2.4,1)(1.9,0.5)(1.5,0.9)
\end{pspicture}$$ Here $\tau_{k+l+2}^{l+1}$ denotes the $(l+1)$st iteration of $\tau_{k+l+2}$, and $\epsilon=(|m^*_1|+|a^*_1|+\dots+|a^*_k|)\cdot
(|m_2|+|a^*_{k+1}|+\dots+|a^*_{k+l}|)$.

A straightforward check shows that if $d^2=0$ and $g^2=0$, then it is also $h^2=0$.

The following proposition identifies algebras over $\textbf{D} (\widehat{\mathcal O^!})$. Its proof is similar to that in [@GK Proposition (4.2.14)].

\[O\_hat\_algebras\] Let $\mathcal O$ be a cyclic quadratic operad, and let $A$ and $M$ be graded vector spaces over $k$, which are finite dimensional in every degree. Then giving $(A,M)$ the structure of an algebra over $\textbf{D} (\widehat{\mathcal O^!})$ is equivalent to the following data:

-   a derivation $d\in \mathrm{Der}(F _{\mathcal O^!}\,A^*[1])$ of degree $1$, with $d^2=0$,

-   a derivation $g\in \mathrm{Der}_d (F_{\mathcal O^!,\,A^*[1]}M^*[1])$ over $d$ of degree $1$, with $g^2=0$, which by definition \[dual-module\] also implies a derivation $h\in \mathrm{Der}_d (F_{\mathcal O^!,\,A^*[1]}M[1])$ over $d$ with $h^2=0$,

-   a module map $f\in \mathrm{Mod}(F_{\mathcal O^!, A^*[1]}M[1], F_{\mathcal
    O^!,A^*[1]}M^*[1])$ of degree $0$ such that $f\circ h = g \circ f$, and satisfying the following symmetry condition: let $f$ be given by maps $f_n:\,M[1]\to \bigoplus_{k+l=n-2} \mathcal O^!(k+l+1)\otimes A^*[1]^{\otimes k}\otimes \,M^*[1] \otimes A^*[1]^{\otimes l}$, then $$\begin{gathered}
    \label{eq:symm}
     f_n(m_2)(\alpha^*;a_1,\dots,a_i,m_1,a'_1,\dots,a'_j)=\\
    = (-1)^\epsilon f_n(m_1)(\alpha^*\circ \tau_{i+j+2}^{j+1}
    ;a'_1,\dots,a'_j,m_2,a_1,\dots,a_i).\end{gathered}$$

where $\epsilon=(|m_2|+|a_1|+\dots+|a_i|+i+1)\cdot (|m_1|+ |a'_1|+\dots+
|a'_j|+j+1)$.

If $\mathcal O$ is cyclic quadratic and Koszul, then, by theorem \[O\_hat\_Koszul\], $\textbf{D}( \widehat{\mathcal O^!} )\cong
\widehat{\mathcal O}$ and we call $(A,M)$ a homotopy $\mathcal O$-algebra with homotopy $\mathcal O$-module and homotopy $\mathcal O$-inner product if there are deriva