Let $\mathcal{C}$ be a fusion category over an algebraically closed field $\mathbb{k}$ of arbitrary characteristic. Two numerical invariants of $\mathcal{C}$ , that is, the Casimir number and the determinant of $\mathcal{C}$ are considered in this paper. These two numbers are both positive integers and admit the property that the Grothendieck algebra $(\mathcal{C})\otimes_{\mathbb{Z}}K$ over any field K is semisimple if and only if any of these numbers is not zero in K. This shows that these two numbers have the same prime factors. If moreover $\mathcal{C}$ is pivotal, it gives a numerical criterion that $\mathcal{C}$ is nondegenerate if and only if any of these numbers is not zero in $\mathbb{k}$ . For the case that $\mathcal{C}$ is a spherical fusion category over the field $\mathbb{C}$ of complex numbers, these two numbers and the Frobenius–Schur exponent of $\mathcal{C}$ share the same prime factors. This may be thought of as another version of the Cauchy theorem for spherical fusion categories.

中文翻译：

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卡西米尔数和融合类别的决定因素

让$\mathcal{C}$是代数闭域上的融合范畴$\mathbb{k}$任意特性。的两个数值不变量$\mathcal{C}$，即卡西米尔数和行列式$\mathcal{C}$本文考虑。这两个数都是正整数并且承认格洛腾迪克代数的性质$(\mathcal{C})\otimes_{\mathbb{Z}}K$在任何领域ķ是半简单的当且仅当这些数字中的任何一个不为零ķ. 这表明这两个数具有相同的质因数。此外，如果$\mathcal{C}$是关键，它给出了一个数值标准$\mathcal{C}$是非退化的当且仅当这些数字中的任何一个不为零$\mathbb{k}$. 对于这种情况$\mathcal{C}$是场上的球面融合类别$\mathbb{C}$复数，这两个数和 Frobenius-Schur 指数$\mathcal{C}$共享相同的质因数。这可以被认为是球面融合类别的柯西定理的另一个版本。