Cohomology and deformations of dendriform algebras, and \(\mathrm{Dend}_\infty \)-algebras. (English) Zbl 1511.17005
A dendriform algebra, originally introduced by J.-L. Loday, is vector space \(A\) equipped with two binary operations, \(\prec\) and \(\succ\), that satisfy several compatibility conditions. In this paper, the author defines cohomology of dendriform algebras with coefficient in a representation. The deformation of a dendriform algebra, in the sense of Gerstenhaber, is then described by the cohomology of \(A\) with coefficient in itself. Then the author studies strongly homotopy dendriform algebras, called \(\mathrm{Dend}_\infty\)-algebras, in which the dendriform identities are true up to homotopies. They can be regarded as splitting of \(A_\infty\)-algebras. The author defines Rota-Baxter operator on \(A_\infty\)-algebras that yields \(\mathrm{Dend}_\infty\)-algebras. Finally, the author classifies skeletal and strict \(\mathrm{Dend}_\infty\)-algebras.
Reviewer: Donald Yau (Newark)
MSC:
| 17A30 | Nonassociative algebras satisfying other identities |
| 16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |
| 18M70 | Algebraic operads, cooperads, and Koszul duality |
Keywords:
cohomology; deformations; dendriform algebras; \(\mathrm{Dend}_\infty \)-algebras; Rota-Baxter operatorsReferences:
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