The Parameterization Of Canonical Bases And Categorification Of Hall Algebras

The main object in this thesis is to study the relation between representation theoryof algebras and the modified forms(U|Ë™) of quantum groups and their canonical bases, andin particular, to study:(1), the relation between root categories and canonical bases of(U|Ë™);(2), the relation between BGP-reflection functors and Lusztig’s symmetries on(U|Ë™);(3), thegeometric realizations of Lusztig’s symmetries.For a symmetrizable Kac-Moody Lie algebra, Lusztig introduced the modified form(U|Ë™) of the corresponding quantum group and its canonical basis in [1]. In this thesis, for asymmetric Lie algebra of finite or afne type, we define a set which depends only on theroot category and prove that there is a bijection between the set and the canonical basis of(U|Ë™). First, we consider the case of finite type. Now, the set of all isomorphism classes in theroot category depends only on the root category, and for any element in this set, we candefine a PBW type element in(U|Ë™). Then we can get a PBW type basis of(U|Ë™). By this basis,we can get a(ˉ)-invariant basis. There is a bijection between the(ˉ)-invariant basis and thecanonical basis. For a symmetric Lie algebra of afne type, basing on the construction ofthe PBW type basis of U~+by Lin, Xiao and Zhang in [2], we also construct a set, whichdepends only on the root category, and prove that there is a bijection between the set andthe canonical basis of(U|Ë™).Lusztig introduced some symmetries on U and(U|Ë™). In view of the realization of U bythe double Ringel-Hall algebra, one can apply the BGP-reflection functors to the doubleRingel-Hall algebra to obtain Lusztig’s symmetries on U and their important properties,for instance, the braid relations. In this thesis, we define a modified formË™of the Ringel-Hall algebra and Lusztig’s symmetries onË™. One can apply the BGP-reflection functorstoË™and get some operators on it. The restriction of these operators on(U|Ë™) realize theLusztig’s symmetries on(U|Ë™). Also these operators can realize the Lusztig’s symmetries on(?).Note that the image of U~+under Lusztig’s symmetry T_iis not contained in U~+.Hence Lusztig[3]studied two subalgebrasiif andf in U~+, whereif is consisted of theelements in U~+, the images of which under T_iare contained in U~+, and~if is the image of_if under T_i. Now the canonical basis induces a basis_iB of_if and a basis~iB of~if. Lusztigalso showed that the operator T_i:_ifâ†'~if induced by T_imaps the basis_iB to the basis ~iB. This is the main theorem in [3] on the relation between canonical basis and Lusztig’ssymmetries. In this thesis, we give the geometric realizations ofif and~if, and simpleperverse sheaves realize the bases_iB and~iB. Then we give the geometric realization ofT_i:_ifâ†'~if and reprove the theorem on the relation between canonical basis and Lusztig’ssymmetries using the geometric realization.