Academic literature on the topic 'Maurer-Cartan element'

A relative Rota–Baxter algebra is a triple ( A, M, T) consisting of an algebra A, an A-bimodule M, and a relative Rota–Baxter operator T. Using Voronov’s derived bracket and a recent work of Lazarev, Sheng, and Tang, we construct an L∞[1]-algebra whose Maurer–Cartan elements are precisely relative Rota–Baxter algebras. By a standard twisting, we define a new L∞[1]-algebra that controls Maurer–Cartan deformations of a relative Rota–Baxter algebra ( A, M, T). We introduce the cohomology of a relative Rota–Baxter algebra ( A, M, T) and study infinitesimal deformations in terms of this cohomology (in low dimensions). As an application, we deduce cohomology of triangular skew-symmetric infinitesimal bialgebras and discuss their infinitesimal deformations. Finally, we define homotopy relative Rota–Baxter operators and find their relationship with homotopy dendriform algebras and homotopy pre-Lie algebras.

AbstractIn a previous work, we associated a complete diòerential graded Lie algebra to any finite simplicial complex in a functorial way. Similarly, we also have a realization functor fromthe category of complete diòerential graded Lie algebras to the category of simplicial sets. We have already interpreted the homology of a Lie algebra in terms of homotopy groups of its realization. In this paper, we begin a dictionary between models and simplicial complexes by establishing a correspondence between the Deligne groupoid of the model and the connected components of the finite simplicial complex.

The purpose of this paper is to introduce and study twisted O-operators on 3-Lie algebras. We construct an L?-algebra whose Maurer-Cartan elements are twisted O-operators and define a cohomology of a twisted O-operator T as the Chevalley-Eilenberg cohomology of a certain 3-Lie algebra induced by T with coefficients in a suitable representation. Then we consider infinitesimal and formal deformations of twisted O-operators.

Weighted Rota–Baxter operators on associative algebras are closely related to modified Yang–Baxter equations, splitting of algebras, and weighted infinitesimal bialgebras and play an important role in mathematical physics. For any λ ∈ k, we construct a differential graded Lie algebra whose Maurer–Cartan elements are given by λ-weighted relative Rota–Baxter operators. Using such characterization, we define the cohomology of a λ-weighted relative Rota-Baxter operator T and interpret this as the Hochschild cohomology of a suitable algebra with coefficients in an appropriate bimodule. We study linear, formal, and finite order deformations of T from cohomological points of view. Among others, we introduce Nijenhuis elements that generate trivial linear deformations and define a second cohomology class to any finite order deformation, which is the obstruction to extend the deformation. In the end, we also consider the cohomology of λ-weighted relative Rota–Baxter operators in the Lie case and find a connection with the case of associative algebras.

<abstract><p>Uchino initiated the investigation of twisted Rota-Baxter operators on associative algebras. Relevant studies have been extensive in recent times. In this paper, we introduce the notion of a twisted Rota-Baxter operator on a Hom-Lie algebra. By utilizing higher derived brackets, we establish an explicit $ L_{\infty} $-algebra whose Maurer-Cartan elements are precisely twisted Rota-Baxter operators on Hom-Lie algebra s. Additionally, we employ Getzler's technique of twisting $ L_\infty $-algebras to establish the cohomology of twisted Rota-Baxter operators. We demonstrate that this cohomology can be regarded as the Chevalley-Eilenberg cohomology of a specific Hom-Lie algebra with coefficients in an appropriate representation. Finally, we study the linear and formal deformations of twisted Rota-Baxter operators by using the cohomology defined above. We also show that the rigidity of a twisted Rota-Baxter operator can be derived from Nijenhuis elements associated with a Hom-Lie algebra.</p></abstract>

Abstract Hodge correlators are complex numbers given by certain integrals assigned to a smooth complex curve. We show that they are correlators of a Feynman integral, and describe the real mixed Hodge structure on the pronilpotent completion of the fundamental group of the curve. We introduce motivic correlators, which are elements of the motivic Lie algebra and whose periods are the Hodge correlators. They describe the motivic fundamental group of the curve. We describe variations of real mixed Hodge structures on a variety by certain connections on the product of the variety by twistor plane. We call them twistor connections. In particular, we define the canonical period map on variations of real mixed Hodge structures. We show that the obtained period functions satisfy a simple Maurer–Cartan type non-linear differential equation. Generalizing this, we suggest a DG-enhancement of the subcategory of Saito’s Hodge complexes with smooth cohomology. We show that when the curve varies, the Hodge correlators are the coefficients of the twistor connection describing the corresponding variation of real MHS. Examples of the Hodge correlators include classical and elliptic polylogarithms, and their generalizations. The simplest Hodge correlators on the modular curves are the Rankin–Selberg integrals. Examples of the motivic correlators include Beilinson’s elements in the motivic cohomology, e.g. the ones delivering the Beilinson–Kato Euler system on modular curves.

A general group theoretic formulation of integrable systems is presented. The approach generalizes the discussion of the KdV equations of Segal and Wilson based on ideas of Sato. The starting point is the construction of commuting flows on the group via left multiplication with elements from an abelian subgroup. The initial data are then coded by elements, called abstract scattering data, in a certain coset space. The resulting equations of motion are then derived from a suitably formulated Maurer-Cartan equation (zero curvature condition) given an abstract Birkhoff factorization. The resulting equations of motion are of the Zakharov-Shabat type. In the case of flows periodic in x-space, the integrals of motion have a natural group theoretic interpretation. A first example is provided by the generalized nonlinear Schrödinger equation, first studied by Fordy and Kulish with integrals of motion which may be local or nonlocal. A suitable reduction gives the mKdV equations of Drinfeld and Sokolov. On the level of abstract scattering data the generalized Miura transformation from solutions of the mKdV equations to the KdV type equations is then just the canonical map from a coset space to a double coset space. This group theoretic approach is related to the algebraic geometric discussion of integrable systems via an affine map from the abelian group describing flows restricted to a suitable set of abstract scattering data, called algebraic geometric, onto a connected component of the Picard variety.

L'objectif de cette thèse est d'étudier les opérateurs de Rota-Baxter relatifs sur les algèbres ternaires de type Lie et de type Jordan. L'étude porte sur leur structure, leur cohomologie, leurs déformations et leur lien avec les équations de Yang-Baxter. Ce travail est divisé en trois parties. La première partie est consacrée à l'étude de l'algèbre de contrôle des systèmes triples de Lie, et à son application à la théorie existante de la cohomologie. De plus, nous introduisons la notion d'opérateur de Rota-Baxter relatif sur les systèmes triples de Lie et construisons une 3-algèbre de Lie comme cas spécial des L∞-algèbres dont les éléments de Maurer-Cartan sont des opérateurs de Rota-Baxter relatifs. Dans la deuxième partie, nous introduisons la notion d'opérateur de Rota-Baxter relatif twisté sur les algèbres 3-Lie et construisons une L∞-algèbre dont les éléments de Maurer-Cartan sont des opérateurs de Rota-Baxter relatifs twistés. Cela nous permet de définir la cohomologie de Chevalley-Eilenberg d'un opérateur de Rota-Baxter relatif twisté. Dans la dernière partie, nous étudions la représentation des algèbres ternaires de Jordan, ce qui nous permet d'introduire la notion d'algèbres ternaires de Jordan cohérentes. Ensuite, les opérateurs de Rota-Baxter relatifs des algèbres ternaires de Jordan sont introduits et les solutions de l'équation de Yang-Baxter de Jordan ternaire sont discutées en impliquant des opérateurs de Rota-Baxter relatifs
The goal of this thesis is to explore relative Rota-Baxter operators in the context of ternary algebras of both Lie and Jordan types. We mainly consider Lie triple systems, 3-Lie algebras and ternary Jordan algebras. The study covers their structure, cohomology, deformations, and their connection with the Yang-Baxter equations. The work is divided into three main parts. The first part aims first to introduce and study a graded Lie algebra whose Maurer-Cartan elements are Lie triple systems. It turns out to be the controlling algebra of Lie triple systems deformations and fits with the adjoint cohomology theory of Lie triple systems introduced by Yamaguti. In addition, we introduce the notion of relative Rota-Baxter operators on Lie triple systems and construct a Lie 3-algebra as a special case of L∞-algebras, where the Maurer-Cartan elements correspond to relative Rota-Baxter operators. In the second part, we introduce the concept of twisted relative Rota-Baxter operators on 3-Lie algebras and construct an L∞-algebra, where the Maurer-Cartan elements are twisted relative Rota-Baxter operators. This allows us to define the Chevalley-Eilenberg cohomology of a twisted relative Rota-Baxter operator. In the last part, we deal with a representation theory of ternary Jordan algebras. In particular, we introduce and discuss the concept of coherent ternary Jordan algebras. We then define relative Rota-Baxter operators for ternary Jordan algebras and discuss solutions ofthe ternary Jordan Yang-Baxter equation involving relative Rota-Baxter operators. Moreover, we investigate ternary pre-Jordan algebras as the underlying algebraic structure of relative Rota-Baxter operators

Cette thèse s’inscrit dans les thèmes de la théorie des opérades et de l’algèbre homotopique. Soient donnés un type d'algèbre, un type de cogèbres et une relation entre ces types de structures algébriques (codés respectivement par une opérade, une coopérade et un morphisme tordant). Il est possible alors de mettre une structure naturelle d’algèbre de Lie à homotopie près sur l’espace des applications linéaires d’une cogèbre C vers une algèbre A. On appelle l’algèbre de Lie `a homotopie près obtenue de cette fac¸on l’algèbre de convolution de A et C. Dans cette thèse, on étudie la théorie des algèbres de convolution et leur compatibilité avec les instruments de l’algèbre homotopique : les infini-morphismes et le théorème de transfert homotopique. Après avoir fait cela, on applique cette théorie à plusieurs domaines, comme la théorie de la déformation dérivée et la théorie de l’homotopie rationnelle. Dans le premier cas, on utilise les instruments développés en construisant une algèbre de Lie universelle qui représente l’espace des éléments de Maurer-Cartan, un objet fondamental de la théorie de la déformation. Dans le deuxième cas, on donne une généralisation d’un résultat de Berglund sur des modèles rationnels pour les espaces de morphismes entre deux espaces pointés
This thesis is inscribed in the topics of operad theory and homotopical algebra. Suppose we are given a type of algebras, a type of coalgebras, and a relationship between those types of algebraic structures (encoded by an operad, a cooperad, and a twisting morphism respectively). Then, it is possible to endow the space of linear maps from a coalgebra C and an algebra A with a natural structure of Lie algebra up to homotopy. We call the resulting homotopy Lie algebra the convolution algebra of A and C. In this thesis, we study the theory of convolution algebras and their compatibility with the tools of homotopical algebra : infinity morphisms and the homotopy transfer theorem. After doing that, we apply this theory to various domains, such as derived deformation theory and rational homotopy theory. In the first case, we use the tools we developed to construct an universal Lie algebra representing the space of Maurer-Cartan elements, a fundamental object of deformation theory. In the second case, we generalize a result of Berglund on rational models for mapping spaces between pointed topological spaces

This chapter illustrates the Maurer-Cartan form. On every Lie group G with Lie algebra g, there is a unique canonically defined left-invariant g-valued 1-form called the Maurer-Cartan form. The chapter describes the Maurer-Cartan form and the equation it satisfies, the Maurer-Cartan equation. The Maurer-Cartan form allows one to define a connection on the product bundle M × G → M for any manifold M. The Lie algebra g of a Lie group G is defined to be the tangent space at the identity. One will often identify the two vector spaces and think of elements of g as left-invariant vector fields on G.