Chevalley-Eilenberg Cohomology Research Articles

Cohomolgies of Rota-Baxter operators on Lie superalgebras and some classifications on Witt superalgebras

In this paper, we introduce the cohomology of Rota-Baxter operators on Lie superalgebras. We define the cohomology of a Rota-Baxter operator R as the Chevalley Eilenberg cohomology of a certain Lie superalgebra induced by R with coefficients in a suitable representation. Then, we give the classification of the homogeneous Rota-Baxter operators of weight 0 on the Witt superalgebras. To apply our approach, we construct pre-Lie superalgebras from the homogeneous Rota-Baxter operators of weight 0 and their subadjacent Lie superalgebras.

Cohomology and deformations of twisted O-operators on 3-Lie algebras

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.

Twisted Rota-Baxter operators on Hom-Lie 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>

Twisted Rota–Baxter operators and Reynolds operators on Lie algebras and NS-Lie algebras

In this paper, we introduce twisted Rota–Baxter operators on Lie algebras as an operator analog of twisted r-matrices. We construct a suitable L∞-algebra whose Maurer–Cartan elements are given by twisted Rota–Baxter operators. This allows us to define the cohomology of a twisted Rota–Baxter operator. This cohomology can be seen as the Chevalley–Eilenberg cohomology of a certain Lie algebra with coefficients in a suitable representation. We study deformations of twisted Rota–Baxter operators from cohomological points of view. Some applications are given to Reynolds operators and twisted r-matrices. Next, we introduce a new algebraic structure, called NS-Lie algebras, that is related to twisted Rota–Baxter operators in the same way pre-Lie algebras are related to Rota–Baxter operators. We end this paper by considering twisted generalized complex structures on modules over Lie algebras.

Characteristic cohomology and observables in higher spin gravity

We give a complete classification of dynamical invariants in 3d and 4d Higher Spin Gravity models, with some comments on arbitrary d. These include holographic correlation functions, interaction vertices, on-shell actions, conserved currents, surface charges, and some others. Surprisingly, there are a good many conserved p-form currents with various p. The last fact, being in tension with ‘no nontrivial conserved currents in quantum gravity’ and similar statements, gives an indication of hidden integrability of the models. Our results rely on a systematic computation of Hochschild, cyclic, and Chevalley-Eilenberg cohomology for the corresponding higher spin algebras. A new invariant in Chern-Simons theory with the Weyl algebra as gauge algebra is also presented.

On Leibniz cohomology

In this paper we prove the Leibniz analogue of Whitehead's vanishing theorem for the Chevalley-Eilenberg cohomology of Lie algebras. As a consequence, we obtain the second Whitehead lemma for Leibniz algebras. Moreover, we compute the cohomology of several Leibniz algebras with adjoint or irreducible coefficients. Our main tool is a Leibniz analogue of the Hochschild-Serre spectral sequence, which is an extension of the dual of a spectral sequence of Pirashvili for Leibniz homology from symmetric bimodules to arbitrary bimodules.

Commutative Lie algebras and commutative cohomology in characteristic 2

We discuss a version of the Chevalley–Eilenberg cohomology in characteristic [Formula: see text], where the alternating cochains are replaced by symmetric ones.

Differential Graded Lie Algebras and Leibniz Algebra Cohomology

Abstract In this note, we interpret Leibniz algebras as differential graded (DG) Lie algebras. Namely, we consider two fully faithful functors from the category of Leibniz algebras to that of DG Lie algebras and show that they naturally give rise to the Leibniz cohomology and the Chevalley–Eilenberg cohomology. As an application, we prove a conjecture stated by Pirashvili in [ 9].

Cohomology of hemistrict Lie 2-algebras

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 Communicated by Jason P. Bell

α-TYPE CHEVALLEY–EILENBERG COHOMOLOGY OF HOM-LIE ALGEBRAS AND BIALGEBRAS

AbstractThe 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.

Deformations and Their Controlling Cohomologies of $${\mathcal{O}}$$ O -Operators

$${\mathcal{O}}$$ -operators are important in broad areas of mathematics and physics, such as integrable systems, the classical Yang–Baxter equation, pre-Lie algebras and splitting of operads. In this paper, a deformation theory of $${\mathcal{O}}$$ -operators is established in consistence with the general principles of deformation theories. On the one hand, $${\mathcal{O}}$$ -operators are shown to be characterized as the Maurer–Cartan elements in a suitable graded Lie algebra. A given $${\mathcal{O}}$$ -operator gives rise to a differential graded Lie algebra whose Maurer–Cartan elements characterize deformations of the given $${\mathcal{O}}$$ -operator. On the other hand, a Lie algebra with a representation is identified from an $${\mathcal{O}}$$ -operator T such that the corresponding Chevalley–Eilenberg cohomology controls deformations of T, thus can be regarded as an analogue of the Andre–Quillen cohomology for the $${\mathcal{O}}$$ -operator. Thereafter, linear and formal deformations of $${\mathcal{O}}$$ -operators are studied. In particular, the notion of Nijenhuis elements is introduced to characterize trivial linear deformations. Formal deformations and extendibility of order n deformations of an $${\mathcal{O}}$$ -operator are also characterized in terms of the new cohomology theory. Applications are given to deformations of Rota–Baxter operators of weight 0 and skew-symmetric r-matrices for the classical Yang–Baxter equation. For skew-symmetric r-matrices, there is an independent Maurer–Cartan characterization of the deformations as well as an analogue of the Andre–Quillen cohomology, which turn out to have an explicit relationship with the ones obtained as $${\mathcal{O}}$$ -operators associated to the coadjoint representations. Finally, linear deformations of skew-symmetric r-matrices and their corresponding triangular Lie bialgebras are studied.

A universal construction of universal deformation formulas, Drinfeld twists and their positivity

In this paper we provide an explicit construction of star products on U(g)-module algebras by using the Fedosov approach. This construction allows us to give a constructive proof to Drinfel'd theorem and to obtain a concrete formula for Drinfel'd twist. We prove that the equivalence classes of twists are in one-to-one correspondence with the second Chevalley-Eilenberg cohomology of the Lie algebra g. Finally, we show that for Lie algebras with K\"ahler structure we obtain a strongly positive universal deformation of *-algebras by using a Wick-type deformation. This results in a positive Drinfel'd twist.

Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology

IN MEMORIAM OF ALEXANDER GROTHENDIECK. THE MAN.

On some formality criteria for DG-Lie algebras

We give some formality criteria for a differential graded Lie algebra to be formal. For instance, we show that a DG-Lie algebra L is formal if and only if the natural spectral sequence computing the Chevalley–Eilenberg cohomology HCE⁎(L,L) degenerates at E2.

On 2 - 3 Matrix Chevalley Eilenberg Cohomology

The main objective of this paper is to provide the tool rather than the classical adjoint representation of Lie algebra; which is essential in the conception of the Chevalley Eilenberg Cohomology. We introduce the notion of representation induced by a 2 - 3 matrix. We construct the corresponding Chevalley Eilenberg differential and we compute all its cohomological groups.

Higher form gauge fields and their nonassociative symmetry algebras

We show that geometric theories with $p$-form gauge fields have a nonassociative symmetry structure, extending an underlying Lie algebra. This nonassociativity is controlled by the same Chevalley-Eilenberg cohomology that classifies free differential algebras, $p$-form generalizations of Cartan-Maurer equations. A possible relation with flux backgrounds of closed string theory is pointed out.

Completeness of quasi-filiform Lie algebras

It is proved that for any quasi-filiform of non-zero rank the solvable Lie algebra obtained by adjoining a maximal torus of outer derivations is complete. Further, for any positive integer m, it is shown that there exist solvable complete Lie algebras with the second Chevalley–Eilenberg cohomology group of arbitrary dimension.

Almost Kähler structures on four dimensional unimodular Lie algebras

Let J be an almost complex structure on a 4-dimensional and unimodular Lie algebra g. We show that there exists a symplectic form taming J if and only if there is a symplectic form compatible with J. We also introduce groups HJ+(g) and HJ−(g) as the subgroups of the Chevalley–Eilenberg cohomology classes which can be represented by J-invariant, respectively J-anti-invariant, 2-forms on g. and we prove a cohomological J-decomposition theorem following Draˇghici et al. (2010) [12]: H2(g)=HJ+(g)⊕HJ−(g). We discover that tameness of J can be characterized in terms of the dimension of HJ±(g), just as in the complex surface case. We also describe the tamed and compatible symplectic cones. Finally, two applications to homogeneous J on 4-manifolds are obtained.

The first Chevalley–Eilenberg Cohomology group of the Lie algebra on the transverse bundle of a decreasing family of foliations

In [L. Lebtahi, Lie algebra on the transverse bundle of a decreasing family of foliations, J. Geom. Phys. 60 (2010), 122–133], we defined the transverse bundle V k to a decreasing family of k foliations F i on a manifold M . We have shown that there exists a ( 1 , 1 ) tensor J of V k such that J k ≠ 0 , J k + 1 = 0 and we defined by L J ( V k ) the Lie Algebra of vector fields X on V k such that, for each vector field Y on V k , [ X , J Y ] = J [ X , Y ] . In this note, we study the first Chevalley–Eilenberg Cohomology Group, i.e. the quotient space of derivations of L J ( V k ) by the subspace of inner derivations, denoted by H 1 ( L J ( V k ) ) .

A note on the Chevalley–Eilenberg cohomology for the Galilei and Poincaré algebras

We construct in a systematic way the complete Chevalley–Eilenberg cohomology at form degrees 2, 3 and 4 for the Galilei and Poincaré groups. The corresponding non-trivial forms belong to certain representations of the spatial rotation (Lorentz) group. In the case of 2-forms they give all possible central and non-central extensions of the Galilei group (and all non-central extensions of the Poincaré group). The procedure developed in this paper can be applied to any spacetime symmetry group.