**Brace algebra** involves higher-order operations related to the study of $A_\infty$-algebras. There is a sequence of canonical functors

dendriform algebras –> brace algebras –> pre-Lie algebras

Hence, in some sense, brace algebras are intermediate between dendriform algebras and pre-Lie algebras.

There is also a notion of a **symmetric brace algebra** which has analogous role for $L_\infty$-algebras instead of $A_\infty$. They are related to post-Lie algebras.

- Tornike V. Kadeishvili,
*The structure of the A(∞)-algebra, and the Hochschild and Harrison cohomologies*, (Russian)Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 91 (1988) 19–27. MR91a:18016

- M. Gerstenkhaber, A. A.Voronov, Higher-order operations on the Hochschild complex, Funktsional. Anal. i Prilozhen.
**29**:1 (1995) 1–6, 96; translation in Funct. Anal. Appl. 29 (1995), no. 1, 1–5. MR96g:18006 - Ezra Getzler,
*Cartan homotopy formulas and the Gauss-Manin connection in cyclic homology*, Quantum deformations of algebras and their representations (Ramat-Gan, 1991/1992; Rehovot, 1991/1992) 65–78, Israel Math. Conf. Proc. 7, 1993 MR95c:19002 - Marcelo Aguiar,
*Infinitesimal bialgebras, pre-Lie and dendriform algebras*, [pdf]

- Tom Lada, Martin Markl,
*Symmetric brace algebras*, Appl Categor Struct 13, 351–370 (2005) doi

category: algebra

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