The universal \(R\)-matrix for quantum untwisted affine Lie algebras
Let \({\mathfrak g}\) be a simple finite dimensional Lie algebra over the complex field. It is well-known that the standard quantum deformation of the universal enveloping algebra \(U_ q({\mathfrak g})\) is a quasitriangular Hopf algebra, i.e. there exists the universal \(R\)-matrix satisfying certain conditions, namely \(\Delta'(x)R=R\Delta(x)\), \(x\in U_ q({\mathfrak g})\), \((\Delta\otimes id)R=R^{13}R^{23}\), \((id\otimes\Delta)R=R^{13}R^{12}\), where \(\Delta\) is the comultiplication in \(V_ q({\mathfrak g})\), \(\Delta'\) is the opposite comultiplication. The explicit formula for \(R\) in the finite dimensional case was known (see references in the paper). The authors deal with an infinite dimensional case. They consider affine untwisted Lie algebras with the symmetrizable Cartan matrix. Let \(\hat{\mathfrak g}\) be such an algebra. Using the so-called normal ordering on the set of positive roots of \(\hat {\mathfrak g}\), one can introduce Cartan-Weil generators in \(U_ q(\hat g)\) and define commutation relations between them. This enables the authors to compute the universal \(R\)-matrix for \(U_ q(\hat{\mathfrak g})\) explicitly.
scientific article; zbMATH DE number 95237