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Zbl 1204.18007
Aloulou, Walid
$(a,b)$-algebras up to homotopy. (Les $(a,b)$-algèbres à homotopie près.) (French)
[J] Ann. Math. Blaise Pascal 17, No. 1, 97-151 (2010). ISSN 1259-1734

For a graded algebra $\cal A$ of type $\cal P$ with a binary multiplication (e.g., an associative multiplication or a Lie bracket) and equipped with a differential $d$, one can define the notion of algebra of type $\cal P$ up to homotopy and construct the corresponding enveloping algebra of $\cal A$ which carries information about homological and cohomological complexes associated with the type of algebras and their modules. When $\cal A$ has two multiplicative operations with certain relations of compatibility, the construction of the enveloping algebra up to homotopy is more complicated. In particular, the complexes of Poisson and Gerstenhaber require to compose structures of a cocommutative cofree coalgebra and of a cofree Lie coalgebra associated with each algebra. Working over the corresponding operads, one associates also the structure of ${\cal P}_{\infty}$-algebra in the Poisson case and of ${\cal G}_{\infty}$-algebra in the Gerstenhaber case. The purpose of the paper under review is to unify the constructions of algebras up to homotopy for Poisson and Gerstenhaber algebras. The author defines the structure of $(a,b)$-algebra up to homotopy. For an algebra with the structure of a commutative and a differential graded Lie algebra and for two shifts degree given by $a$ and $b$, he presents an explicit construction of the associated algebra up to homotopy and clarifies the relationship between $(a,b)$-algebras and algebras over the operad of little $n+1$-dimensional cubes. The Gerstenhaber and Poisson cases are obtained for $a=0$, $b=-1$ and $a=b=0$, respectively.
[Vesselin Drensky (Sofia)]

MSC 2000:: *18G55 Nonabelian homotopical algebra
16T05
16T15
17B63 Poisson algebras
16E45 Differential graded algebras and applications

Keywords: homotopical algebras; coalgebras; Poisson algebras; graded differential algebras

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