Lie algebras arising from two-periodic projective complex and derived categories
数学专题报告
报告题目(Title)：Lie algebras arising from two-periodic projective complex and derived categories
报告人(Speaker)：方杰鹏（香港大学）
地点(Place)：后主楼1220
时间(Time)：2026年1月14日（周三）16:15-17:15
邀请人(Inviter)：肖杰、覃帆、周宇
报告摘要
Let $A$ be a finite-dimensional $\mathbb{C}$-algebra of finite global dimension and $\mathcal{A}$ be the category of finitely generated right $A$-modules. By using of the category of two-periodic projective complexes $\mathcal{C}_2(\mathcal{P})$, we construct the motivic Bridgeland's Hall algebra for $\mathcal{A}$, where the structure constants are given by Poincar\'{e} polynomials in $t$, and then construct a $\mathbb{C}$-Lie algebra $\mathfrak{g}$. For the stable category $\mathcal{K}_2(\mathcal{P})$ of $\mathcal{C}_2(\mathcal{P})$, we construct a $\mathbb{C}$-Lie algebra $\tilde{\mathfrak{g}}$. We prove that the natural functor $\mathcal{C}_2(\mathcal{P})\rightarrow \mathcal{K}_2(\mathcal{P})$ induces a Lie algebra isomorphism $\mathfrak{g}\cong\tilde{\mathfrak{g}}$. This makes clear that the structure constants at $t=-1$ provided by Bridgeland in terms of exact structure of $\mathcal{C}_2(\mathcal{P})$ precisely equal to that given by Peng-Xiao in terms of triangulated category structure of $\mathcal{K}_2(\mathcal{P})$. This is a joint work with Yixin Lan and Jie Xiao.
