Title：Divided power Hopf algebras（I-III）

Abstract：

In this series of lectures I will outline the following topics:

(1) As a motivation, I will outline Tate's construction of a free resolution of $R/I$ as an example of strict commutative dg algebra with a divided power structure, computing homology and cohomology of Noetherian rings.

(2) Introduction to general divided power structures and their differential graded extensions. This is useful in crystalline cohomology theory.

(3) Extending the construction of the algebra of symmetric tensors of free modules over commutative modules (also called shuffle algebras) to different graded cases by using trace maps.

(4) Using the functoriality of symmetric tensor constructions, one gets a natural Hopf algebra structure on the algebra of symmetric tensors.

(5) Extending Roby's construction of divided power structures to dg setting.

(6)  Milnor-Moore  theorem on structures of Hopf algebras and Andre theorem on Hopf algebras with divided powers. This algebra will play the role of free $\Gamma$-dg algebras.

(7) Homotopy Lie algebras and its applications to cohomological support varieties of sheaves as associated varieties for representations of Lie algebras.

Speaker：Zongzhu Lin, Kansas State University, Professor. Lin works on representation theory，his results are published in Invent. Math., Adv. Math, Comm. Math. Phys, etc.

Date：2025-6-25（9:00am-11:00am）、6-28（3:00pm-5:00pm）、6-30（9:00am-11:00am）

Venue: 617

Organizer：School of Mathematical Science

Inviter：Libin Li, Zhiqiang Yu

Students and teachers are welcome.