Some aspects of a theory of locally convex Hopf algebras

分析学研讨班

题目：Some aspects of a theory of locally convex Hopf algebras

报告人：Wang Hua(王华） 

时间：2024年12月11日（星期三），14:30-16:00

地点：明德楼B区201-1

Zoom会议，会议号：947 0981 8605，密码：477439

摘要： 

We will give an overview of a recent theory of locally convex Hopf algebras, with an emphasis on the duality aspect. After motivating and introducing the theoretical framework, we will describe how Hopf algebras, compact and discrete quantum groups, locally compact groups, Lie groups, as well as their respective duals, can all give rise to examples in various ways. Then the emphasis shifts to the non-locally compact aspects of theory. The main result is that under the very weak assumption that the topology of a topological group $G$ is compactly generated, one can fully describe $G$ using only the locally convex Hopf algebra of continuous functions on it. This goes much beyond the previously well-known locally compact case, and includes interesting examples such as all metrizable groups, all topological groups that admit a cellular decomposition, etc.  On the non-commutative side, we will describe some new interesting topological quantum groups that arise naturally but seem to go beyond the current Kustermans-Vaes framework. Time permitting, exploiting some deep structural resuts related to Hilbert's fifth problem, we shall mention how one can describe a second countable locally compact group using a variant of Bruhat's notion of regular (smooth) functions on it.