On the coloured graph structure of Lusztig's canonical basis
Let \({\mathfrak g}\) be a semisimple finite dimensional Lie algebra over the complex numbers, \(\Delta\) its Dynkin diagram, \(U\) the associated quantized enveloping algebra and \(U^+\) its positive part. Lusztig has shown the existence of what is called a canonical basis \({\mathfrak B}\) of \(U^+\). From Kashiwara's theory of crystal bases one gets a coloured graph structure on \({\mathcal B}\), which contains all the crystal graphs of finite dimensional simple representations. The paper under review gives a handy description of the coloured graph structure of \({\mathfrak B}\) under the additional assumption that \({\mathfrak g}\) does not contain a component of type \(E_8\). The method is based on \textit{C. M. Ringel}'s [Invent. Math. 101, 583-591 (1990; Zbl 0735.16009)] description of \(U^+\) as a Hall algebra of a quiver. There is a bijection between elements of the canonical basis \({\mathfrak B}\) and isomorphism classes of representations of certain quivers which are oriented graphs with underlying diagram \(\Delta\). Ringel defined a multiplicative structure on these representations. Thus the action of Kashiwara's operators can be given an interpretation within the category of representations of a quiver. Indecomposable representations and homomorphisms or extensions between them are conveniently encoded in the (finite) Auslander-Reiten quiver \(\Gamma\). Since the orientation of \(\Delta\) can be chosen arbitrarily, one has a choice between several Auslander-Reiten quivers to work with. For the cases \(A_n\), \(D_n\), \(E_6\) and \(E_7\), the author finds an orientation of \(\Delta\) such that the space of homomorphisms from any indecomposable module to a simple one has dimension zero or one. (The cases \(B_n\), \(C_n\), \(F_4\) and \(G_2\) are covered by a slight modification of the method.) This implies that the crucial set of representations to be considered in this context becomes a partially ordered set. The computation of the coloured graph structure of the canonical basis is reduced to a consideration of extensions by simple modules which in turn is reduced to the consideration of the above mentioned partially ordered sets inside the Auslander-Reiten quiver. In fact, the extensions by simple modules which have to be considered can be described by antichains in these partially ordered sets. These antichains are studied in detail. Altogether, this implies a quite efficient recipe (theorem 7.1) for computing the coloured graph structure. In the case of type \(A_n\), the result both on the canonical basis and on the crystal graphs of any simple finite dimensional representation can be stated easily within nine lines (see the introduction of the paper under review).
scientific article; zbMATH DE number 1031515