Algebraic & Geometric Topology

Koszul duality theory for operads over Hopf algebras

Olivia Bellier

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The transfer of the generating operations of an algebra to a homotopy equivalent chain complex produces higher operations. The first goal of this paper is to describe precisely the higher structure obtained when the unary operations commute with the contracting homotopy. To solve this problem, we develop the Koszul duality theory of operads in the category of modules over a cocommutative Hopf algebra. This gives rise to a simpler category of homotopy algebras and infinity morphisms, which allows us to get a new description of the homotopy category of algebras over such operads. The main example of this theory is given by Batalin–Vilkovisky algebras.

Article information

Algebr. Geom. Topol., Volume 14, Number 1 (2014), 1-35.

Received: 25 February 2013
Revised: 6 June 2013
Accepted: 6 June 2013
First available in Project Euclid: 19 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18D50: Operads [See also 55P48] 18G55: Homotopical algebra
Secondary: 16W30 55P48: Loop space machines, operads [See also 18D50]

operads Batalin–Vilkovisky algebras Koszul duality theory homotopical algebra


Bellier, Olivia. Koszul duality theory for operads over Hopf algebras. Algebr. Geom. Topol. 14 (2014), no. 1, 1--35. doi:10.2140/agt.2014.14.1.

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