The q-bracket product and quantum enveloping algebras of classical types
Let \(\hat U\) be the quantum enveloping algebra over a commutative ring R corresponding to a symmetrizable generalized Cartan matrix. \(\hat U\) is generated by \(e_ i,f_ i,k_ i,k_ i^{-1}\) for \(1\leq i\leq n\), with relations deforming the (unquantized) Serre relations. One of these involves only the \(e_ i\), and another only the \(f_ i\). Let \(\hat U^+\) and \(\hat U^-\) be the algebras generated by the \(e_ i\) and the \(f_ i\) respectively, subject to these relations respectively. The author considers whether these algebras are polynomial algebras, i.e., there is a totally ordered (finite) set of generators such that the ordered monomials in them are a basis for the algebra. The author proves this for the classical types \(A_ n,B_ n,C_ n\) and \(D_ n\), provided that, in each case, certain polynomials in the quantum parameter q in R be invertible in R. These generators are certain iterated q- bracket products in the \(e_ i\) and \(f_ i\) respectively, and a specific total ordering must be used on them. Lusztig had obtained this when R is a field of characteristic 0 with q transcendental over \({\mathbb{Q}}\), and the type \(A_ n\) case had been studied by Rosso and by Yamane.
scientific article; zbMATH DE number 4198260