Real simple modules over simply-laced quantum affine algebras and categorifications of cluster algebras
报告题目：Real simple modules over simply-laced quantum affine algebras and categorifications of cluster algebras
报告专家：段冰 （兰州大学）
报告时间：2023年6月9日15:30-16:30
报告地点：数学学院东409
报告摘要：
Let $\mathscr{C}$ be the category of finite-dimensional modules over a simply-laced quantum affine algebra $U_q(\widehat{\mathfrak{g}})$. For any height function $\xi$ and $\ell\in \mathbb{Z}_{\geq 1}$, we introduce certain subcategories $\mathscr{C}^{\leq \xi}_\ell$ of $\mathscr{C}$, and prove that the quantum Grothendieck ring $K_t(\mathscr{C}^{\leq \xi}_\ell)$ of $\mathscr{C}^{\leq \xi}_\ell$ admits a quantum cluster algebra structure. Using $F$-polynomials and monoidal categorifications of cluster algebras, we classify all real simple modules in $\mathscr{C}^{\leq \xi}_1$ in terms of their highest $l$-weight monomials, among them the families of type $D$ and type $E$ are new. For any $\ell$, inspired by Hernandez and Leclerc's work, we propose two conjectures for the study of real simple modules, and prove them for the subcategories $\mathscr{C}^{\leq \xi}_\ell$ whose Grothendieck rings are cluster algebras of finite type. This is a joint work with Professor Ralf Schiffler.
专家简介：
段冰，兰州大学数学与统计学院副教授。主要从事丛代数和量子仿射代数、表示论等相关方面的研究，目前在IMRN、J. Lond. Math. Soc.、Math. Z.、J. Algebra等国际重要期刊上发表论文数篇。
邀请人：付昌建
