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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?

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    $\begingroup$ This is not a great answer, but you can try to take the factorization homology of the $e_n$-algebra over $S^n$ (perhaps you need to assume that the algebra is framed). Then, since $S^n$ has an action of $SO(n+1)$, you could take the homotopy orbits (or fixed points) of the result. When $n=1$, this returns the usual definition of cyclic homology (or negative cyclic homology). But it's not obvious that this can be expressed in the terms that you're asking for above. $\endgroup$
    – Craig Westerland
    Commented Feb 14, 2013 at 21:40
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    $\begingroup$ You might want to look at this paper : arxiv.org/abs/1104.0181 by Jon Francis. He shows that Hochschild cohomology of an e_n algebra is the Lie algebra of some derived algebraic group. Of course this doesn't really answer the question. $\endgroup$
    – Geoffroy Horel
    Commented Feb 19, 2013 at 0:31
  • $\begingroup$ The thing you want to have an analogy with doesn't have anything to do with a commutative structure. In fact, there is an analog of cyclic homology for associative algebras, TC. This is pretty hard to compute. There is also an analog of HH that is specific to $E_n$ algebras, iterated THH. I don't know of anyone who has investigated cycltomic structures on iterated THH in a way that remembers that it is iterated. That might be interesting. $\endgroup$
    – Sean Tilson
    Commented Sep 19, 2013 at 18:34
  • $\begingroup$ I think I have definitely lied to you. See Covering Homology by Brun, Carlsson, and Dundas or higher topological cyclic homology by Carlsson, Douglas, and Dundas. These might answer your question. $\endgroup$
    – Sean Tilson
    Commented Sep 20, 2013 at 12:21

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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 $\int_{S^n} A$.

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