Maurer-Cartan Elements Research Articles

Deformations and homotopy theory for Rota–Baxter family algebras

The concept of Rota–Baxter family algebra is a generalization of Rota–Baxter algebra. It appears naturally in the algebraic aspects of renormalizations in quantum field theory. Rota–Baxter family algebras are closely related to dendriform family algebras. In this paper, we first construct an [Formula: see text]-algebra whose Maurer–Cartan elements correspond to Rota–Baxter family algebra structures. Using this characterization, we define the cohomology of a given Rota–Baxter family algebra. As an application of our cohomology, we study formal and infinitesimal deformations of a given Rota–Baxter family algebra. Next, we define the notion of a homotopy Rota–Baxter family algebra structure on a given [Formula: see text]-algebra. We end this paper by considering the homotopy version of dendriform family algebras and their relations with homotopy Rota–Baxter family algebras.

Crossed modules, non-abelian extensions of associative conformal algebras and wells exact sequences

In this paper, we introduce the notions of crossed modules of associative conformal algebras, two-term strongly homotopy associative conformal algebras, and discuss the relationship between them and the third Hochschild cohomology of associative conformal algebras. We classify the non-abelian extensions by introducing the non-abelian cohomology. We show that non-abelian extensions of an associative conformal algebra can be viewed as Maurer–Cartan elements of a suitable differential graded Lie algebra, and prove that there is a one-one correspondence between the Deligne groupoid of this differential graded Lie algebra and the non-abelian cohomology. For any two associative conformal algebras [Formula: see text] and [Formula: see text], we provide the condition that a pair of automorphisms in Aut(A) × Aut(B) can be extended to an automorphisms of the non-abelian extension algebra of [Formula: see text] by [Formula: see text], and give the fundamental sequence of Wells in the context of associative conformal algebras.

Deformations and abelian extensions of compatible pre-Lie algebras

In this paper, first, we give the notion of a compatible pre-Lie algebra and its representation. We study the relation between compatible Lie algebras and compatible pre-Lie algebras. We also construct a new bidifferential graded Lie algebra whose Maurer-Cartan elements are compatible pre-Lie structures. We give the bidifferential graded Lie algebra which controls deformations of a compatible pre-Lie algebra. Then, we introduce a cohomology of a compatible pre-Lie algebra with coefficients in itself. We study infinitesimal deformations of compatible pre-Lie algebras and show that equivalent infinitesimal deformations are in the same second cohomology group. We further give the notion of a Nijenhuis operator on a compatible pre-Lie algebra. We study formal deformations of compatible pre-Lie algebras. If the second cohomology group H2(g;g) is trivial, then the compatible pre-Lie algebra is rigid. Finally, we give a cohomology of a compatible pre-Lie algebra with coefficients in arbitrary representation and study abelian extensions of compatible pre-Lie algebras using this cohomology. We show that abelian extensions are classified by the second cohomology group.

The cohomology and deformations of [formula omitted]-operators on BiHom-associative algebras

We first generalize the cohomology of O-operators on BiHom-associative algebras by construct a graded Lie-algebra, in which the Maurer-Cartan elements are characterized by the given O-operator, and show that the cohomology represents the Hochschild cohomology of a certain BiHom-associative algebra with coefficients in a bimodule. Next, we study the linear and formal deformations of O-operators on BiHom-associative algebras, which are controlled by the Hochschild cohomology. Finally, as applications, we introduce the deformations of BiHom-associative r-matrices and infinitesimal BiHom-bialgebras on certain regular BiHom-associative algebras.

Cohomologies and deformations of relative Rota-Baxter operators on n-Lie superalgebras

In this paper, we characterize a relative Rota-Baxter operator on an n-Lie superalgebra with a representation (called an n-LSRep pair) as a Maurer-Cartan element of a certain Lie n-algebra. Then, we introduce the notions of n-pre-Lie superalgebras and symplectic n-Lie superalgebras and give some related results based on relative Rota-Baxter operators. In the next part, we give the cohomologies of relative Rota-Baxter operators on n-LSRep pairs and study infinitesimal deformations and extensions of finite order deformations of relative Rota-Baxter operators through the cohomology groups of relative Rota-Baxter operators. Finally, we build the relationship between the cohomology groups of relative Rota-Baxter operators on n-LSRep pairs and those on ( n + 1 ) -LSRep pairs by certain linear functions.

3-pre-Leibniz Algebras, Deformations and Cohomologies of Relative Rota-Baxter Operators on 3-Leibniz Algebras

In this paper, first we introduce the notions of 3-pre-Leibniz algebras and relative Rota-Baxter operators on 3-Leibniz algebras. We show that a 3-pre-Leibniz algebra gives rise to a 3-Leibniz algebra and a representation such that the identity map is a relative Rota-Baxter operator. Conversely, a relative Rota-Baxter operator naturally induces a 3-pre-Leibniz algebra. Then we construct a Lie 3-algebra, and characterize relative Rota-Baxter operators as its Maurer-Cartan elements. Consequently, we obtain the L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_\\infty$$\\end{document}-algebra that controls deformations of relative Rota-Baxter operators on 3-Leibniz algebras. Next we define the cohomology of relative Rota-Baxter operators on 3-Leibniz algebras and show that infinitesimal deformations of a relative Rota-Baxter operator are classified by the second cohomology group. Finally, we construct an L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_\\infty$$\\end{document}-algebra whose Maurer-Cartan elements are relative Rota-Baxter 3-Leibniz algebra structures, and define the cohomology of relative Rota-Baxter 3-Leibniz algebras.

Deformations, cohomologies and abelian extensions of compatible 3-Lie algebras

In this paper, first we give the notion of a compatible 3-Lie algebra and construct a bidifferential graded Lie algebra whose Maurer-Cartan elements are compatible 3-Lie algebras. We also obtain the bidifferential graded Lie algebra that governs deformations of a compatible 3-Lie algebra. Then we introduce a cohomology theory of a compatible 3-Lie algebra with coefficients in itself and show that there is a one-to-one correspondence between equivalent classes of infinitesimal deformations of a compatible 3-Lie algebra and the second cohomology group. We further study 2-order 1-parameter deformations of a compatible 3-Lie algebra and introduce the notion of a Nijenhuis operator on a compatible 3-Lie algebra, which could give rise to a trivial deformation. At last, we introduce a cohomology theory of a compatible 3-Lie algebra with coefficients in arbitrary representation and classify abelian extensions of a compatible 3-Lie algebra using the second cohomology group.

Deformations and cohomology theory of Ω-Rota-Baxter algebras of arbitrary weight

In this paper, we introduce the concepts of relative and absolute Ω-Rota-Baxter algebras of weight λ, which can be considered as a family algebraic generalization of relative and absolute Rota-Baxter algebras of weight λ. We study the deformations of relative and absolute Ω-Rota-Baxter algebras of arbitrary weight. Explicitly, we construct an L∞[1]-algebra via the method of higher derived brackets, whose Maurer-Cartan elements correspond to relative Ω-Rota-Baxter algebra structures of weight λ. For a relative Ω-Rota-Baxter algebra of weight λ, the corresponding twisted L∞[1]-algebra controls its deformations, which leads to the cohomology theory of it, and this cohomology theory can interpret the formal deformations of the relative Ω-Rota-Baxter algebra. Moreover, we also obtain the corresponding results for absolute Ω-Rota-Baxter algebras of weight λ from the relative version.

L ∞ -structures and cohomology theory of compatible O-operators and compatible dendriform algebras

The notion of O-operator is a generalization of the Rota–Baxter operator in the presence of a bimodule over an associative algebra. A compatible O-operator is a pair consisting of two O-operators satisfying a compatibility relation. A compatible O-operator algebra is an algebra together with a bimodule and a compatible O-operator. In this paper, we construct a graded Lie algebra and an L∞-algebra that respectively characterize compatible O-operators and compatible O-operator algebras as Maurer–Cartan elements. Using these characterizations, we define cohomology of these structures and as applications, we study formal deformations of compatible O-operators and compatible O-operator algebras. Finally, we consider a brief cohomological study of compatible dendriform algebras and find their relationship with the cohomology of compatible associative algebras and compatible O-operators.

Pre-Leibniz algebras

The notion of pre-Leibniz algebras was recently introduced in the study of Rota-Baxter operators on Leibniz algebras. In this paper, we first construct a graded Lie algebra whose Maurer-Cartan elements are pre-Leibniz algebras. Using this characterization, we define the cohomology of a pre-Leibniz algebra with coefficients in a representation. This cohomology is shown to split the Loday-Pirashvili cohomology of Leibniz algebras. As applications of our cohomology, we study formal and finite order deformations of a pre-Leibniz algebra. Finally, we define homotopy pre-Leibniz algebras and classify some special types of homotopy pre-Leibniz algebras.

Deformations and abelian extensions of compatible pre-Lie superalgebras

In this paper, we give cohomologies and deformations theory, as well as abelian extensions for compatible pre-Lie superalgebras. Explicitly, we first introduce the notation of a compatible pre-Lie superalgebra and its representation. We also construct a new bidfferential graded Lie superalgebra whose Maurer–Cartan elements are compatible pre-Lie structures. We give the bidifferential graded Lie superalgebra which controls deformations of a compatible pre-Lie superalgebra. Then, we introduce a cohomology of a compatible pre-Lie superalgebra with coefficients in itself. We study infinitesimal deformations of compatible pre-Lie superalgebras and show that equivalent infinitesimal deformations are in the same second cohomology group. We further give the notion of a Nijenhuis operator on a compatible pre-Lie algebra. We study formal deformations of compatible pre-Lie algebras. If the second cohomology group [Formula: see text] is trivial, then the compatible pre-Lie superalgebra is rigid. Additionally, we give a cohomology of a compatible pre-Lie superalgebra with coefficients in arbitrary representation and study abelian extensions of compatible pre-Lie superalgebras using this cohomology. We show that abelian extensions are classified by the second cohomology group. Finally, we study the relation between compatible Lie superalgebras and compatible pre-Lie superalgebras and we give some examples of compatible pre-Lie superalgebras.

Nijenhuis-operator on Hom-Lie conformal algebras

The aim of this paper is to study the Nijenhuis operators on Hom-Lie conformal algebras. We construct a graded Lie algebra whose Maurer-Cartan elements are given by Nijenhuis operators with the aid of Frölicher-Nijenhuis bracket. After constructing graded Lie algebra, we define the cohomology associated with a Nijenhuis operator. We further introduce Hom-NS Lie conformal algebra as an algebraic structure behind Nijenhuis operators on Hom-Lie conformal algebra. We provide various examples of Hom-NS Lie conformal algebras. Finally, as an application to cohomology, we study formal deformations of Nijenhuis operators.

On compatible Leibniz algebras

In this paper, we study compatible Leibniz algebras. We explore the classification of complex Leibniz algebras in dimensions 2 and 3 to provide compatible pairs. We characterize compatible Leibniz algebras in terms of Maurer–Cartan elements of a suitable differential graded Lie algebra. Moreover, we define a cohomology theory of compatible Leibniz algebras, which in particular controls one-parameter formal deformations of this algebraic structure. Motivated by a classical application of cohomology, we moreover study abelian extensions of compatible Leibniz algebras.

Embedding tensors on Lie ∞-algebras with respect to Lie ∞-actions

Given two Lie ∞ -algebras E and V, any Lie ∞ -action of E on V defines a Lie ∞ -algebra structure on E ⊕ V . Some compatibility between the action and the Lie ∞ -structure on V is needed to obtain a particular Loday ∞ -algebra, the non-abelian hemisemidirect product. These are the coherent actions. For coherent actions it is possible to define non-abelian homotopy embedding tensors as Maurer-Cartan elements of a convenient Lie ∞ -algebra. Generalizing the classical case, we see that a non-abelian homotopy embedding tensor defines a Loday ∞ -structure on V and is a morphism between this new Loday ∞ -algebra and E.

An introduction to L∞-algebras and their homotopy theory for the working mathematician

In this paper, we give a detailed introduction to the theory of (curved) [Formula: see text]-algebras and [Formula: see text]-morphisms, avoiding the concept of operads and providing explicit formulas. In particular, we recall the notion of (curved) Maurer–Cartan elements, their equivalence classes and the twisting procedure. The main focus is then the study of the homotopy theory of [Formula: see text]-algebras and [Formula: see text]-modules. In particular, one can interpret [Formula: see text]-morphisms and morphisms of [Formula: see text]-modules as Maurer–Cartan elements in certain [Formula: see text]-algebras, and we show that twisting the morphisms with equivalent Maurer–Cartan elements yields homotopic morphisms. We hope that these notes provide an accessible entry point to the theory of [Formula: see text]-algebras.

Twisting theory, relative Rota-Baxter type operators and L∞-algebras on Lie conformal algebras

Based on Nijenhuis-Richardson bracket and bidegree on the cohomology complex for a Lie conformal algebra, we develop a twisting theory of Lie conformal algebras. By using derived bracket constructions, we construct L∞-algebras from (quasi-)twilled Lie conformal algebras. And we show that the result of the twisting by a C[∂]-module homomorphism on a (quasi-)twilled Lie conformal algebra is also a (quasi-)twilled Lie conformal algebra if and only if the C[∂]-module homomorphism is a Maurer-Cartan element of the L∞-algebra. In particular, we show that relative Rota-Baxter type operators on Lie conformal algebras are Maurer-Cartan elements. Besides, we propose a new algebraic structure, called NS-Lie conformal algebras, that is closely related to twisted relative Rota-Baxter operators and Nijenhuis operators on Lie conformal algebras. As an application of twisting theory, we give the cohomology of twisted relative Rota-Baxter operators and study their deformations.

Maurer-Cartan characterizations and cohomologies of crossed homomorphisms on Lie triple systems

In this paper, first we give the notion of a crossed homomorphism on a Lie triple system with respect to an action on another Lie triple system. We also construct an -algebra using the higher derived brackets, whose Maurer-Cartan elements are crossed homomorphisms on Lie triple systems. Consequently, we obtain the twisted -algebra that controls deformations of a given crossed homomorphism on Lie triple systems. Next we construct a cohomology theory for a crossed homomorphism on Lie triple systems and classify infinitesimal deformations of crossed homomorphisms using the second cohomology group. Moreover we consider the relationship between cohomology groups of crossed homomorphisms on Lie algebras and associated Lie triple systems.

(Co)homology of compatible associative algebras

In this paper, we define and study (co)homology theories of a compatible associative algebra. At first, we construct a new graded Lie algebra whose Maurer-Cartan elements are given by compatible associative structures on a vector space. Then we define the cohomology of a compatible associative algebra A and as applications, we study extensions, deformations and extensibility of finite order deformations of A. We end this paper by considering compatible presimplicial vector spaces and the homology of compatible associative algebras.

Cohomologies and deformations of O-operators on Lie triple systems

In this paper, first, we provide a graded Lie algebra whose Maurer–Cartan elements characterize Lie triple system structures. Then, we use it to study cohomology and deformations of O-operators on Lie triple systems by constructing a Lie 3-algebra whose Maurer–Cartan elements are O-operators. Furthermore, we define a cohomology of an O-operator T as the Lie–Yamaguti cohomology of a certain Lie triple system induced by T with coefficients in a suitable representation. Therefore, we consider infinitesimal and formal deformations of O-operators from a cohomological viewpoint. Moreover, we provide relationships between O-operators on Lie algebras and associated Lie triple systems.

Cohomology and deformations of compatible Hom-Lie algebras

In this paper, we consider compatible Hom-Lie algebras as a twisted version of compatible Lie algebras. Compatible Hom-Lie algebras are characterized as Maurer-Cartan elements in a suitable bidifferential graded Lie algebra. We also define a cohomology theory for compatible Hom-Lie algebras generalizing the recent work of Liu, Sheng and Bai. As applications of cohomology, we study abelian extensions and deformations of compatible Hom-Lie algebras.