Verlinde模性范畴上的Casimir数及其应用

王志华, 李立斌

数学学报 ›› 2018, Vol. 61 ›› Issue (1) : 59-66.

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数学学报 ›› 2018, Vol. 61 ›› Issue (1) : 59-66. DOI: 10.12386/A2018sxxb0005
论文

Verlinde模性范畴上的Casimir数及其应用

    王志华1,2, 李立斌3
作者信息 +

The Casimir Number of a Verlinde Modular Category and Its Applications

    Zhi Hua WANG1,2, Li Bin LI3
Author information +
文章历史 +

摘要

本文计算了秩为n+1的一类特殊的Verlinde模性范畴l的Casimir数,计算结果表明该Casimir数为2n+4.作为应用,由Higman定理知域K上的Grothendieck代数Gr (l)⊗Z K是半单代数当且仅当2n+4在域K中不为零.这也给出了第二类型n+1次Dickson多项式En+1X)在K[X]中无重因式的一个等价刻画.如果2n+4在域K中为零,借助于Dickson多项式的有关因式分解定理,本文完全给出了Grothendieck代数Gr (l)⊗Z K的Jacobson根.

Abstract

In this paper the Casimir number of a special kind of Verlinde modular category l of rank n+1 is calculated to be 2n+4. As an application it follows from Higman's theorem that the Grothendieck algebra Gr(l) ⊗Z K over a field K is semisimple if and only if 2n+4 is a unit in K. This is equivalent to saying that the (n+1)-th Dickson polynomial En+1(X) of the second kind has no multiple factors in K[X]. If 2n+4 is zero in K, we use the factorizations of Dickson polynomials to describe the Jacobson radical of Gr(l) ⊗Z K explicitly.

关键词

Grothendieck环 / Verlinde模性范畴 / Casimir数 / Jacobson根 / Dickson多项式

Key words

Grothendieck ring / Verlinde modular category / Casimir number / Jacobson radical / Dickson polynomial

引用本文

导出引用
王志华, 李立斌. Verlinde模性范畴上的Casimir数及其应用. 数学学报, 2018, 61(1): 59-66 https://doi.org/10.12386/A2018sxxb0005
Zhi Hua WANG, Li Bin LI. The Casimir Number of a Verlinde Modular Category and Its Applications. Acta Mathematica Sinica, Chinese Series, 2018, 61(1): 59-66 https://doi.org/10.12386/A2018sxxb0005

参考文献

[1] Higman D. G., On orders in separable algebras, Canad. J. Math., 1955, 7:509-515.
[2] Lorenz M., Some applications of Frobenius algebras to Hopf algebras, Contemp. Math., 2011, 537:269-289.
[3] Bhargava M., Zieve M. E., Factoring Dickson polynomials over finite fields, Finite Fields Appl., 1999, 5(2):103-111.
[4] Chou W. S., The factorization of Dickson polynomials over finite fields, Finite Fields Appl., 1997, 3:84-96.
[5] Etingof P., Nikshych D., Ostrik V., On fusion categories, Annals of Mathematics, 2005, 162:581-642.
[6] Bakalov B., Kirillov A. A., Lectures on Tensor Categories and Modular Functors, University Series Lectures, Vol. 21, AMS, 2001.
[7] Etingof P., Gelaki S., Nikshych D., Ostrik V., Tensor Categories, Mathematical Surveys and Monographs, 205, AMS, 2015.
[8] Drinfeld V., Gelaki S., Nikshych D., Ostrik V., On braided fusion categories I, Selecta Mathematica (N. S.), 2010, 16:1-119.

基金

国家自然科学基金资助项目(11471282);中国博士后科学基金资助项目(2017M610316)

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