TITLE: Analogue of cyclic homology for e_n-algebras?
QUESTION [8 upvotes]: Cyclic homology may be defined as the primitive part (with respect to a natural product) of the homology of the Lie algebra associated with the "stabilization" of an associative algebra $A$. Here the "stabilization" is the matrix algebra $Mat_n(A)$, where $n$ goes to infinity. This definition explains another name of cyclic homology - "additive K-theory".
I believe that there exists an analogous notion for $e_n$-algebras. For a $e_n$-algebra $A$ it must be the primitive part of the homology of the Lie algebra associated with a $e_n$-algebra, which is an appropriate "stabilization" of $A$. 
Whether anyone has some ideas how this thing could be defined? What is the homology of the trivial algebra? Is it an additive version of something?

REPLY [3 votes]: In the paper mentioned in the comments Francis defines a candidate for this. If $A$ is an $E_n$-algebra one has an equivalence of associative algebras $B = \int_{S^{n-1}} A \simeq (\int_{S^{n-1}} A)^{\rm op} = B^{\rm op}$. Thus, any left $B$-module has a canonical right $B$-module structure. Identifying $B$ with the $E_n$-enveloping algebra ${\rm Env}(A)$ this gives $A$ the natural structure of a left $\int_{S^{n-1}}A$-module structure. Letting $A^\tau$ be the induced right-module we can form
$$
A^\tau \otimes_{\int_{S^{n-1}} A} A
$$
which he calls the $E_n$-Hochschild homology of $A$. Using excision you can prove that $\int_{S^1} A = A \otimes_{A \otimes A^{\rm op}} A$, recovering classical Hochshild homology. Further if $A$ is commutative, or at least admits a $E_{n+1}$-refinement, then the above definition coincides with $\int_{S^n} A$.