Koszul duality, minimal model and \(L_\infty \)-structure for differential algebras with weight.(English)Zbl 1544.18025
Summary: A differential algebra with weight is an abstraction of both the derivation (weight zero) and the forward and backward difference operators (weight \(\pm 1\)). In [Georgian Math. J. 17, No. 2, 347–372 (2010;Zbl 1237.18007)]J.-L. Loday established the Koszul duality for the operad of differential algebras of weight zero. He did not treat the case of nonzero weight, noting that new techniques are needed since the operad is no longer quadratic. This paper continues Loday’s work and establishes the Koszul duality in the case of nonzero weight. In the process, the minimal model and the Koszul dual homotopy cooperad of the operad governing differential algebras with weight are determined. As a consequence, a notion of homotopy differential algebras with weight is obtained and the deformation complex as well as its \(L_\infty \)-algebra structure for differential algebras with weight are deduced.
MSC:
| 18M70 | Algebraic operads, cooperads, and Koszul duality |
| 12H05 | Differential algebra |
| 18M65 | Non-symmetric operads, multicategories, generalized multicategories |
| 12H10 | Difference algebra |
| 16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |
| 16S80 | Deformations of associative rings |
| 18M60 | Operads (general) |
Keywords:
cohomology;deformation;differential algebra;homotopy cooperad;homotopy differential algebra;Koszul dual;\( L_\infty \)-algebra;minimal model;operadCitations:
Zbl 1237.18007Cite
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