Projective module description of the q-monopole

Projective modules of $0$-Schur algebras

We study the structure of the $0$-Schur algebra $S_0(n, r)$ following the geometric construction of $S_0(n, r)$ by Jensen and Su \cite{JS}. The main results are the construction and classification of indecomposable projective modules. In addition, we construct bases of these modules and their homomorphism spaces. We also give a filtration of projective modules, which leads to a decomposition of $S_0(n,r)$ into indecomposable left modules.

Regular representation on the big cell and big projective modules in the category O

We study the algebra of regular functions on the big cell of the Gauss decomposition of a simple complex Lie group G. We prove that it is spanned by the matrix elements of big projective modules in the BGG category O, and admits a decomposition of Peter-Weyl type. The subspace of Whittaker vectors in the regular representation on the big cell gives a realization of the big projective modules in O, similar to the Borel-Weil theorem. These results can be extended to the quantum groups.

Finitely-Generated Projective Modules over the Theta-deformed 4-sphere

We investigate the "theta-deformed spheres" C(S^{3}_{theta}) and C(S^{4}_{theta}), where theta is any real number. We show that all finitely-generated projective modules over C(S^{3}_{theta}) are free, and that C(S^{4}_{theta}) has the cancellation property. We classify and construct all finitely-generated projective modules over C(S^{4}_θ) up to isomorphism. An interesting feature is that if theta is irrational then there are nontrivial "rank-1" modules over C(S^{4}_θ). In that case, every finitely-generated projective module over C(S^{4}_θ) is a sum of a rank-1 module and a free module. If t…

射影直线的手形同调与顶点代数模的扩张

Chiral homology of the projective line and extensions of vertex algebra modules

该研究建立了顶点代数两模的扩张向量空间与对应手性代数系数下一维射影直线的一阶手形同调间的同构关系，推进了手形同调与顶点代数模扩张的关联研究。

Braid group action on projective quantum $\mathfrak{sl}(2)$ modules

We define a family of the braid group representations via the action of the $R$-matrix (of the quasitriangular extension) of the restricted quantum $\mathfrak{sl}(2)$ on a tensor power of a simple projective module. This family is an extension of the Lawrence representation specialized at roots of unity. Although the center of the braid group has finite order on the specialized Laurence representations, this action is faithful for our extension.

A large family of indecomposable projective modules for the Khovanov-Kuperberg algebra of $sl_3$-webs

We recall a construction of Mackaay, Pan and Tubbenhauer of the algebras $K^ε$ which allow to understand the $sl_3$ homology for links in a local way (i.e. for tangles). Then, by studying the combinatorics of the Kuperberg bracket, we give a large family of non-elliptic webs whose associated projective $K^ε$-modules are indecomposable.