Commutative Poisson algebras $A$ can be thought of as commutative algebras equipped with a first-order deformation into a noncommutative algebra given by the Poisson bracket. A simple example is the symmetric algebra $S(\mathfrak{g})$ of a Lie algebra, which can be deformed into the universal enveloping algebra $U(\mathfrak{g})$.


Today I learned that Poisson algebras also appear [in algebraic topology](http://arxiv.org/abs/math/0610236) as follows:


* If $X$ is a pointed space, the iterated loop space $\Omega^d(X)$ is an algebra over the little $d$-disks operad.
* Hence the homology $H\_{\ast}(\Omega^d(X))$ is an algebra over the homology of the little $d$-disks operad.
* The homology of the little $d$-disks operad is an operad $\text{Pois}^d$ whose algebras are (graded commutative) Poisson algebras where the bracket has degree $1 - d$.


(I may have that last statement slightly wrong.)



> 
> Can these two points of view be related?
> 
> 
> 


For example, is it known whether $H\_{\ast}(\Omega^d(X))$ is naturally the associated graded of some filtered algebra $F(X)$ such that the Poisson bracket arises from the (super) commutator in $F(X)$? 



Poisson algebras as deformations vs. Poisson algebras in algebraic topology
Let me start by rephrasing what is already in the answers of David Ben-Zvi and Theo Johnson-Freyd. The DG $\mathbb{Q}$-linear operad $\mathbb{E}\_n:=C\_{-\bullet}(E\_n,\mathbb{Q})$ is filtered. For $n\geq2$ the filtration is the degree filtration, and thus $gr(\mathbb{E}\_n)=H\_{-\bullet}(E\_n,\mathbb{Q})={\rm Pois}^n$.


The situation for $n=1$ is a bit different. We know that $\mathbb{E}\_1\cong {\rm As}$ (this is the formality theorem for $E\_1$ which, contrary to the case when $n\geq2$, is easy to prove). The operad ${\rm As}$ of associative algebras is also filtered, but in a less obvious way. To be short, one assigns the following two-step filtration onto ${\rm As}(2)=\mathbb{Q}[\Sigma\_2]$ (which generates ${\rm As}$):
$$
F^0{\rm As}(2)=\mathbb{Q}(1-\sigma)\subset F^1{\rm As}(2)={\rm As}(2).
$$
It then an exercise to check that $gr({\rm As})={\rm Pois}^1$.


Then, in order to relate the two stories, I have the feeling that one does not need to invoke the formality of $E\_n$ for $n\geq2$. Given a filtered $\mathbb{E}\_n$-algebra $A$ (i.e. a filtered DG $\mathbb{Q}$-vector space equipped with an action of $\mathbb{E}\_n$ that is compatible with the above filtration), then $gr(A)$ is a ${\rm Pois}^n$-algebra.


Concerning the last example in the question, one has to take $A=C\_{-\bullet}(\Omega^d(X),\mathbb{Q})$ equipped with the degree filtration. Then $gr(A)=H\_{-\bullet}(\Omega^d(X),\mathbb{Q})$ is going to be a ${\rm Pois}^d$-algebra.


**Side remark:** Observe that the story for $E\_0$ is even more degerated. Nevertheless,deformation theory of $E\_0$-algebras is still very interesting (for a discussion about this issue and its relation to the BV formalisms, see Costello-Gwilliam work-in-progress [http://math.northwestern.edu/~costello/factorization\_public.html](http://math.northwestern.edu/%7Ecostello/factorization_public.html) - especially 5b and 5c).


