Let $A$ be a unital commutative associative algebra over a field of characteristic zero, $\k$ be a Lie algebra, and $\z$ a vector space, considered as a trivial module of the Lie algebra $\g := A \otimes \k$. In this paper we give a description of the cohomology space $H^2(\g,\z)$ in terms of well accessible data associated to $A$ and $\k$. We also discuss the topological situation, where $A$ and $\k$ are locally convex algebras.
We study sympathetic Lie algebras, namely perfect and complete Lie algebras. They arise among other things in the study of adjoint Lie algebra cohomology. This is motivated by a conjecture of Pirashvili, which says that a non-trivial finite-dimensional complex perfect Lie algebra is semisimple if and only if its adjoint cohomology vanishes. We prove several results on sympathetic Lie algebras and the adjoint Lie algebra cohomology of Lie algebras in general, using the Hochschild-Serre formula. For certain semidirect products we obtain explicit results for the adjoint cohomology.
The paper studies the cohomology of Lie algebras and quadratic Lie algebras. Firstly, we propose to describe the cohomology of $MD(n,1)$-class which was introduced in \cite{LHNCN16}. This class contains Heisenberg Lie algebras. In 1983, L. J. Santharoubane \cite{San83} computed the cohomology of Heisenberg Lie algebras. In this paper, we will completely describe the cohomology of the other ones of $MD(n, 1)$-class. Finally, we will be concerned about the cohomology of quadratic Lie algebras. In 1985, A. Medina and P. Revoy \cite{MR85} computed the second Betti number of the generalized real di…
In this paper, we study Hom-Lie superalgebras of Heisenberg type. For 3-dimensional Heisenberg Hom-Lie superalgebras, we describe their Hom-Lie super structures, compute the cohomology spaces and characterize their infinitesimal deformations.
In this paper, we study the cohomology theory of Hom-Lie triple systems generalizing the Yamaguti cohomology theory of Lie triple systems. We introduce the central extension theory for Hom-Lie triple systems and show that there is a one-to-one correspondence between equivalent classes of central extensions of Hom-Lie triple systems and the third cohomology group. We develop the 1-parameter formal deformation theory of Hom-Lie triple systems, and prove that it is governed by the cohomology group.
The purpose of this paper is define the representation and the cohomology of n-ary-Nambu-Lie superalgebras. Morever we study extensions and provide the computation of the derivations and second cohomology group of super w_1 3-algebra
This paper defines compatible BiHom-Lie algebras by twisting the compatible Lie algebras by two linear commuting maps. We show the characterization of compatible BiHom-Lie algebra as a Maurer-Cartan element in a suitable bidifferential graded Lie algebra. We also define a cohomology theory for compatible BiHom-Lie algebras.
In this paper, all (super)algebras are over a field $\mathbb{F}$ of characteristic different from $2, 3$. We construct the so-called 5-sequences of cohomology for central extensions of a Lie superalgebra and prove that they are exact. Then we prove that the multipliers of a Lie superalgebra are isomorphic to the second cohomology group with coefficients in the trivial module for the Lie superalgebra under consideration.
Suppose the ground field $\mathbb{F}$ is an algebraically closed field of characteristic different from 2, 3. We determine the Betti numbers and make a decomposition of the associative superalgebra of the cohomology for the model filiform Lie superalgebra. We also describe the associative superalgebra structures of the (divided power) cohomology for some low-dimensional filiform Lie superalgebras.
We prove that the Gerstenhaber bracket on the Hochschild cohomology of the group algebra of a cyclic group over a field of positive characteristic is not trivial. In this case, we relate the Lie algebra structure on the odd degrees of the Hochschild cohomology with a Witt-type algebra.
In this paper, we introduce the notion of crossed homomorphisms between Lie-Yamaguti algebras and establish the cohomology theory of crossed homomorphisms via the Yamaguti cohomology. Consequently, we use this cohomology to characterize linear deformations of crossed homomorphisms between Lie-Yamaguti algebras. We show that if two linear or formal deformations of a crossed homomorphism are equivalent, then their infinitesimals are in the same cohomology class in the first cohomology group. Moreover, we show that an order $n$ deformation of a crossed homomorphism can be extended to an order $n+…
In this paper, we study the cup products and Betti numbers over cohomology superspaces of two-step nilpotent Lie superalgebras with coefficients in the adjoint modules over an algebraically closed field of characteristic zero. As an application, we prove that the cup product over the adjoint cohomology superspaces for Heisenberg Lie superalgebras is trivial and we also determine the adjoint Betti numbers for Heisenberg Lie superalgebras by means of Hochschild-Serre spectral sequences.
We show that the central representation is nontrivial for all one-dimensional central extensions of nilpotent Lie algebras possessing a codimension one abelian ideal.
A cohomology theory, associated to a $n$-Lie algebra and a representation space of it, is introduced. It is observed that this cohomology theory is qualified to encode the generalized derivation extensions, and that it coincides, for $n=3$, with the known cohomology of $n$-Lie algebras. The abelian extensions and infinitesimal deformations of $n$-Lie algebras, on the other hand, are shown to be characterized by the usual cohomology of $n$-Lie algebras. Furthermore, the Hochschild-Serre spectral sequence of the Lie algebra cohomology is upgraded to the level of $n$-Lie algebras, and is applied …
In this paper, we introduce the notion of a BiHom-Lie conformal algebra and develop its cohomology theory. Also, we discuss some applications to the study of deformations of regular BiHom-Lie conformal algebras. Finally, we introduce derivations of multiplicative BiHom-Lie conformal algebras and study their properties.
This paper aims to study the low dimensional cohomology of Hom-Lie algebras and q-deformed W(2,2) algebra. We show that the q-deformed W(2,2) algebra is a Hom-Lie algebra. Also, we establish a one-to-one correspondence between the equivalence classes of one dimensional central extensions of a Hom-Lie algebra and its second cohomology group, leading us to determine the second cohomology group of the q-deformed W(2,2) algebra. In addition, we generalize some results of derivations of finitely generated Lie algebras with values in graded modules to Hom-Lie algebras. As application we compute all …
In this paper, first we show that under the assumption of the center of h being zero, diagonal non-abelian extensions of a regular Hom-Lie algebra g by a regular Hom-Lie algebra h are in one-to-one correspondence with Hom-Lie algebra morphisms from g to Out(h). Then for a general Hom-Lie algebra morphism from g to Out(h), we construct a cohomology class as the obstruction of existence of a non-abelian extension that induce the given Hom-Lie algebra morphism.
Suppose the ground field $\mathbb{F}$ is an algebraically closed field characteristic of $p>2$. In this paper, we investigate the restricted cohomology theory of restricted Lie superalgebras. Algebraic interpretations of low dimensional restricted cohomology of restricted Lie superalgebra are given. We show that there is a family of restricted model filiform Lie superalgebra $L_{p,p}^λ$ structures parameterized by elements $λ\in \mathbb{F}^{p}.$ We explicitly describe both the $1$-dimensional ordinary and restricted cohomology superspaces of $L_{p,p}^λ$ with coefficients in the $1$-dimensional…
This paper is primarily concerned with the first cohomology of $\frak{gl}_{0|2}$ with coefficients in the generalized Witt Lie superalgebra, where $\frak{gl}_{0|2}$ is a subalgebra of the general linear Lie superalgebra. The derivations and inner derivations from $\frak{gl}_{0|2}$ into submodules of the generalized Witt Lie superalgebra are represented by matrices, respectively. Then the first cohomology of $\frak{gl}_{0|2}$ with coefficients in the generalized Witt Lie superalgebra is completely determined by matrices.
By definition, a quadratic Lie superalgebra is a Lie superalgebra endowed with a non-degenerate supersymmetric bilinear form which satisfies the even and invariant properties. In this paper we calculate all of the second cohomology group of elementary quadratic Lie superalgebras which have been classified in \cite{DU14} by applying the super-Poisson bracket on the super exterior algebra. Besides, we give the classification of 8-dimensional solvable quadratic Lie superalgebras having 6-dimensional indecomposable even part. The method is based on the double extension and classification results o…
Given a nilpotent Lie algebra $\mathfrak n$ we construct a spectral sequence which is derived from a filtration of its Chevalley-Eilenberg differential complex and converges to the Lie algebra cohomology of $\mathfrak n$. The limit of this spectral sequence gives a grading for the Lie algebra cohomology, except for the cohomology groups of degree 0, 1, $\dim \mathfrak n-1$ and $\dim\mathfrak n$ as we shall prove. We describe the spectral sequence associated to a nilpotent Lie algebra which is a direct sum of two ideals, one of them of dimension one, in terms of the spectral sequence of the co-…
In this paper, first we give the notion of a crossed homomorphism on a 3-Lie algebra with respect to an action on another 3-Lie algebra, and characterize it using a homomorphism from a Lie algebra to the semidirect product Lie algebra. We also establish the relationship between crossed homomorphisms and relative Rota-Baxter operators of weight 1 on 3-Lie algebras. Next we construct a cohomology theory for a crossed homomorphism on 3-Lie algebras and classify infinitesimal deformations of crossed homomorphisms using the second cohomology group. Finally, using the higher derived brackets, we con…
In this paper, we first discuss cohomology and a one-parameter formal deformation theory of Lie-Yamaguti algebras. Next, we study finite group actions on Lie-Yamaguti algebras and introduce equivariant cohomology for Lie-Yamaguti algebras equipped with group actions. Finally, we study an equivariant one-parameter formal deformation theory and show that our equivariant cohomology is the suitable deformation cohomology.
We study representations of hemistrict Lie 2-algebras and give a functorial construction of their cohomology. We prove that both the cohomology of an injective hemistrict Lie 2-algebra $L$ and the cohomology of the semistrict Lie 2-algebra obtained from skew-symmetrization of $L$ are isomorphic to the Chevalley-Eilenberg cohomology of the induced Lie algebra $L_{\operatorname{Lie}}$.
A morphism Lie algebra is a triple $(\mathfrak{g}, \mathfrak{h}, ϕ)$ consisting of two Lie algebras $\mathfrak{g}, \mathfrak{h}$ and a Lie algebra homomorphism $ϕ: \mathfrak{g} \rightarrow \mathfrak{h}$. We define representations and cohomology of morphism Lie algebras. As applications of our cohomology, we study some aspects of deformations, abelian extensions of morphism Lie algebras and classify skeletal morphism sh Lie algebras. Finally, we consider the cohomology of morphism Lie groups and find a relation with the cohomology of morphism Lie algebras.
This note presents a general theorem about the cohomology of finite dimensional Lie algebras of arbitrary characteristic. As an application we compute the cohomology of the Borel subalgebra of sl(N).
In this paper, we define representations and cohomology of weighted Rota-Baxter Lie algebras. As applications of cohomology, we study abelian extensions and formal $1$-parameter deformations weighted Rota-Baxter Lie algebras. Finally, we consider weighted Rota-Baxter paired operators that induces a weighted Rota-Baxter Lie algebra and a representation of it. We define suitable cohomology for such paired operators that govern deformation.
The purpose of this paper is to define an $α$-type cohomology, which we call $α$-type Chevalley-Eilenberg cohomology, for Hom-Lie algebras. We relate it to the known Chevalley-Eilenberg cohomology and provide explicit computations for some examples. Moreover, using this cohomology we study formal deformations of Hom-Lie algebras, where the bracket as well as the structure map $α$ are deformed. Furthermore, we provide a generalization of the Grand Crochet and study, in a particular case, the $α$-type cohomology for Hom-Lie bialgebras.
We compute the cohomology with trivial coefficients of two graded infinite-dimensional Lie algebras of maximal class, give explicit formulas for their representative cocycles. Also we discuss the relations with combinatorics and representation theory.
The aim of this paper is to study the cohomology of Hom-Leibniz superalgebras. We construct the $q$-deformed Heisenberg-Virasoro superalgebra of Hom-type and provide as application the computations of the derivations and second cohomology group. Moreover, we extend to graded case the Takhtajan's construction of a cohomology of $n$-ary Hom-Nambu-Lie algebras starting from cohomology of Hom-Leibniz algebras.
We develop the cohomology theory of color Lie superalgebras due to Scheunert--Zhang in a framework of nonhomogeneous quadratic Koszul algebras. In this approach, the Chevalley--Eilenberg complex of a color Lie algebra becomes a standard Koszul complex for its universal enveloping algebra. As an application, we calculate cohomologies with trivial coefficients of 3-dimensional color Lie superalgebras.
In this paper, we establish the cohomology of relative Rota-Baxter operators on Lie-Yamaguti algebras via the Yamaguti cohomology. Then we use this type of cohomology to characterize deformations of relative Rota-Baxter operators on Lie-Yamaguti algebras. We show that if two linear or formal deformations of a relative Rota-Baxter operator are equivalent, then their infinitesimals are in the same cohomology class in the first cohomology group. Moreover, an order $n$ deformation of a relative Rota-Baxter operator can be extended to an order $n+1$ deformation if and only if the obstruction class …
The main result of this work is a new proof and generalization of Lazard's comparison theorem of locally analytic group cohomology with Lie algebra cohomology for K-Lie groups, where K is a finite extension of the p-adic numbers. We show the following theorem: Let K be a finite extension of the p-adic numbers and let G be a K-Lie group. Then there exists an open subgroup U of G such that the Lazard morphism, which is induced by differentiating cochains, is an isomorphism. The proof of this theorem is independent of the proof of Lazard's comparison result. Our strategy to prove the comparison i…
The aim of this paper is to provide cohomologies of $n$-ary Hom-Nambu-Lie algebras governing central extensions and one parameter formal deformations. We generalize to $n$-ary algebras the notions of derivations and representation introduced by Sheng for Hom-Lie algebras. Also we show that a cohomology of $n$-ary Hom-Nambu-Lie algebras could be derived from the cohomology of Hom-Leibniz algebras.
A LieYRep pair consists of a Lie-Yamaguti algebra and its representation. In this paper, we establish the cohomology theory of LieYRep pairs and characterize their linear deformations by the second cohomology group. Then we introduce the notion of relative Rota-Baxter-Nijenhuis structures on LieYRep pairs, investigate their properties, and prove that a relative Rota-Baxter-Nijenhuis structure gives rise to a pair of compatible relative Rota-Baxter operators under a certain condition. Finally, we show the equivalence between $r$-matrix-Nijenhuis structures and Rota-Baxter-Nijenhuis structures o…
We introduce the concept of 3-Lie-Rinehart superalgebra and systematically describe a cohomology complex by considering coefficient modules. Furthermore, we study the relationships between a Lie-Rinehart superalgebra and its induced 3-Lie-Rinehart superalgebra. The deformations of 3-Lie-Rinehart superalgebra are considered via the cohomology theory.
We discuss Massey products in a N-graded Lie algebra cohomology. One of the main examples is the positive part L_1 of the Witt algebra $W$. We consider an associated graded algebra m_0 of L_1 with respect to the descending central series and prove that H*(m_0) is generated with respect to non-trivial Massey products by one cohomology H^1(m_0).
We give an interpretation of the Brauer group of a purely inseparable extension of exponent 1, in terms of restricted Lie-Rinehart cohomology. In particular, we define and study the category $p$-$\rm{LR}(A)$ of restricted Lie-Rinehart algebras over a commutative algebra $A$. We define cotriple cohomology groups $H_{p-LR}(L,M)$ for $L\in p$-$\rm{LR}(A)$ and $M$ a Beck $L$-module. We classify restricted Lie-Rinehart extensions. Thus, we obtain a classification theorem for regular extensions considered by Hoshschild.
We develop a cohomology theory for Jordan triples, including the infinite dimensional ones, by means of the cohomology of TKK Lie algebras. This enables us to apply Lie cohomological results to the setting of Jordan triples. Some preliminary results for von Neumann algebras are obtained.
We compute explicitly the adjoint cohomology of two N-graded Lie algebras of maximal class (infinite dimensional filiform Lie algebras) m_0 and m_2. It is known that up to an isomorphism there are only three N-graded Lie algebras of the maximal class. The third algebra from this list is the "positive" part L_1 of the Witt (or Virasoro) algebra and its adjoint cohomology was computed earlier by Feigin and Fukhs. We show that the total space H*(m_j,m_j) is "almost" isomorphic to the completed tensor product of the algebra m_j by scalar cohomology space H^*(m_j), j=0,2.
The main purpose of this paper is to define representations and a cohomology of color Hom-Lie algebras and to study some key constructions and properties. We describe Hartwig-Larsson-Silvestrov Theorem in the case of $Γ$-graded algebras, study one-parameter formal deformations, discuss $α^{k}$-generalized derivation and provide examples.
In this dissertation, we investigate the cohomology theory of restricted Lie algebras. The representation theory of restricted Lie algebras is reviewed including a description of the restricted universal enveloping algebra. In the case of an abelian restricted Lie algebra, we construct an augmented complex of free modules over the enveloping algebra that is exact in dimensions less than p and hence define the cohomology theory of these algebras in dimension less than p. In the non-abelian case, we explicitly construct cochain spaces for any coefficient module in dimensions less than 3, and giv…
This paper is devoted to studying deformation, cohomology theory of Rota-Baxter pre-Lie algebras of arbitrary weights. First we give the notion of a new representation of a Rota-Baxter pre-Lie algebra of arbitrary weight and define the cohomology theory of a Rota-Baxter pre-Lie algebra of arbitrary weight. Then we study formal deformations by lower degree cohomology groups. Finally, we classify abelian extensions of Rota-Baxter pre-Lie algebras of arbitrary weight using the second cohomology group, and classify skeletal Rota-Baxter pre-Lie 2-algebra of arbitrary weight using the third cohomolo…
We consider the extension problem for Lie algebroids over schemes over a field. Given a locally free Lie algebroid Q over a scheme (X,O), and a sheaf of finitely generated Lie O-algebras L, we determine the obstruction to the existence of extensions 0 --> L --> E --> Q --> 0, and classify the extensions in terms of a suitable Lie algebroid hypercohomology group. In the preliminary sections we study free Lie algebroids and recall some basic facts about Lie algebroid hypercohomology.
In this paper, we introduce the notion of BiHom-Lie conformal superalgebras. We develop its representation theory and define the cohomology group with coefficients in a module. Finally, we introduce conformal derivations of BiHom-Lie conformal superalgebras and study some of their properties.
We introduce the notion of compatible actions in the context of split extensions of finite dimensional Lie algebras over a field. Using compatible actions, we construct a new resolution to compute the cohomology of semi-direct products of Lie algebras. We also give an alternative way to construct the Hochschild-Serre spectral sequence associated to a split extension of finite dimensional Lie algebras and obtain a sharper bound for the length of this spectral sequence.
We compare by a very elementary approach the second adjoint and trivial Leibniz cohomology spaces of a Lie algebra to the usual ones. Examples are given of coupled cocycles. Some properties are deduced as to Leibniz deformations. We also consider the class of Lie algebras for which the Koszul 3-form is zero, and prove that it contains all quotients of Borel subalgebras, or of their nilradicals, of finite dimensional semisimple Lie algebras. Finally, a list of Kac-Moody types for indecomposable nilpotent Lie algebras of dimension $\leqslant 7$ is given.
