Homology, Homotopy and Applications

Volume 19 (2017)

Number 1

Koszul duality and homotopy theory of curved Lie algebras

Pages: 319 – 340

DOI: https://dx.doi.org/10.4310/HHA.2017.v19.n1.a16


James Maunder (Department of Mathematics and Statistics, Lancaster University, Lancaster, United Kingdom)


This paper introduces the category of marked curved Lie algebras with curved morphisms, equipping it with a model structure. This model structure is—when working over an algebraically closed field of characteristic zero—Quillen equivalent to a model category of pseudo-compact unital commutative differential graded algebras; extending known results regarding the Koszul duality of unital commutative differential graded algebras and differential graded Lie algebras. As an application of the theory developed within this paper, algebraic deformation theory is extended to functors over pseudo-compact, not necessarily local, commutative differential graded algebras. Further, these deformation functors are shown to be representable.


curved Lie algebra, homotopy, Koszul duality, deformation functor

2010 Mathematics Subject Classification

18G55, 55U99

Published 6 June 2017