Geometric Realizations Of Algebras Associated To Categories

The main object of this thesis is to study geometric realizations of Lie algebras and associative algebras associated to categories.Let CF((?)bi_A)denote the vector space over Q of constructible functions on a given stack (?)bi_A for an abelian category A.In[1],Joyce proved that CF((?)bi_A)is an associative Q-algebra via the convolution multiplication and the subspace CFind((?)bi_A)consisting of constructible functions supported on indecomposables is a Lie subalgebra of CF((?)bi_A).Let CFfin((?)bi_A)be the subspace consisting of constructible functions with finite sup-port and CFfin ind((?)bi_A)the intersection of CFfin((?)bi_A)and CFind((?)bi_A).Suppose that the isomorphism classes of extensions of Y by X are finite for all X,Y ? Obj((?)bi_A),then Joyce proved that CFfin((?)bi_A)is isomorphic to the enveloping algebra of the Lie algebra CFfin ind((?)bi_A).Furthermore,Joyce generalized CF((?)bi_A)to SF((?)bi_A)and proved that SF((?)bi_A)is an associative algebra,where the stack function is a universal generaliza-tion of the constructible function and SF((?)bi_A)is defined to be the vector space over Q generated by stack functions.Kontsevich and Soibelman[2],Bridgeland[3].considered the algebras that are generated by stack functions on other categories,which are called'motivic Hall algebras'.The main results of this thesis are summarized as follows:(1)for a Krull-Schmidt exact K-category A,the vector spaces CF((?)bi_A)and CFing((?)bi_A)are also an associa-tive algebra and a Lie subalgebra of CF((?)bi_A)respectively.(2)There exists a subalgebra CFKS((?)bi_A)of CF((?)bi_A)isomorphic to the universal enveloping algebra of CFind((?)bi_A).(3)There is a comultiplication on CFKs((?)bi_A)which is compatible with the multiplica-tion,namely we obtain a degenerate form of Green's formula.These refines Joyce's results,as well as results of[4].(4)For the motivic Hall algebra MH(A)of a hereditary abelian category A we construct a comultiplication and a counit on MH(A)and prove Green's formula on MH(A).Namely,the multiplication and the comultiplication on MH(A)are compatible.Hence MH(A)can be endowed with a bialgebraic structure.