
Verlinde模性范畴上的Casimir数及其应用
The Casimir Number of a Verlinde Modular Category and Its Applications
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+1(X)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.
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.
Grothendieck环 / Verlinde模性范畴 / Casimir数 / Jacobson根 / Dickson多项式 {{custom_keyword}} /
Grothendieck ring / Verlinde modular category / Casimir number / Jacobson radical / Dickson polynomial {{custom_keyword}} /
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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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