Difference $2$-algebras and difference $A_\infty$-algebras
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917531158052864 |
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| author | Das, Apurba |
| author_facet | Das, Apurba |
| contents | A difference operator on an associative algebra is an algebraic abstraction of the forward and backward difference operators. In this paper, we first introduce difference operators on associative $2$-algebras and consider the category of difference associative $2$-algebras. Subsequently, we also introduce difference operators on a given $A_\infty$-algebra in terms of their Maurer-Cartan characterization. We prove that the category of difference associative $2$-algebras and the category of $2$-term difference $A_\infty$-algebras are equivalent. We characterize skeletal and strict $2$-term difference $A_\infty$-algebras by respectively third cocycles and crossed modules of difference algebras. Finally, we define the notion of a $2$-term bimodule up to homotopy over a difference algebra, which in turn yields a construction of a $2$-term difference $A_\infty$-algebra. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_25587 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Difference $2$-algebras and difference $A_\infty$-algebras Das, Apurba Rings and Algebras Quantum Algebra Representation Theory 16W99, 18N25, 18N40 A difference operator on an associative algebra is an algebraic abstraction of the forward and backward difference operators. In this paper, we first introduce difference operators on associative $2$-algebras and consider the category of difference associative $2$-algebras. Subsequently, we also introduce difference operators on a given $A_\infty$-algebra in terms of their Maurer-Cartan characterization. We prove that the category of difference associative $2$-algebras and the category of $2$-term difference $A_\infty$-algebras are equivalent. We characterize skeletal and strict $2$-term difference $A_\infty$-algebras by respectively third cocycles and crossed modules of difference algebras. Finally, we define the notion of a $2$-term bimodule up to homotopy over a difference algebra, which in turn yields a construction of a $2$-term difference $A_\infty$-algebra. |
| title | Difference $2$-algebras and difference $A_\infty$-algebras |
| topic | Rings and Algebras Quantum Algebra Representation Theory 16W99, 18N25, 18N40 |
| url | https://arxiv.org/abs/2605.25587 |