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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:

  1. 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?

  2. What is the significance of the solutions of the Maurer--Cartan equation (using the higher commutator) to the given $E_n$-algebra?

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I've always assumed the answer to 1 is "yes". I don't know a reference, however. One place these MC equations arise in deformation theory is the following. Any $E_{n-1}$ algebra $B$ has an "endomorphism ring" $A = \mathrm{End}(B)$ which is an $E_n$ algebra: it is a certain Hochschild complex, and by design MC elements in $A$ are "the same" as deformations of $B$ as an $E_{n-1}$ algebra. I've left this as a comment because someone with more expertise should be the one to leave an answer. – Theo Johnson-Freyd Nov 8 '13 at 3:38

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" 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", 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.

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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.

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If the bracket is of the wrong degree, then we can fix this by shifting $A$. It seems like the MC equation then does make sense. Or are you referring to other "degree reasons"? – John Pardon Nov 11 '13 at 1:40
Sorry, you're right, the only problem is that MC elements have degree other than $0$: $|d(a)|=|a|+1$ and $|1/2[a,a]|=2|a|+n-1$ so the MC equation has sense for $|a|=2-n$. I'd say it's wiser to include higher brackets in the MC equation. The MC equation for L-infinity algebras has been much studied (I remember a paper by Getzler in Annals). However I don't know how to interpret solutions in this case. Maybe also as deformations by general Koszul duality principles? It looks like sonething that somebody must have looked at! :-) – Fernando Muro Nov 11 '13 at 7:24
See… – Fernando Muro Nov 11 '13 at 7:43
As you may expect, solutions are probably unrelated to the rest of the E-infinity structure (the commutative algebra part) – Fernando Muro Nov 11 '13 at 7:44

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