The Research Of Categorifications Of Representations,BiHom–algebras And N–th Differential Modules

This dissertation mainly studies some problems related to categori?cation of representations of the universal enveloping algebra, Bi Hom–algebras and Gorenstein theory of n-th differential modules. The spin representations and vector representation of the universal enveloping algebra U(so(8, C)) of Lie algebra with type D4 are categori?ed; The de?nition of σ–presymmetric matrix is introduced and the properties of σ–presymmetric matrices over polynomial ring K[x1, · · ·, xn]are described. Then an associative algebra T with indentity is de?ned by a simple laces unoriented diagram Γ, the relation between T and a class of special KLR algebras is discussed. At last, the de?nitions of Bi Hom–type algebras are given and some problems of Bi Hom–algebras are described. The de?nition of n-th differential modules is introduced, Gorenstein theory for n-th differential modules is described. The details are as follows:(1) We study the categori?cations of universal enveloping algebra U(so(8, C)).Firstly, we categorify the n-th tensor powers of the vector representation V of U(so(8, C)) by using certain subcategories of BGG category O. Then we de?ne some projective functors which categorify the actions of the generators ei, fi, hion theZ(Vn). Finally, we categorify the de?ning relations of the generators ei, fi, hi via projective functors de?ned above. Similarly, we can give the categori?cations of n-th tensor powers Vn±spof spin representations V±spof U(so(8, C)).(2) We give the de?nition of σ–presymmetric matrix, prove that all the σ–presymmetric matrices over polynomial ring K[x1, · · ·, xn] form a Jordan algebra.Then we introduce a σ–presymmetric matrix T by a simple laces unoriented diagram Γ, de?ne a Z–graded associative algebra T by T and classify all the indecomposable projective representations of T in a combinational way. At last, we show that T is isomorphic to some KLR algebra R.(3) We give the de?nition of Bi Hom–associative algebras which are the deformation of Hom–associative algebras, and give a method of constructing Bi Hom–associative algebras from usual associative algebras. We also introduce the definition of Bi Hom–Lie algebras which are the deformation of Hom–Lie algebras.Similar to constructing Lie algebras from associative algebras, Hom–Lie algebras from Hom–associative algebras, we can get a Bi Hom–Lie algebra from a Bi Hom–associative algebra via the commutator bracket. Hence we obtain a functor G from the category of Bi Hom–associative algebras to the category of Bi Hom–Lie algebras. Moreover, we construct the Bi Hom–associative enveloping algebra U(L)of a Bi Hom–Lie algebra L by the theories of planar binary trees. In addition, we prove that the enveloping functor U is the left adjoint of the functor G.(4) The de?nition of n-th differential modules is introduced. It is shown that an n-th differential module(M, δM, n) is Gorenstein projective(resp. injective)if and only if M is Gorenstein projective(resp. injective). It is established that the relations between Gorenstein homological dimensions of an n-th differential module and the ones of its underlying module.