On bases of quantum affine algebras

Let U be the positive part of the quantum group U associated with a generalized Cartan matrix A. In the case of finite type, Lusztig constructed the canonical basisB ofU via two approaches ([9]). The first one is an elementary algebraic construction via RingelHall algebra realization of U. The isomorphism classes of representations of the corresponding Dynkin quiver form a PBW-type basis of U. By a lemma (Lemma 3.1) of Lusztig, one can construct a bar-invariant basis, which is the canonical basis B. A remarkable characteristic in his construction is that Lusztig reveals the triangular relations among three kind of bases: PBW-basis, monomial basis and canonical basis. The second one is a geometric construction. In [9] and [10], Lusztig gave a geometric realization of U via the category of some semisimple complexes on the variety Eν consisting of representations with dimension vector ν ∈ NI of the corresponding quiver Q. The set of the isomorphism classes of simple perverse sheaves gives the canonical basis B of U. The geometric construction of canonical basis was generalized to the cases of all types in [10] (see also [12]). Furthermore, Lusztig in [11] gave the construction of affine canonical bases by the perverse sheaves associated with tame quivers, an important feature is that those perverse sheaves are indexed by the classes of aperiodic modules of tame quivers and irreducible modules of symmetric groups.