Jingheng Zhou | Searcharxiv

arXiv2026

The weak bialgebra structures on $\mathbb{k}^{\oplus n}$

Weak bialgebras are natural generalizations of bialgebras that play a fundamental role in quantum groups, tensor categories, and representation theory. In this paper, we systematically classify all (weak) bialgebra structures on the finite-dimensional semisimple algebra $\Bbbk^{\oplus n}$, with a distinguished basis of primitive orthogonal idempotents. For the bialgebra case, we prove that such structures are in bijection with finite monoids, and that bialgebra isomorphisms correspond exactly to monoid isomorphisms. Extending the result to weak bialgebras, we show that every weak bialgebra structure uniquely determines a finite small category.

math.QA↗

arXiv2020

The applications of probability groups on Hopf algebras

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 $Z(H)$ of $H$ and the product of two class sums is an integral combination up to a factor of $\dim (H)^{-1}$ of the class sums of $H$. We classify all the 2-integral probability groups with 2 or 3 elements.

math.RA↗