Triangulated Categories And Geometric Realization Of Infinite Dimensional Lie Algebras

The construction of Ringel-Hall algebras via abelian category was firstly introduced by Ringel and was applied to representation category of Dynkin quiver in order to realize positive parts of semisimple Lie algebras. Peng Liangang and Xiao Jie used the method of Hall algebras to realize the whole Kac-Moody algebras of any type from 2-periodic orbit category of the derived category. Their method also applies to constructing infinite dimensional Lie algebras of other type via derived category. On the other hand, Lusztig built the geometric model for Hall algebras associated to representation categories of quivers, in which the structure constants are given by the Euler characteristics of the corresponding varieties. Ringel realized positive parts of semisimple Lie algebras in the framework of Ringel-Hall algebras. The main result of this thesis is to build a geometric and topological model over triangulated categories such as derived categories and stable module categories of repetitive algebras. We defines a Lie bracket by Euler characteristics of constructible subsets and thus realizes infinite dimensional Lie algebras of various types with non-degenerated bilinear form.Firstly, we establish a topological space associated to the derived category of a finite dimensional algebra with finite global dimension, which is a generalization of module variety and complex variety. There are two actions on this topological space, one is the group G of isomorphisms between complexes, the other is the translation action T induced by contractible complexes. These two actions determine isomorphisms in derived category and used to define the moduli space over derived category. We focus on a particular class of subsets in the above topological space which is named support-bounded constructible subset as an analogue of constructible subsets in module variety and is invariant under derived equivalence. We also define Euler characteristic for moduli space by the concept of geometric quotient. Furthermore, we define a convolution multiplication between characteristic functions of constructible subsets by using push-forward functor from the category of algebraic varieties over C to the category of spaces of constructible functions. We construct geometric model for "intrinsic symmetry" of the octahedral axiom in a triangulated category. Using it, we deduce the multiplication satisfies the Jacobi identity of Lie algebra and then realize infinite dimensional Lie algebras.Finally, we show two applications of our result. One is to realize a Kac-moody Lie algebra of affine type globally. The other is to realize a Lie algebra globally from the stable module category of a repetitive algebra. The latter is essentially derived from the geometric realization of Happel's triangulated equivalence between stable module category of repetitive algebra and bounded derived category of finite dimensional algebra. In terms of this realization, we deduce that the Lie algebra realized by derived category of a finite dimensional algebra is isomorphic to the Lie algebra realized by stable module category of the corresponding repetitive algebra.