标题： 自投射代数上的协结合结构 显示英文标题

标题： Coassociative structures on self-injective algebras

摘要： 对于一般的有限维自投射代数$A$，我们构造了一族内射的余结合余乘法$A\to A\otimes A$，所有$A$-双模态射。 特别是这样的结构总是存在，证实了Hernandez、Walton和Yadav的猜想。 余积由$\{1,\cdots,m(i)\}\times \{1,\cdots,m(\nu^{-1}i)\}$的子集索引，其中$A\cong \mathrm{End}_{\Lambda}(M)$是在基本 Frobenius$\Lambda$的基础上自内射代数的一般形式，$m(i)$，$1\le i\le n$是不可分解投射$\Lambda$-模在$M$中的重数，而$\nu$是$\Lambda$的 Nakayama 排列。 我们还根据这种组合数据来描述通过这种方式引入的余积中那些具有单位元的余积。 显示英文摘要

摘要： For general finite-dimensional self-injective algebra $A$ we construct a family of injective coassociative coproducts $A\to A\otimes A$, all $A$-bimodule morphisms. In particular such structures always exist, confirming a conjecture of Hernandez, Walton and Yadav. The coproducts are indexed by subsets of $\{1,\cdots,m(i)\}\times \{1,\cdots,m(\nu^{-1}i)\}$, where $A\cong \mathrm{End}_{\Lambda}(M)$ is the general form of a self-injective algebra in terms of a basic Frobenius $\Lambda$, the $m(i)$, $1\le i\le n$ are the multiplicities of the indecomposable projective $\Lambda$-modules in $M$, and $\nu$ is the Nakayama permutation of $\Lambda$. We also characterize those among the coproducts introduced in this fashion, in terms this combinatorial data, which are counital.