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We obtain a new interpretation of the cohomological Hall algebra $\mathcal{H}_Q$ of a symmetric quiver $Q$ in the context of the theory of vertex algebras. Namely, we show that the graded dual of $\mathcal{H}_Q$ is naturally identified with the underlying vector space of the principal free vertex algebra associated to the Euler form of $Q$. Properties of that vertex algebra are shown to account for the key results about $\mathcal{H}_Q$. In particular, it has a natural structure of a vertex bialgebra, leading to a new interpretation of the product of $\mathcal{H}_Q$. Moreover, it is isomorphic to the universal enveloping vertex algebra of a certain vertex Lie algebra, which leads to a new interpretation of Donaldson--Thomas invariants of $Q$ (and, in particular, re-proves their positivity). Finally, it is possible to use that vertex algebra to give a new interpretation of CoHA modules made of cohomologies of non-commutative Hilbert schemes., Comment: 43 pages

The celebrated Diamond Lemma of Bergman gives an effectively verifiable criterion of uniqueness of normal forms for term rewriting in associative algebras. We present a new way to interpret and prove this result from the viewpoint of homotopical algebra. Our main result states that every multiplicative free resolution of an algebra with monomial relations gives rise to its own Diamond Lemma, so that Bergman's condition of "resolvable ambiguities" becomes the first non-trivial component of the Maurer--Cartan equation in the corresponding tangent complex. The same approach works for many other algebraic structures, emphasizing the relevance of computing multiplicative free resolutions of algebras with monomial relations., 28 pages, comments are welcome

Recent work of the first author and Tamaroff implies that the embedding of the Lie operad into the operad of pre-Lie algebras has the PBW property, and consequently, a functorial PBW type theorem holds for universal enveloping pre-Lie algebras of Lie algebras, generalising Segal's PBW theorem. We compute the corresponding Schur functor in terms of combinatorics of rooted trees.

We show how to use Groebner bases for operads to prove various freeness theorems: freeness of certain operads as nonsymmetric operads, freeness of an operad Q as a P-module for an inclusion P into Q, freeness of a suboperad. This gives new proofs of many known results of this type and helps to prove some new results., Comment: 15 pages, no figures, corrected typos and changed in parts the structure of the paper

We discuss a new notion of pattern avoidance motivated by the operad theory: pattern avoidance in planar labelled trees. It is a generalisation of various types of consecutive pattern avoidance studied before: consecutive patterns in words, permutations, coloured permutations etc. The notion of Wilf equivalence for patterns in permutations admits a straightforward generalisation for (sets of) tree patterns; we describe classes for trees with small numbers of leaves, and give several bijections between trees avoiding pattern sets from the same class. We also explain a few general results for tree pattern avoidance, both for the exact and the asymptotic enumeration., 27 pages, corrected various misprints, added an appendix explaining the operadic context

We present an implementation of the algorithm for computing Groebner bases for operads due to the first author and A. Khoroshkin. We discuss the actual algorithms, the choices made for the implementation platform and the data representation, and strengths and weaknesses of our approach., Comment: 18 pages, 6 figures

We have constructed the parafermionic chiral algebra with the principal parafermionic fields \Psi,\Psi^{+} having the conformal dimension \Delta_{\Psi}=8/3 and realizing the symmetry Z_{3}., Comment: 6 pages, no figure

We have considered a particular conformal model, called WD3, where the disorder effects are non trivial and can be studied analytically. In a renormalisation group approach, we have determined the infrared limit of this model with bond disorder and we have studied the associated problem of N coupled models. A particular attention has been paid to analyse the restauration of the permutational symmetry.Then, we have constructed the unknown representations of a class of parafermionic chiral algebras which realise the ciclic symmetry ZN. It is natural to associate the correspondent theories to the multi-critical points of the statistical models with symmetry ZN., Nous avons d'abord considere un modele theorique,appele modele WD3, ou les effets du desordre sont nontriviaux mais peuvent etre determines analytiquement. Enutilisant un calcul de groupe de renormalisation, nousavons determine la limite infrarouge du modele desordonne et etudie le probleme annexe de N modeles couples.Puis nous avons construit les representations d'une classe d' algebres de type parafermionique.L'hypothese la plus naturelle est que les theories correspondantes sont associees aux points multi-critiques auto duaux des systemes de spins avec symetrie Z_N.

This thesis concerns the analysis of two-dimensional systems with randomness using conformal field theory techniques. The first chapter is devoted to a brief review of different methods in conformal field theory. The second chapter is a presentation of different types of disordered systems which will be studied. One will found also a brief summary of know results. In the next chapters, we present applications of these methods to the case of Ising, Potts and multicritical models. Finaly in the last chapter we analyse the effect of disorder in some systems with a first order phase transition., Cette thèse a pour objet l'étude de la théorie des transitions de phases dans des systèmes désordonnés en dimension deux en utilisant les outils des théories conformes. Le premier chapitre est consacré a un exposé des différentes méthodes et techniques de calcul en théories conformes. Le deuxième chapitre est une présentation des différents types de systèmes désordonnés qui seront etudiés. On y trouvera aussi un bref resumé des résultats les plus connus dans l'étude de ces systèmes. L'application des méthodes de l'invariance conforme aux systèmes désordonnés se fera dans les chapitres 4, 5 et 6, ou l'on calcule les effets que produit un désordre faible dans les modèles d'Ising, de Potts et multicritiques respectivement. Finalement, dans le dernier chapitre, nous analysons les effets du désordre sur certains systèmes qui ont une transition de phases du premier ordre.