Invariance of the Drinfeld pairing of a quantum group

Yetter-Drinfeld algebras over a pairing of multiplier Hopf algebras

In the present work, we study Yetter-Drinfeld algebras over a pairing of multiplier Hopf algebras. Our main motivation is the construction of a self-dual theory of (C*-)algebraic quantum transformation groupoids. Instead of the standard characterization of Yetter-Drinfeld algebras given in the case of Hopf algebras, we develop an equivalent "only coaction" characterization in the framework of multiplier Hopf algebras. Finally, as a special case, we focus on Yetter-Drinfeld structures over Van Daele's algebraic quantum groups.

包络代数是李代数自同构群上正则函数Hopf代数的Yetter--Drinfeld模代数

Enveloping algebra is a Yetter--Drinfeld module algebra over Hopf algebra of regular functions on the automorphism group of a Lie algebra

本文构造了泛包络代数与自同构群正则函数Hopf代数间的退化Hopf配对，赋予包络代数Yetter--Drinfeld模代数结构，进而构造Hopf代数胚，并推广至Leibniz代数情形。

Generating Functions with $τ$-Invariance and Vertex Representations of Quantum Affine Algebras $U_{r,s}(\widehat{\mathfrak{g}})$ (I): Simply-laced Cases

We put forward the exact version of two-parameter generating functions with $τ$-invariance, which allows us to give a unified and inherent definition for the Drinfeld realization of two-parameter quantum affine algebras for all the untwisted types. As verification, we first construct their level-one vertex representations of $U_{r,s}(\widehat{\mathfrak{g}})$ for simply-laced types, which in turn well-detect the effectiveness of our definitions both for $(r,s)$-generating functions and $(r,s)$-Drinfeld realization in the framework of establishing the two-parameter vertex representation theory.

Note on Invariance and Finiteness for the Exponent of Hopf Algebras

There are two notions of exponent of finite-dimensional Hopf algebras introduced and studied in the literature. In this note, we discuss and compare their properties including invariance and finiteness in this note. Specifically, one notion is invariant under twisting and taking the Drinfeld double, just like the other one. We also find that if the non-cosemisimplicity and dual Chevalley property hold, both exponents are infinite in characteristic $0$ but finite in positive characteristic.

具有量子对称性的代数与范畴的内射性

Injectivity for algebras and categories with quantum symmetry

本文证明了含幺Yetter-Drinfeld C*-代数及相关双模范畴的内射包存在性，并由此得出紧量子群Drinfeld double边界作用的幺单不变性。

来自Hopf幺半群与缎带图的映射类群作用

Mapping class group actions from Hopf monoids and ribbon graphs

本研究证明对称幺半范畴中的主Hopf幺半群可导出有向曲面映射类群作用，结合缎带图边滑动给出具体刻画，并在范畴有限完备余完备时得到闭曲面情形的对应作用。