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Let $A$ be a graded quadratic algebra over a field $k$, and suppose that it admits the Koszul dual $A^!$. I want to know if there is a natural pairing $A\otimes A^!\to k$ or something similar to this. I found a vague statement on this line at the beginning of Section 1.4 of this note by Yanki Lekili, but there are no references there and I couldn't find any.

I'm also interested in operads, and I welcome any analogous statement in operadic setting (my original motivation is coming from there).

Thank you.

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    $\begingroup$ That vague statement does not really promise a pairing, it promises some analogue of a pairing. I do not think there is anything reasonable to expect. Already in the classical case of symmetric/exterior algebras, what would such a pairing be, in your opinion? Symmetric powers do not really pair with exterior powers. $\endgroup$
    – Vladimir Dotsenko
    Commented May 19 at 8:25
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    $\begingroup$ One expects a pairing like this in the case when $A$ is equivalent to its own Koszul dual algebra, but not generally. Since you are specifically interested in the operad case, I would recommend Ching-Salvatore. They do this for the $E_n$ operad. They talk a fair amount about operad-cooperad pairings if that is what you are interested in. $\endgroup$
    – Connor Malin
    Commented May 19 at 17:56
  • $\begingroup$ In the case of (right) modules over an augmented algebra $A \rightarrow k$, one does expect a pairing like you suggest. This is because if $R$ is an $A$-module, $K(R)$ has an underlying object $R/\mathrm{Decom}(R)^\vee \simeq \mathrm{RMod}_A(R,k)$, in some appropriate derived sense. And so one can pair $R$ and $K(R)$ by evaluation, though I don't have any references for this pairing. $\endgroup$
    – Connor Malin
    Commented May 19 at 18:04

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