Hall Algebras Of Tame Type And Affine Canonical Bases

This thesis aims to construct the affine canonical basis for U+,where U+is the positive part of the quantum group U of affine type.There are three maj or parts in the thesis:(1)The construction of the affine canonical basis for a symmetric Cartan matrix of affine type;(2)The construction of the affine canonical basis for any symmetrizable Cartan matrix of affine type;(3)The parametrization of the canonical basis of the quantized enveloping algebra U.For a symmetric Cartan matrix C of affine type,there is a tame quiver Q,such that U+ corresponding to C is isomorphic to the composition algebra C*of the Ringel-Hall algebra H*of the quiver Q.By using the relations of Hall multiplication among preprojective,preinjective,non-homogeneous regular and homogeneous regular modules,we construct an algebra H0 called the extended composition algebra,such that C*(?)H0(?)H*,and give a PBW basis of H0.This PBW basis has nice properties such as generality,almost orthogonality and integral property.Furthermore,by using the method of DengDu-Xiao[33],we construct a monomial basis,a PBW basis and a bar-invariant basis for C*.Finally,we prove that the bar-invariant basis is exactly the canonical basis geometrically constructed by Lusztig[4]and the bijection is explicitly given.For general case where the Cartan matrix is symmetrizable,there is a valued tame quiver Γ,such that U+corresponding to C is isomorphic to the composition algebra C*of the Ringel-Hall algebra H*of the valued quiver Γ.Similarly,we construct the extended composition algebra H0 such that C*(?)H0(?)H*,and give a PBW basis of H0.Then we construct a monomial basis,a PBW basis and a bar-invariant basis for C*.Since the elements in the canonical basis geometrically constructed by Lusztig are characterized by perverse sheaves with automorphisms,we prove a relation between the monomial basis and the signed canonical basis in terms of quivers with automorphisms.Finally,we prove that the bar-invariant basis is exactly the signed canonical basis and the bijecti on is explicitly given.For the modified quantized enveloping algebra U,we define a similar form D0 by Drinfel’d double.The algebra D0 has a PBW basis,based on which we can construct a monomial basis and a bar-invariant basis.Since the root category is independent of the orientation of the quiver,we construct an index set depending only on the root category and give a parametrization of the canonical basis of U.