On the universal \(R\)-matrix for \(U_ q(sl_{r+1})\) (Q1180736)

In the original quantum universal enveloping algebra construction, the \(R\)-matrix is obtained using a suitable linear basis in the ``upper triangular'' subalgebra of raising operators and of the dual subalgebra of lowering operators, and in the case of \(U_ q(s\ell_ n)\), one has the Birkhoff-Witt property. On the other hand, several people have given a multiplicative formula for \(R\) using the \(q\)-Weyl group and the Lusztig automorphisms. The authors give formulas relating the two approaches. As an application, a Borel sub-quantum group of \(SL_ q(r+1)\) is obtained, and related to the finite-dimensional representations of \(U_ q(s\ell_{r+1})\) for generic \(q\).