Computing with Quantized Enveloping Algebras: PBW-Type Bases, Highest-Weight Modules and R-Matrices

Quantized enveloping algebras have been widely studied, almost exclusively by theoretical means (see, for example, De Concini and Procesi, 1993; Jantzen, 1996; Lusztig, 1993). In this paper we consider the problem of computing with a quantized enveloping algebra. For this we need a basis of it, along with a method for computing the product of two basis elements. To this end we will use so-called PBW-type bases. The main subject of this paper will be an algorithm for computing the product of two elements of such a PBWtype basis. We use this to construct highest-weight modules over quantized enveloping algebras, and the corresponding R-matrices. Below we recall some notation and definitions. Here, as in the rest of this paper, we borrow heavily from Jantzen (1996). A citation as Jantzen (1996, 4.15(3)) refers to formula (3) in paragraph 4.15 of Jantzen (1996). Let g be a split semisimple Lie algebra over Q, with root system Φ. We let V be the vector space over R spanned by Φ. By W (Φ) we denote the Weyl group of Φ. On V we fix a W (Φ)-invariant inner product ( , ) such that (α, α) = 2 for all short roots α. This means that (α, α) = 2, 4, 6, where the last possibility only occurs if α comes from a component of type G2. We work over the field Q(q). For α ∈ Φ we set