Deformation theory of representation of prop(erad)s I

By Bruno Vallette, S. Merkulov. 2009

163 citations recorded by the source corpus.

In this paper and its follow-up, we study the deformation theory of morphisms of properads and props thereby extending Quillen's deformation theory for commutative rings to a non-linear framework. The associated chain complex is endowed with an L_infinity-algebra 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 the 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 an L_infinity-algebra structure in general and a Lie algebra structure only in the Koszul case. In particular, we make the deformation complex of morphisms from the properad of associative bialgebras explicit. 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 an L_infinity-algebra structure on the Gerstenhaber-Schack bicomplex associated to the deformations of associative bialgebras.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bc9fd3f7d851e425a901cb322684abfa3b503d72

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Citations in the corpus

On the operad of associative algebras with derivation (2009, methodology): “So one needs new techniques to extend Koszul duality to this case, see [15].”
Operadic Twisting -- with an application to Deligne's conjecture (2012, background): “For the proof we refer the reader to [1] and [11].”
Combinatorial Hopf algebras (2008, background): “Some of these operads are known, the other ones are variations of known types: • MB : multibrace algebra, also called B∞-algebra in [42], non-differential B∞ in [19], Hirsch algebra in [30], LR-algebra in [45], see 2.”
Cohomology and Deformations of Hom-algebras (2010, background): “For global deformations and more general works involving operads and where deformation theory is described using a certain differential graded Lie algebras one may see [3, 4, 5, 6, 7, 13, 20, 24].”
M. Kontsevich’s graph complex and the Grothendieck–Teichmüller Lie algebra (2010, background): “Then following [17] one can define a deformation complex of f :”
Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots (2011, methodology): “has a natural interpretation from the point of view of the deformation theory of operads. This theory was initiated by Kontsevich and Soibelman [19] and was further developed by Merkulov and Vallette [27]. Given m<n, one has inclusion of the operads of little discs: Bm i →Bn. inducing morphism in”
On the rational homology of high dimensional analogues of spaces of long knots (2011, background): “In the differential graded context this structure was called ”infinitesimal bimodule” by Merkulov and Vallette [19].”