine High Energy Physics School (Theoretical Physics and Cosmology), 3-7 April 2004 (Algeria)
arxiv created 2004/05/25 · arxiv updated 2009/12/01
Recently, a class of transformations of Rq-matrices was introduced such that the q → 1 limit gives explicit nonstandard Rh-matrices. The transformation matrix is singular as q → 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 Rq matrices of all dimensions of Uq(sl(N)), UQ(osp(1|2)) and Uq(sl(2|1)) algebras. Explicit constructions are presented for Uq(sl(2)), Uq(sl(3)), Uq(osp(1|2)) and Uq(sl(2|1)) algebras, while choosing Rq matrix for (fund. rep.) ⊗ (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 Uh(sl(2)), Uh(sl(3)), Uh(osp(1|2)) and Uh(sl(2|1)). These Hopf (super)algebras are equipped with a remarkably simpler coalgebraic structure. Generalizing our results on Uh(sl(3)), we give the higher dimensional Jordanian (super)algebras Uh(sl(N)) for all N. The universal Rh matrices are also given.