School of Mathematical Sciences Academic Report [2025] No. 020

(High-Level University Construction Series Report No. 1043)

Lecture Title: Two Quantum de Rham Super Complexes and Stratification Theorem

Speaker: Prof. Hu Naihong (East China Normal University)  

Date & Time: 4:00–5:00 PM, March 27, 2025  

Venue: Huiwen Building Room 1420, Shenzhen University  

Abstract: In order to study the “modular” representation theory of quantum gl(m|n) at root of unity, we introduce the quantum Manin supersapce and quantum (dual) Grassmann superalgebra with quantum divided power structure, and develop a kind of quantum differential calculus over them, and construct two kinds of quantum de Rham super complexes: one is of infinite length which is the quantized version of the classical analogue due to Manin-Deligne-Morgan in their early study of supermanifolds from gauge field theory, another is of finite length which has no classical analogue to our knowledge. For the latter, we prove the Poincare lemma for nontruncated complex, while for the truncated case, in order to calculate all the qauntum de Rham cohomologies we need to develop a specific technique to overcome the complicated difficulties encountered in the quantum supercase. I'll also talk about the “l-adic phenomenon” occurred in a kind of indecomposable modules in the root of unity case which originally were irreducible modules in the generic case. This talk is based on a series of our joint work with Dr. Ge Feng, and Prof. Marc Rosso.

Biography: Prof. Hu Naihong is a Professor and Ph.D. Supervisor at the School of Mathematical Sciences, East China Normal University, serving as Executive Director of the LIA Sino-French Laboratory for Basic Mathematics and an Alexander von Humboldt Fellow. Her research focuses on Lie theory, classification of quantum groups and Hopf algebras, and representation theory, with recent interests in finite tensor category theory and topological quantum computation. She has received awards including the Second Prize of the Ho Yingdong Young Teacher Award (Research Category), the Third Ministry of Education Outstanding Teaching and Research Award, and Shanghai Qianren Plan grants. She has led multiple General Projects of the National Natural Science Foundation of China (NSFC), Doctoral Program Fund projects, and participated in two NSFC Key Projects. Collaborating with Prof. Jing Naihuan (North Carolina State University), she received support from the NSFC Overseas Young Scholars Collaborative Research Fund (Class B Distinguished Young Scholars Fund). She has published 65 papers in international journals such as Crelle J.、Comm. Math. Phys.、Israel J. Math.、J. Algebra、JPAA、Pacific J. Math.

All faculty and students are welcome to attend!

Invited by: Xiao Wei

School of Mathematical Sciences

March 26, 2025