Abstract In this work, we use probability groups, introduced by Harrison in 1979, as a tool to study a semisimple Hopf algebra H with a commutative character ring and prove that the algebra generalized by the dual probability group is the center of H and the product of two class sums is an integral combination up to a factor of of the class sums of H. We classify all the 2-integral probability groups with two or three elements.

中文翻译：

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概率群在Hopf代数上的应用

摘要 在这项工作中，我们使用 Harrison 在 1979 年引入的概率群作为工具，研究了一个具有可交换字符环的半简单 Hopf 代数 H，并证明了由对偶概率群推广的代数是 H 的中心和乘积两个类和的整数组合是 H 的类和的因子的整数组合。我们用两个或三个元素对所有 2 整数概率组进行分类。