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Symmetrization of brace algebras. (English) Zbl 1162.18302
Čadek, Martin (ed.), The proceedings of the 25th winter school “Geometry and physics”, Srní, Czech Republic, January 15–22, 2006. Palermo: Circolo Matemático di Palermo. Supplemento ai Rendiconti del Circolo Matemático di Palermo. Serie II 79, 75-86 (2006).
Summary: We show that the symmetrization of a brace algebra structure yields the structure of a symmetric brace algebra. We also show that the symmetrization of the natural brace structure on $$\bigoplus_{k\geq 1}\operatorname{Hom}(V^{\otimes k},V)$$ coincides with the natural symmetric brace structure on $$\bigoplus_{k\geq 1}\operatorname{Hom}(V^{\otimes k},V)^{as}$$, the direct sum of spaces of antisymmetric maps $$V^{\otimes k}\to V$$.
For the entire collection see [Zbl 1103.53001].

##### MSC:
 18D50 Operads (MSC2010) 55S20 Secondary and higher cohomology operations in algebraic topology 16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
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