Higher commutators in E_n algebras and the Maurer--Cartan equation Let $A$ be an associative algebra in $dgVect_k$.  Then the commutator $[\cdot,\cdot]:A\otimes A\to A$ defined by $[x,y]=xy-(-1)^{|x||y|}yx$ gives $A$ the structure of a (dg-)Lie algebra.  The Maurer--Cartan equation:
$$d\alpha+\frac 12[\alpha,\alpha]=0$$
for $\alpha\in A$ (necessarily of degree $1$) plays an important role in deformation theory.

Does anyone recognize the  following generalization of the above construction from associative algebras (that is, $E_1$-algebras) to $E_n$-algebras?

Let $A$ be an $E_n$-algebra in $dgVect_k$.  The $E_n$-algebra structure gives a map $A\otimes A\otimes C_\bullet(Conf_2(D^n))\to A$ where $Conf_2(D^n)\simeq S^{n-1}$ is the configuration space of two distinct points in $D^n$.  By picking a cycle in $C_\bullet(Conf_2(D^n))$ representing $[S^{n-1}]$, we get a pairing $[\cdot,\cdot]:A\otimes A\to A$ which I will think of as a sort of "higher commutator" (of course, this recovers the usual notion of commutator when $n=1$).
Now for some questions:


*

*Does this "higher commutator" endow $A$ with the structure of a dg-Lie algebra (or, more likely, an $L_\infty$-algebra), or is it something more exotic?

*What is the significance of the solutions of the Maurer--Cartan equation (using the higher commutator) to the given $E_n$-algebra?
 A: This operation appears prominently in the theory of $n$-fold loop spaces, where it is called a Browder operation and is related by suspension to the Samelson product.  In characteristic $p$, this operation accompanies Dyer-Lashof operations, and these operations together give enough structure to compute  $H_*(\Omega^n \Sigma^n X;\mathbf{F}_p)$ as an explicit functor of $H_*(X;\mathbf{F}_p)$ for any space $X$.  This was part of Fred Cohen's 1972 PhD thesis and appears in "The homology of iterated loop spaces" http://www.math.uchicago.edu/~may/BOOKS/homo_iter.pdf.
This does not directly answer the questions, but I thought the history of these operations might be of some interest.  The answer to question 1 in characteristic zero is well-known and is summarized in Section 5 of "Operads, algebras, and modules", http://www.math.uchicago.edu/~may/PAPERS/mayi.pdf.  The algebras over the homology of an $E_{n+1}$-operad $\mathcal{C}_{n+1}$ are $n$-braid algebras, which are commutative algebras and $n$-Lie algebras that satisfy the Poisson formula.  When $n=1$, we see Batalin-Vilkovisky algebras.  The free $n$-braid algebras are described explicitly in Theorem 5.6 op cit, where the description is deduced from topology. 
A: Over a fiel of characteristic zero, that bracket operation is the initial part of an L-infinity structure. This follows from the fact that the little discs operads are formal. It seems that the Maurer-Cartan equation doesn't make sense for higher brackets for degree reasons.
