Abstract: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.
| Comments: | 19 pages; comments are welcome |
| Subjects: | Rings and Algebras (math.RA); Quantum Algebra (math.QA); Representation Theory (math.RT) |
| MSC classes: | 16W99, 18N25, 18N40 |
| Cite as: | arXiv:2605.25587 [math.RA] |
| (or arXiv:2605.25587v1 [math.RA] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25587 arXiv-issued DOI via DataCite (pending registration) |
From: Apurba Das [view email]
[v1]
Mon, 25 May 2026 08:35:53 UTC (23 KB)
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