1) In [HIGHER OPERATIONS ON HOCHSCHILD COMPLEX], Gerstenhaber and Voronov showed that the Hochschild complex $C_1(\mathcal A)$ of any associative algebra (or e_1 algebra) $\mathcal A$ is naturally endowed with a brace algebra structure $Br_2$. In cohomology, the brace algebra structure reduces to the Gerstenhaber structure (or e_2 algebra) discovered by Gerstenhaber in [The Cohomology Structure of an Associative Ring].

2a) In hep-th/9409063, the two authors further show that to any operad $\mathscr P$, one can associate a brace algebra on the suspension $V[1]$ of the graded vector space $V=\bigoplus_{n\geq0}\mathscr P(n)$.

2b)In the particular case where $\mathscr P$ is the endomorphism operad $End_{\mathcal A}$ on the vector space $\mathcal A$, then $V[1]$ is the Hochschild complex associated to $\mathcal A$ and the induced brace algebra is the one of 1).

3) In 1310.4605, Calaque and Willwacher introduce a notion of higher braces (or $Br_n$-structure) such that an ordinary brace algebra coincides with the case $n=2$. They also show that the deformation complex $C_n$ of an $e_n$ algebra is naturally endowed with a $Br_{n+1}$-structure. In the case, $n=1$, this reproduces the result of Gerstenhaber and Voronov 1).


a) Does the deformation complex $C_n$ of a $e_n$ algebra admits a construction (similar to the one of 2b)) as the (shifted by [n]?) graded vector space $V$ constructed from the $\mathscr P(m)$ spaces of a suitable operad?

b) Is there a generalisation of 2a) to the $n$ case? In other words, is it possible to associate to any (generalisation of) operad a $Br_n$ structure?

  • 3
    $\begingroup$ Aren't your questions answered in the Calaque--Willwacher paper? For any Hopf cooperad $C$ there's a notion of a $C$-operad. $Hom(C\{n\}, P)$ for any operad $P$ and any number $n$ is a $C$-operad. For any $C$-operad $O$, $\oplus O(m)^{S_m}$ carries a $preLie_C$-action. For $C=e_1^*$ you get exactly the Gerstenhaber--Voronov construction. $\endgroup$ Aug 5, 2018 at 9:37


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