AI 中文总结
本文对有限维半单代数ℂ^⊕ⁿ上的（弱）双代数结构进行分类，证明双代数结构对应有限幺半群、弱双代数结构对应有限小范畴，明确了两类结构的对应关系。
AI 中文摘要
弱双代数是双代数的自然推广，在量子群、张量范畴和表示论中具有重要作用。本文系统地对有限维半单代数ℂ^⊕ⁿ上的所有（弱）双代数结构进行分类，该代数具有本原正交幂等元的特殊基。对于双代数情形，我们证明这类结构与有限幺半群一一对应，且双代数同构恰好对应幺半群同构。将结果推广到弱双代数，我们表明每个弱双代数结构唯一确定一个有限小范畴。
英文摘要
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.
