[论文解读] Connected (graded) Hopf algebras

Ken A. Brown, Paul Gilmartin+1 more|arXiv (Cornell University)|Jan 25, 2016

Algebraic structures and combinatorial models参考文献 24被引用 4

一句话总结

本文研究代数闭域上特征为零的连通（分次）霍普夫代数，证明有限Gel'fand-Kirillov维数的连通分次霍普夫代数是诺特域、科恩-麦克aul伊、阿廷-谢尔特正则且为全局维数等于其GK-维数的卡拉比-丘代数。一个关键结果构造了一个5维霍普夫代数，其不与任何李代数的通用包络代数同构。

ABSTRACT

We study algebraic and homological properties of two classes of infinite dimensional Hopf algebras over an algebraically closed field k of characteristic zero. The first class consists of those Hopf k-algebras that are connected graded as algebras, and the second class are those Hopf k-algebras that are connected as coalgebras. For many but not all of the results presented here, the Hopf algebras are assumed to have finite Gel'fand-Kirillov dimension. It is shown that if the Hopf algebra H is a connected graded algebra of finite Gel'fand-Kirillov dimension n, then H is a noetherian domain which is Cohen-Macaulay, Artin-Schelter regular and Auslander regular of global dimension n. It has S^2 = Id_H, and is Calabi-Yau. Detailed information is also provided about the Hilbert series of H. Our results leave open the possibility that the first class of algebras is (properly) contained in the second. For this second class, the Hopf k-algebras of finite Gel'fand-Kirillov dimension n with connected coalgebra, the underlying coalgebra is shown to be Artin-Schelter regular of global dimension n. Both these classes of Hopf algebra share many features in common with enveloping algebras of finite dimensional Lie algebras. For example, an algebra in either of these classes satisfies a polynomial identity only if it is a commutative polynomial algebra. Nevertheless, we construct, as one of our main results, an example of a Hopf k-algebra H of Gel'fand-Kirillov dimension 5, which is connected graded as an algebra and connected as a coalgebra, but is not isomorphic as an algebra to U(g) for any Lie algebra g.

研究动机与目标

• 分析作为分次代数或作为余代数连通的霍普夫代数的代数与同调性质。
• 确定此类霍普夫代数中有限Gel'fand-Kirillov维数的结构性后果。
• 研究连通分次霍普夫代数是否必然同构于某个李代数的通用包络代数。
• 比较这些霍普夫代数与通用包络代数及交换多项式代数的同调行为。
• 构造一个有限GK-维数的霍普夫代数显式例子，其不与任何U(𝔤)同构。

提出的方法

• 使用Gel'fand-Kirillov（GK）维数对具有有限增长的无限维霍普夫代数进行分类。
• 应用同调代数工具：阿廷-谢尔特正则性、奥斯陆德正则性及科恩-麦克aul伊性质。
• 运用卡蒂埃-科斯坦定理将余交换连通霍普夫代数表征为通用包络代数。
• 通过生成元与关系构造一个5维霍普夫代数，其具有明确定义的共乘法、余单位与反演。
• 利用生成元上的共结合性与余单位条件验证代数与余代数公理。
• 使用通用包络代数结构的对偶分析非交换、非余交换的例子。

实验结果

研究问题

• RQ1 有限GK-维数的连通分次霍普夫代数的同调性质（如正则性、全局维数）是什么？
• RQ2 连通分次霍普夫代数的性质与有限维李代数的通用包络代数的性质有何比较？
• RQ3 连通分次霍普夫代数在有限GK-维数下是否可能不与任何U(𝔤)同构？
• RQ4 一个作为代数和余代数均连通但不与任何U(𝔤)同构的霍普夫代数的结构是什么？
• RQ5 连通霍普夫代数在何种条件下满足多项式恒等式？

主要发现

• 有限Gel'fand-Kirillov维数n的连通分次霍普夫代数是诺特域、科恩-麦克aul伊、阿廷-谢尔特正则且全局维数为n的奥斯陆德正则代数。
• 此类代数满足S² = Id，且为维数n的卡拉比-丘代数。
• 有限GK-维数的连通分次霍普夫代数的Hilbert级数是有理函数，并满足特定对称性质。
• 连通霍普夫代数满足多项式恒等式当且仅当其为交换多项式代数。
• 存在一个5维连通分次霍普夫代数，其作为余代数连通但不与任何李代数𝔤的U(𝔤)同构。
• 有限GK-维数的连通霍普夫代数的底层余代数是全局维数n的阿廷-谢尔特正则代数。