nLab brace algebra

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.

Literature

Brace 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]

Symmetric brace algebras

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

category: algebra