Constructing Canonical Bases of Quantized Enveloping Algebras

Since the invention of canonical bases of quantized enveloping algebras, one of the main problems has been to establish what they look like. Explicit formulas are only known in a few cases corresponding to root systems of low rank, namely A1 (trivial), A2 ([Lusztig 90]), A3 ([Xi 99a]), and B2 ([Xi 99b]). Furthermore, there is evidence suggesting that for higher ranks the formulas become so complicated that an explicit description is virtually impossible (see [Carter 97]). Therefore, it is natural to start less ambitiously, and try to find part of the canonical basis, say the part consisting of all elements of a given weight. We will describe an algorithm for computing the elements of a given weight of the canonical basis of the negative part of the quantized enveloping algebra of a finite-dimensional semisimple Lie algebra. We will also give an algorithm for computing the canonical basis of a finite-dimensional module. To the best of my knowledge, two algorithms for computing canonical bases of modules are known. In [Leclerc and Toffin 00] an algorithm is described for computing the canonical basis of a Uq(sln)-module, and [Marsh 96] contains an algorithm for computing the canonical basis of a fundamental module, when the root system is of type A − D. Our approach differs from the ones taken in [Leclerc and Toffin 00], [Marsh 96] in that we work with PBW-type bases. This leads to algorithms that are more generally applicable: They work for any finitedimensional module of the quantized enveloping algebra