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 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 commutator in $F(X)$?