Categorification and Group Extensions

We review several known categorification procedures, and introduce a functorial categorification of group extensions (Section 4.1) with applications to non-Abelian group cohomology (Section 4.2). The obstruction to the existence of group extensions (Section 4.2.4, Equation (9)) is interpreted as a coboundary condition (Proposition 4.5).     

A diagrammatic categorification of the fermion algebra

In this paper, we study the diagrammatic categorification of the fermion algebra. We construct a graphical category corresponding to the one-dimensional （1D） fermion algebra, and we investigate the properties of this category. The categorical analogues of the Fock states are some kind of 1-morphisms in our category, and the dimension of the vector space of 2-morphisms is exactly the inner product of the corresponding Fock states. All the results in our categorical framework coincide exnetlv with those in normal quantum mechanics.     

Categorification and Quasi-Hopf Algebras

We categorify the notion of coalgebras by imposing a co-associative law up to some isomorphisms on the co-multiplication map and requiring that these isomorphisms satisfy certain law of their own, which is called the co-pentagon identity. We also set up a 2-category of 2-coalgebras. The purpose of this study is from the idea of reconsidering the quasi-Hopf algebras by the categorification process, so that we can study the theory of quasi-Hopf algebras and their representations in some new framework of higher category theory in natural ways.     

A diagrammatic categorification of q-boson and q-fermion algebras

In this paper,we study the diagrammatic categorification of q-boson algebra and also q-fermion algebra.We construct a graphical category corresponding to q-boson algebra.q-Fock states correspond to some kind of 1-morphisms,and the graded dimension of the graded vector space of 2-morphisms is exactly the inner product of the corresponding q-Fock states.We also find that this graphical category can be used to categorify q-fermion algebra.     

$D_4$型李代数包络代数旋模的范畴化

本文我们定义复数域$C$上一般线性李代数${\rm gl}_n$ BGG 范畴的若干子范畴及其上的投射函子,利用这些子范畴和投射函子范畴化了$D_4$型李代数包络代数旋模的$n$-次张量积.     

$U({\mathfrak {so}}(7,{\mathbb C}))$的向量表示的范畴化

The aim of this paper is to categorify the n-th tensor power of the vector representation of U (so(7, C)). The main tools are certain singular blocks and projective functors of the BGG category of the complex Lie algebra gl n .     

Cluster algebra structure on the finite dimensional representations of affine quantum group U_q(_3)

In this paper,we prove one case of conjecture given by Hernandez and Leclerc.We give a cluster algebra structure on the Grothendieck ring of a full subcategory of the finite dimensional representations of affine quantum group Uq(3).As a conclusion,for every exchange relation of cluster algebra,there exists an exact sequence of the full subcategory corresponding to it.