AI 中文总结
该研究探讨张量范畴 ind-完备化中三类代数的关系，证明适度增长对称张量范畴的交换代数中三类代数重合，给出反例说明一般情况的分化，还补充了两个附录的相关成果。
AI 中文摘要
我们研究半单代数、阿廷单代数与阿廷绝对平坦代数在张量范畴（刚性阿贝尔幺半范畴）的 ind-完备化中的关系。主要结果表明，若限定在具有适度增长的对称张量范畴中的交换代数，这三类代数重合；若接受该领域的主要猜想，则所有此类范畴均成立。我们还给出例子说明，与域上代数的经典情形相反，一般情况下这些概念趋于分化。在一个附录中，我们给出了据我们所知首个具有有限维态射空间但对象长度无限的张量范畴的例子；在第二个附录中，我们推广了 Deligne 的定理，即 tannakian 范畴在代数闭域上是中性的。
英文摘要
We study the relation between semisimple algebras, artinian simple algebras, and artinian absolutely flat algebras in ind-completions of tensor categories (rigid abelian monoidal categories). Our main results show that the three types of algebras coincide, if we restrict to commutative algebras in certain (all, if we accept the main conjectures in the field) symmetric tensor categories of moderate growth. We also provide examples to show that in general these notions tend to diverge, contrary to the classical case of algebras over a field. In one appendix we give the first, to the best of our knowledge, example of a tensor category with finite-dimensional morphism spaces but objects of infinite length. In a second appendix we establish a generalisation of Deligne's theorem that tannakian categories are neutral over algebraically closed fields.
Comments37 pages; comments welcome!
