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\centerline {\bf Strong homotopy algebra over an operad}
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A quadratic algebra $A$ is a quotient of the tensor algebra $T(V)$ by an ideal $(R)$ 
generated by $R\subset V\otimes V$, so $A = T(V)/(R)$. Since the ideal is homogeneous, this is a graded
algebra. For such algebras there is a notion of dual algebra $A^!$ canonically associated to $A$, which is
constructed as follows. Let $V^*$ be the linear dual of $V$. If we denote by $R^{\perp}$ the kernel of $V^*\otimes
V^*\to R^*$, then $A^!:=T(V^*)/(R^{\perp})$. 

To any quadratic algebra $A$ there is associated a bigraded complex $(A^{!*}\otimes A, d)$ called the Koszul
complex of $A$. If this complex is acyclic, then $A$ is said to be a {\it Koszul algebra}.

Let $B(A)$ be the bar construction on $A$. It is the (graded) cofree coalgebra on the suspension of $A$. The
boundary map is the unique coderivation of degree $-1$ which agrees with the multiplication of $A$ on $A\otimes A$.
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\noindent {\bf Theorem.} {\it If the quadratic algebra $A$, over a characteristic zero field, is
Koszul, then $B(A^!)^*$ is the minimal model of $A$ in the category of d.g. associative algebras.}
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Recall that the minimal model is a resolution which is free as an algebra and whose differential
is quadratic.
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These constructions and result hold in any tensor category over a characteristic zero field. In
particular, it works in the category of algebraic operads, since an algebraic operad ${\cal P}$ is
an associative algebra (i.e. a monad) in the tensor category of analytical functors.

By definition a strong homotopy ${\cal P}$-algebra is an algebra over the minimal model of ${\cal
P}$ (in the category of d.g. algebraic operads).
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\noindent {\bf Theorem.} {\it If the quadratic operad ${\cal P}$ is Koszul, then a strong
homotopy ${\cal P}$-algebra is a $B({\cal P}^!)^*$-algebra.}
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For associative algebras, Lie algebras, commutative algebras, this gives precisely the notion of
$A_{\infty}$-algebra, $L_{\infty}$-algebra and  $C_{\infty}$-algebra, respectively.
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\noindent {\bf Ariane's thread: }V.  Ginzburg,  M.M. Kapranov, {\it Koszul duality for operads},
Duke Math. J.  76 (1994), 203--272.  

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\hfill ${}_{\rm JLL/mek/SHAO/02.99 }$
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