Categorifical And Geometric Construction Of Infinite Dimensional Lie Algebra And Quantized Enveloping Algebra

This thesis aims to study the categorifical and geometric construction of infinite dimensional Lie algebra and quantized enveloping algebra.There are three major parts in the thesis:(1)the constructions of Lie algebras for the category of two periodic projective complexes and its stable category(triangulated category);(2)the categorification of Green’s formula;(3)the categorifical construction of Bridgeland’s Hall algebra and the generalization of Lusztig’s perverse sheaf and dual canonical basis theory to the category of two periodic projective complexes.For the finite dimensional module category of a finite dimensional algebra over C which is of finite global dimension,on the one hand,on the category of two periodic projective complexes C~2(P),by using Bridgeland’s construction of the localization algebra of Hall algebra over finite field and Joyce’s definition of motivic Hall algebra,we construct the motivic form of Bridgeland’s Hall algebra,and prove that there is a C-subspace g in the classical limit,which is closed under the Lie bracket.Hence it is a Lie algebra.On the other hand,on the stable category of two periodic projective complexes K~2(P),by using of Peng-Xiao’s construction of Lie algebra for two periodic triangulated category over finite field and Xiao-Xu-Zhang’s generalization to C,there is a Lie algebra g.We prove that the natural functor C~2(P)→K~2(P)induces an isomorphism of Lie algebras g(?)g.Based on the following results:Ringel-Hall algebra for the category of quiver representations over finite field,Green’s formula and Lusztig’s categorification theory for quantum group,we prove a formula about Lusztig’s induction functor and restriction functor,which generalizes a formula of Lusztig to more general case and establish the categorification of Green’s formula.With the aim of the categorifical construction of Bridgeland’s Hall algebra,we generalize Lusztig’s theory for the category of quiver representations to the category of two periodic projective complexes,and define the algebraic variety of two periodic projective complexes,projective flag variety,induction functor and restriction functor.We study the properties of induction functor and restriction functor.In particular,for the category of two periodic projective complexes of the category of quiver representations,we prove a formula about restriction functors and construct an associative algebra which admits a basis having positivity.