代数学研讨会II

2022年9月23日

会议报告摘要

Embeddings among quantum affine sln

量子仿射sln之间的嵌入

李毅强

纽约州立大学布法罗分校

We establish an explicit embedding of a quantum affine sln into a quantum affine sln+1. This embedding serves as a common generalization of two natural, but seemingly unrelated, embeddings, one on the quantum affine Schur algebra level and the other on the non-quantum level. The embedding on the quantum affine Schur algebras is used extensively in the analysis of canonical bases of quantum affine sln and gln. The embedding on the non-quantum level is used crucially in a work of Riche and Williamson on the study of modular representation theory of general linear groups over a finite field. The same embedding is also used in a work of Maksimau on the categorical representations of affine general linear algebras. We further provide a more natural compatibility statement of the embedding on the idempotent version with that on the quantum affine Schur algebra level. A gln-variant of the embedding is also established.

（我们给出量子仿射sln到量子仿射sln+1的嵌入。该嵌入时两种嵌入的推广，一种为量子仿射Schur代数层面，一种为非量子层面。量子仿射Schur代数的嵌入最近被广泛应用到研究量子仿射sln和gln的典范基中。非量子层面的嵌入在Riche and Williamson研究有限域上的一般线性群的模表示起到了重要的作用。同样的嵌入被Maksimau应用到范畴话仿射一般线性代数的表示上。）

From (derived) Hall algebras to acyclic quantum cluster algebras

（从导出Hall代数到无圈量子丛代数）

陈学庆

威斯康星大学白水分校

Inspired by Caldero-Keller’s discovery of the similarity between the multiplication formulas in a cluster algebra and that in a (dual) Hall algebra, we firstly discuss an algebra homomorphism from the dual Hall algebra associated to Rep(Q) (category of representations of an acyclic quiver Q) to the corresponding quantum cluster algebra. Then we address the connection from two certain quotients of subalgebras of the derived Hall algebras of Rep(Q) to acyclic quantum cluster algebra. Finally, we give cluster multiplication formulas via the above derived Hall algebras. This talk is based on the joint works with Ming Ding and Fan Xu, and with Ming Ding and Haicheng Zhang respectively.

（受到Caldero-Keller工作的启发，我们首先给出Rep(Q)对应对偶Hall代数到Q对应量子丛代数的一个代数同态，接下来，我们给出Hall代数的两个子商代数到无圈量子丛代数的关系。最后，我们给出上述导出Hall代数的乘法公式。）