Algorithms to obtain the canonical basis in some fundamental modules of quantum groups
Let \(U\) be the quantized universal enveloping algebra of a finite-dimensional complex simple Lie algebra. \textit{M. Kashiwara} [Duke Math. J. 63, 465-516 (1991; Zbl 0739.17005)] proved the existence and uniqueness of a crystal basis for every finite-dimensional simple \(U\)-module using certain operators and defines its crystal graph which encodes how these operators act on the crystal basis. \textit{M. Kashiwara} and \textit{T. Nakashima} [J. Algebra 165, 295-345 (1994; Zbl 0808.17005)] gave explicit descriptions of these crystal graphs in the case that \(U\) is of type \(A, B, C\), or \(D\). On the other hand, \textit{G. Lusztig} [J. Am. Math. Soc. 3, 447-498 (1990; Zbl 0703.17008)] showed the existence of a canonical basis for every finite-dimensional simple \(U\)-module. The author of the paper under review observes that if for each element of the crystal basis a certain sequence of Kashiwara's operators can be found, then the canonical basis can be written in a very nice way in terms of these sequences. Then he presents algorithms which generate sequences with the required properties for most of the fundamental modules in types \(A, B, C\), or \(D\). Moreover, he shows that with his choice of sequences there is a close link between the crystal and canonical bases of the fundamental modules in type \(C\) with the corresponding bases in type \(A\) and a similar connection also occurs for type \(D\) and type \(B\). By a different method the author gives in [Commun. Algebra 25, 1365-1386 (1997); see the review Zbl 0874.17013 below] an explicit description of the canonical bases for the fundamental \(U\)-modules when \(U\) is of type \(A\).
scientific article; zbMATH DE number 1013956