Deformation theory of representations of prop(erad)s

Abstract

We study the deformation theory of morphisms of properads and props thereby extending to a non-linear framework Quillen's deformation theory for commutative rings. The associated chain complex is endowed with a Lie algebra up to homotopy structure. Its Maurer-Cartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these results. To do so, we endow the category of prop(erad)s with a model category structure. We provide a complete study of models for prop(erad)s. A new effective method to make minimal models explicit, that extends Koszul duality theory, is introduced and the associated notion is called homotopy Koszul. As a corollary, we obtain the (co)homology theories of (al)gebras over a prop(erad) and of homotopy (al)gebras as well. Their underlying chain complex is endowed with a canonical Lie algebra up to homotopy structure in general and a Lie algebra structure only in the Koszul case. In particular, we explicit the deformation complex of morphisms from the properad of associative bialgebras. For any minimal model of this properad, the boundary map of this chain complex is shown to be the one defined by Gerstenhaber and Schack. As a corollary, this paper provides a complete proof of the existence of a Lie algebra up to homotopy structure on the Gerstenhaber-Schack bicomplex associated to the deformations of associative bialgebras.

Version 4 : Statement about the properad of (non-commutative) Frobenius bialgebras fixed in Section 4. [82 pages]

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• Deformation theory of representations of prop(erad)s
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Cited by in corpus (56)

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• Infinity Properads and Infinity Wheeled Properads
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• A tale of three homotopies
• The Frobenius properad is Koszul
• Homotopy Batalin-Vilkovisky algebras
• Homotopy units in A-infinity algebras
• Toric varieties of Loday's associahedra and noncommutative cohomological field theories
• Two-colored noncommutative Gerstenhaber formality and infinity Duflo isomorphism
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• Wheeled props in algebra, geometry and quantization
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• The L_\infty-deformation complex of diagrams of algebras
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• A short note on infinity-groupoids and the period map for projective manifolds
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• The minimal model for the Batalin-Vilkovisky operad
• Delooping totalization of a multiplicative operad
• Chains(R) does not admit a geometrically meaningful properadic homotopy Frobenius algebra structure
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• Tree- versus graph-level quasilocal Poincare duality on S^1
• The odd origin of Gerstenhaber brackets, Batalin-Vilkovisky operators, and master equations
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• Quantizations of Lie bialgebras, duality involution and oriented graph complexes
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• Derived Commutator Complete Algebras and Relative Koszul Duality for Operads
• Operads and Maurer-Cartan spaces
• Homotopy theory of curved operads and curved algebras
• Getzler-Kapranov complexes and moduli stacks of curves
• Cohomology and Versal Deformations of Hom-Leibniz algebras
• The equifibered approach to $\infty$-properads
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• Cylinders for non-symmetric DG-operads via homological perturbation theory
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• Minimal models of some differential graded modules
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• Combinatorics and algebra of tensor calculus
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• Hochschild and cotangent complexes of operadic algebras
• A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids