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Assume that $A$ is a complex algebra. By $C^{n}(A)$ we mean the space of all $n-$linear map $\phi:A^n \to \mathbb{C}$. An alternative $k-$ form is an element $\phi \in C^{k}(A)$ with $\phi(a_{\sigma(1))},a_{\sigma(2)},\ldots,a_{\sigma(n)})=sign(\sigma)\phi(a_1,a_2,\ldots,a_k)$ for every permutation $\sigma$. The standard Hochschild coboundary $b:C^n(A) \to C^{n+1}(A)$ is defined by $$b\phi(a_0,a_1,\ldots,a_n)=\sum_{j=0}^{n-1}{(-1)}^{j}\phi(a_0,a_1,\ldots a_ja_{j+1},\ldots a_n)+{(-1)}^n\phi(a_na_0,a_1,\ldots,a_{n-1}) $$

Question: Is there a unital complex algebra $A$ with the following property?:

For every alternative $k-$form $\phi: A^k\to \mathbb{C}$, $b\phi:A^{k+1}\to \mathbb{C}$ is an alternative $(k+1)-$ form

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    $\begingroup$ Loday's "Cyclic homology" has precise formulas about relationship between the deRham and Hochschild coboundaries $\endgroup$
    – მამუკა ჯიბლაძე
    Commented May 27, 2017 at 6:54
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    $\begingroup$ Lemma 2.6.5 (in the second edition) $\endgroup$
    – მამუკა ჯიბლაძე
    Commented May 27, 2017 at 7:36
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    $\begingroup$ @July I don't know whether OP had this in mind but examples of such are Frobenius manifolds $\endgroup$
    – მამუკა ჯიბლაძე
    Commented May 28, 2017 at 12:52
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    $\begingroup$ @July as it is pointed out in the above comment and the following answer, Frobenius manifolds are examples of this situation mathoverflow.net/questions/266497/… $\endgroup$
    – Ali Taghavi
    Commented May 28, 2017 at 13:18
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    $\begingroup$ @მამუკაჯიბლაძე and Ali Taghavi: thank you for the references! $\endgroup$
    – M.G.
    Commented May 28, 2017 at 13:30

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