Martin Markl

Models for operads

We study properties of differential graded (dg) operads modulo weak equivalences, that is, modulo the relation given by the existence of a chain of dg operad maps including a homology isomorphism. ...

Operads and PROPs

Abstract We review definitions and basic properties of operads, PROPs and algebras over these structures.

Cotangent cohomology of a category and deformations

Abstract We construct, for a general k-linear equationally given category, a cohomology theory controlling deformations of objects of this category.

PROPped-Up Graph Cohomology

We consider graph complexes with a flow and compute their cohomology. More specifically, we prove that for a PROP generated by a Koszul dioperad, the corresponding graph complex gives a minimal model of the PROP. We also give another proof of the existence of a minimal model of the bialgebra PROP from [14]. These results are based on the useful notion of a \(\frac{1}{2}\)PROP introduced by Kontsevich in [9].

Wheeled PROPs, graph complexes and the master equation

Abstract We introduce and study wheeled PROPs, an extension of the theory of PROPs which can treat traces and, in particular, solutions to the master equations which involve divergence operators. We construct a dg free wheeled PROP whose representations are in one-to-one correspondence with formal germs of SP-manifolds, key geometric objects in the theory of Batalin–Vilkovisky quantization. We also construct minimal wheeled resolutions of classical operads Com and Ass as non-trivial extensions of the well-known dg operads Com ∞ and Ass ∞ . Finally, we apply the above results to a computation of cohomology of a directed version of Kontsevich’s complex of ribbon graphs.

Symmetric Brace Algebras

We develop a symmetric analog of brace algebras and discuss the relation of such algebras to L∞-algebras. We give an alternate proof that the category of symmetric brace algebras is isomorphic to the category of pre-Lie algebras. As an application, symmetric braces are used to describe transfers of strongly homotopy structures. We then explain how these symmetric brace algebras may be used to examine the L∞-algebras that result from a particular gauge theory for massless particles of high spin.

IDEAL PERTURBATION LEMMA

We explain the essence of perturbation problems. The key to understanding is the structure of chain homotopy equivalence – the standard one must be replaced by a finer notion which we call a strong chain homotopy equivalence. We formulate an Ideal Perturbation Lemma and show how both new and classical (including the Basic Perturbation Lemma) results follow from this ideal statement.

CROSSED INTERVAL GROUPS AND OPERATIONS ON THE HOCHSCHILD COHOMOLOGY

We prove that the operad B of natural operations on the Hochschild cohomology has the homotopy type of the operad of singular chains on the little disks operad. To achieve this goal, we introduce crossed interval groups and show that B is a certain crossed interval extension of an operad T whose homotopy type is known. This completes the investigation of the algebraic structure on the Hochschild cochain complex that has lasted for several decades.

Modular envelopes, OSFT and nonsymmetric (non-

Our aim is to introduce and advocate non-

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