Coassociative structures on self-injective algebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915514539835392 |
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| author | Chirvasitu, Alexandru |
| author_facet | Chirvasitu, Alexandru |
| contents | 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 $\{1,\cdots,m(i)\}\times \{1,\cdots,m(ν^{-1}i)\}$, where $A\cong \mathrm{End}_Λ(M)$ is the general form of a self-injective algebra in terms of a basic Frobenius $Λ$, the $m(i)$, $1\le i\le n$ are the multiplicities of the indecomposable projective $Λ$-modules in $M$, and $ν$ is the Nakayama permutation of $Λ$. We also characterize those among the coproducts introduced in this fashion, in terms this combinatorial data, which are counital. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_21435 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Coassociative structures on self-injective algebras Chirvasitu, Alexandru Rings and Algebras Category Theory Quantum Algebra Representation Theory 16D50, 16L60, 16T15, 18M05, 16U40, 16D40, 16D60, 16D20 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 $\{1,\cdots,m(i)\}\times \{1,\cdots,m(ν^{-1}i)\}$, where $A\cong \mathrm{End}_Λ(M)$ is the general form of a self-injective algebra in terms of a basic Frobenius $Λ$, the $m(i)$, $1\le i\le n$ are the multiplicities of the indecomposable projective $Λ$-modules in $M$, and $ν$ is the Nakayama permutation of $Λ$. We also characterize those among the coproducts introduced in this fashion, in terms this combinatorial data, which are counital. |
| title | Coassociative structures on self-injective algebras |
| topic | Rings and Algebras Category Theory Quantum Algebra Representation Theory 16D50, 16L60, 16T15, 18M05, 16U40, 16D40, 16D60, 16D20 |
| url | https://arxiv.org/abs/2509.21435 |