Differential Graded Lie Algebra Research Articles

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.

Multi-derivations of Poisson algebras

The purpose of this paper is to study multi-derivations of Poisson algebras. First, we show that the space of multi-derivations of a Lie algebra [Formula: see text] with the Nijenhuis–Richardson bracket is a differential graded Lie algebra. Then we introduce the notion of a multi-derivation of a Poisson algebra, and show that the space of multi-derivations of a Poisson algebra with the Nijenhuis–Richardson bracket is also a graded Lie algebra. Finally, we study multi-derivations of three concrete Poisson algebras coming from linear Poisson manifolds, symplectic manifolds and Poisson manifolds with vanishing first cohomology groups. We establish the relationship between multi-derivations of a Lie algebra [Formula: see text] and multi-derivations of the linear Poisson manifold [Formula: see text], and show that a two-derivation of a connected symplectic manifold [Formula: see text] is [Formula: see text], where [Formula: see text] is the Poisson bivector field induced by the symplectic structure [Formula: see text] and [Formula: see text] is a real number. We also prove that a two-derivation of a connected Poisson manifold [Formula: see text] with trivial Poisson cohomology group [Formula: see text] is [Formula: see text], where [Formula: see text] is a Casimir function.

Tensor Products and Crossed Differential Graded Lie Algebras in the Category of Crossed Complexes

The study of algebraic structures endowed with the concept of symmetry is made possible by the link between Lie algebras and symmetric monoidal categories. This relationship between Lie algebras and symmetric monoidal categories is useful and has resulted in many areas, including algebraic topology, representation theory, and quantum physics. In this paper, we present analogous definitions for Lie algebras within the framework of whiskered structures, bimorphisms, crossed complexes, crossed differential graded algebras, and tensor products. These definitions, given for groupoids in existing literature, have been adapted to establish a direct correspondence between these algebraic structures and Lie algebras. We show that a 2-truncation of the crossed differential graded Lie algebra, obtained from our adapted definitions, gives rise to a braided crossed module of Lie algebras. We also construct a functor to simplicial Lie algebras, enabling a systematic mapping between different Lie algebraic categories, which supports the validity of our adapted definitions and establishes their compatibility with established categories.

English

In the formulation of his celebrated Formality conjecture, M. Kontsevich introduced a universal version of the deformation theory for the Schouten algebra of polyvector fields on affine manifolds. This universal deformation complex takes the form of a differential graded Lie algebra of graphs, denoted $\mathsf{fGC}_2$, together with an injective morphism towards the Chevalley-Eilenberg complex associated with the Schouten algebra. The latter morphism is given by explicit local formulas making implicit use of the supergeometric interpretation of the Schouten algebra as the algebra of functions on a graded symplectic manifold of degree $1$. The ambition of the present work is to generalise Kontsevich's construction to graded symplectic manifolds of arbitrary degree $n\geq1$. The corresponding graph model is given by the full Kontsevich graph complex $\mathsf{fGC}_d$ where $d=n+1$ stands for the dimension of the associated AKSZ type $\sigma$-model. This generalisation is instrumental to classify universal structures on graded symplectic manifolds. In particular, the zeroth cohomology of the full graph complex $\mathsf{fGC}_{d}$ is shown to act via $\mathsf{Lie}_\infty$-automorphisms on the algebra of functions on graded symplectic manifolds of degree $n$. This generalises the known action of the Grothendieck-Teichm\"{u}ller algebra $\mathfrak{grt}_1\simeq H^0(\mathsf{fGC}_2)$ on the space of polyvector fields. This extended action can in turn be used to generate new universal deformations of Hamiltonian functions, generalising Kontsevich flows on the space of Poisson manifolds to differential graded manifolds of higher degrees. As an application of the general formalism, universal deformations of Courant algebroids via trivalent graphs are presented.

Lie n-algebras and cohomologies of relative Rota-Baxter operators on n-Lie algebras

Based on the differential graded Lie algebra controlling deformations of an n-Lie algebra with a representation (called an n-LieRep pair), we construct a Lie n-algebra, whose Maurer-Cartan elements characterize relative Rota-Baxter operators on n-LieRep pairs. The notion of an n-pre-Lie algebra is introduced, which is the underlying algebraic structure of the relative Rota-Baxter operator. We give the cohomology of relative Rota-Baxter operators and study infinitesimal deformations and extensions of order m deformations to order m+1 deformations of relative Rota-Baxter operators through the cohomology groups of relative Rota-Baxter operators. Moreover, we build the relation between the cohomology groups of relative Rota-Baxter operators on n-LieRep pairs and those on (n+1)-LieRep pairs by certain linear functions.

Homotopy Rota–Baxter operators and post-Lie algebras

Rota–Baxter operators and the more general $\mathcal{O}$-operators, together with their interconnected pre-Lie and post-Lie algebras, are important algebraic structures, with Rota–Baxter operators and pre-Lie algebras instrumental in the Connes–Kreimer approach to renormalization of quantum field theory. This paper introduces the notions of a homotopy Rota–Baxter operator and a homotopy $\mathcal{O}$-operator on a symmetric graded Lie algebra. Their characterization by Maurer–Cartan elements of suitable differential graded Lie algebras is provided. Through the action of a homotopy $\mathcal{O}$-operator on a symmetric graded Lie algebra, we arrive at the notion of an operator homotopy post-Lie algebra, together with its characterization in terms of Maurer–Cartan elements. A cohomology theory of post-Lie algebras is established, with an application to 2-term skeletal operator homotopy post-Lie algebras.

Nonabelian embedding tensors

In this paper, first we introduce the notion of a nonabelian embedding tensor, which is a nonabelian generalization of an embedding tensor. Then, we introduce the notion of a Leibniz–Lie algebra, which is the underlying algebraic structure of a nonabelian embedding tensor, and can also be viewed as a nonabelian generalization of a Leibniz algebra. Next using the derived bracket, we construct a differential graded Lie algebra, whose Maurer–Cartan elements are exactly nonabelian embedding tensors. Consequently, we obtain the differential graded Lie algebra that governs deformations of a nonabelian embedding tensor. Finally, we define the cohomology of a nonabelian embedding tensor and use the second cohomology group to characterize linear deformations.

From Lie algebra crossed modules to tensor hierarchies

The present paper, though inspired by the use of tensor hierarchies in theoretical physics, establishes their mathematical credentials, especially as genetically related to Lie algebra crossed modules. Gauging procedures in supergravity rely on a pairing – the embedding tensor – between a Leibniz algebra and a Lie algebra. Two such algebras, together with their embedding tensor, form a triple called a Lie-Leibniz triple, of which Lie algebra crossed modules are particular cases. This paper is devoted to showing that any Lie-Leibniz triple induces a differential graded Lie algebra – its associated tensor hierarchy – whose restriction to the category of Lie algebra crossed modules is the canonical assignment associating to any Lie algebra crossed module its corresponding unique 2-term differential graded Lie algebra. This shows that Lie-Leibniz triples form natural generalizations of Lie algebra crossed modules and that their associated tensor hierarchies can be considered as some kind of ‘lie-ization’ of the former. We deem the present construction of such tensor hierarchies clearer and more straightforward than previous derivations. We stress that such a construction suggests the existence of further well-defined Leibniz gauge theories.

Non-abelian extensions of pre-Lie algebras

In this paper, we study non-abelian extensions of pre-Lie algebras. First we introduce the non-abelian cohomology for pre-Lie algebras by which we classify non-abelian extensions of pre-Lie algebras. Then we construct the bimultipliers for pre-Lie algebras, which naturally gives rise to a strict Lie 2-algebra. We show that there is a one-to-one correspondence between isomorphic classes of non-abelian extensions of pre-Lie algebras and homotopy classes of homomorphisms from the subadjacent Lie algebra to the strict Lie 2-algebra obtained from the bimultipliers. Finally we classify non-abelian extensions of pre-Lie algebras by Maurer-Cartan elements of a differential graded Lie algebra.

The strong homotopy structure of Poisson reduction

In this paper, we propose a reduction scheme for multivector fields phrased in terms of $L\_\infty$-morphisms. Using well-known geometric properties of the reduced manifolds, we perform a Taylor expansion of multivector fields, which allows us to build up a suitable deformation retract of differential graded Lie algebras (DGLAs). We first obtained an explicit formula for the $L\_\infty$-projection and -inclusion of generic DGLA retracts. We then applied this formula to the deformation retract that we constructed in the case of multivector fields on reduced manifolds. This allows us to obtain the desired reduction $L\_\infty$-morphism. Finally, we perform a comparison with other reduction procedures.

Cohomology and deformations of weighted Rota–Baxter 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.

Nonlinear realizations of Lie superalgebras

For any decomposition of a Lie superalgebra into a direct sum of a subalgebra and a subspace without any further resctrictions on and we construct a nonlinear realization of on The result generalizes a theorem by Kantor from Lie algebras to Lie superalgebras. When is a differential graded Lie algebra, we show that it gives a construction of an associated -algebra.

Amplitudes in YM and GR as a Minimal Model and Recursive Characterization

We study the tree scattering amplitudes of Yang–Mills and General Relativity as functions of complex momenta, using a homological and geometrical approach. This approach uses differential graded Lie algebras, one for YM and one for GR, whose Maurer Cartan equations are the classical field equations. The tree amplitudes are obtained as the L_infty minimal model brackets, given by a trivalent Feynman tree expansion. We show that they are sections of a sheaf on the complex variety of momenta, and that their residues factor in a characteristic way. This requires classifying the irreducible codimension one subvarieties where poles occur; constructing dedicated gauges that make the factorization manifest; and proving a flexible version of gauge independence to be able to work with different gauges. The residue factorization yields a simple recursive characterization of the tree amplitudes of YM and GR, by exploiting Hartogs’ phenomenon for singular varieties. This is similar to and inspired by Britto–Cachazo–Feng–Witten recursion, but avoids BCFW’s trick of shifting momenta, hence avoids conditions at infinity under such shifts.

Lie models of homotopy automorphism monoids and classifying fibrations

Given X a finite nilpotent simplicial set, consider the classifying fibrationsX→BautG⁎(X)→BautG(X)andX→Z→Bautπ⁎(X) where G and π denote, respectively, subgroups of the free and pointed homotopy classes of free and pointed self homotopy equivalences of X which act nilpotently on H⁎(X) and π⁎(X). We give algebraic models, in terms of complete differential graded Lie algebras (cdgl's), of the rational homotopy type of these fibrations. Explicitly, if L is a cdgl model of X, there are connected sub cdgl's DerGL and DerΠL of the Lie algebra of derivations of L such that the geometrical realizations of the sequences of cdgl morphismsL→adDerGL→DerGLטsLandL→LטDerΠL→DerΠL have the rational homotopy type of the above classifying fibrations. Among the consequences we also describe in cdgl terms the Malcev Q-completion of G and π together with the rational homotopy type of the classifying spaces BG and Bπ.

DIFFERENTIAL GRADED LIE GROUPS AND THEIR DIFFERENTIAL GRADED LIE ALGEBRAS

In this paper we discuss the question of integrating differential graded Lie algebras (DGLA) to differential graded Lie groups (DGLG). We first recall the classical problem of integration in the context, and recollect the known results. Then, we define the category of differential graded Lie groups and study its properties. We show how to associate a differential graded Lie algebra to every differential graded Lie group and vice versa. For the DGLA → DGLG direction, the main “tools” are graded Hopf algebras and Harish-Chandra pairs (HCP)—we define the category of graded and differential graded HCPs and explain how those are related to the desired construction. We describe some near-at-hand examples and mention possible generalizations.

Cohomology and deformations of Filippov algebroids

In this article, we study the deformations of Filippov algebroids. First, we define a differential graded Lie algebra for a Filippov algebroid by introducing the notion of Filippov multiderivations for a vector bundle. We then discuss deformations of a Filippov algebroid in terms of low-dimensional cohomology associated with this differential graded Lie algebra. We define Nijenhuis operators on Filippov algebroids and characterize trivial deformations of Filippov algebroids in terms of these operators. Finally, we define finite order deformations and discuss the problem of extending a given finite order deformation to a deformation of a higher order.

Batalin-Vilkovisky formality for Chern-Simons theory

We prove that the differential graded Lie algebra of functionals associated to the Chern-Simons theory of a semisimple Lie algebra is homotopy abelian. For a general field theory, we show that the variational complex in the Batalin-Vilkovisky formalism is a differential graded Lie algebra.

Duality Hierarchies and Differential Graded Lie Algebras

The gauge theories underlying gauged supergravity and exceptional field theory are based on tensor hierarchies: generalizations of Yang-Mills theory utilizing algebraic structures that generalize Lie algebras and, as a consequence, require higher-form gauge fields. Recently, we proposed that the algebraic structure allowing for consistent tensor hierarchies is axiomatized by ‘infinity-enhanced Leibniz algebras’ defined on graded vector spaces generalizing Leibniz algebras. It was subsequently shown that, upon appending additional vector spaces, this structure can be reinterpreted as a differential graded Lie algebra. We use this observation to streamline the construction of general tensor hierarchies, and we formulate dynamics in terms of a hierarchy of first-order duality relations, including scalar fields with a potential.

Formality conjecture for minimal surfaces of Kodaira dimension 0

Let $\mathcal {F}$ be a polystable sheaf on a smooth minimal projective surface of Kodaira dimension 0. Then the differential graded (DG) Lie algebra $R\operatorname {Hom}(\mathcal {F},\mathcal {F})$ of derived endomorphisms of $\mathcal {F}$ is formal. The proof is based on the study of equivariant $L_{\infty }$ minimal models of DG Lie algebras equipped with a cyclic structure of degree 2 which is non-degenerate in cohomology, and does not rely (even for K3 surfaces) on previous results on the same subject.

L ∞ -algebras of Einstein–Cartan–Palatini gravity

We give a detailed account of the cyclic L∞-algebra formulation of general relativity with a cosmological constant in the Einstein–Cartan–Palatini formalism on spacetimes of arbitrary dimension and signature, which encompasses all symmetries, field equations, and Noether identities of gravity without matter fields. We present a local formulation as well as a global covariant framework, and an explicit isomorphism between the two L∞-algebras in the case of parallelizable spacetimes. By duality, we show that our L∞-algebras describe the complete Batalin-Vilkovisky-Becchi-Rouet-Stora-Tyutin formulation of Einstein–Cartan–Palatini gravity. We give a general description of how to extend on-shell redundant symmetries in topological gauge theories to off-shell correspondences between symmetries in terms of quasi-isomorphisms of L∞-algebras. We use this to extend the on-shell equivalence between gravity and Chern–Simons theory in three dimensions to an explicit L∞-quasi-isomorphism between differential graded Lie algebras, which applies off-shell and for degenerate dynamical metrics. In contrast, we show that there is no morphism between the L∞-algebra underlying gravity and the differential graded Lie algebra governing BF theory in four dimensions.