The article examines twisted cohomologies on algebraic and analytic varieties.
problem Understanding and comparing twisted cohomologies on algebraic and analytic varieties.
method Comparison and definition of twisting parameters in both categories, algebraic and analytic.
result Reviewed isomorphisms of twisted cohomologies for cohomologous twisting parameters.
Study cohomology of hemistrict Lie 2-algebras, proving isomorphic results.
problem Understanding cohomology of hemistrict Lie 2-algebras.
method Functorial construction and isomorphism proof of cohomology.
result Cohomology of hemistrict Lie 2-algebras is isomorphic to Chevalley-Eilenberg cohomology.
This study introduces a unified cohomology theory for braided algebras.
problem Classifying infinitesimal deformations of braided algebras.
method Developed a cohomology theory unifying Hochschild and Yang-Baxter cohomology.
result The second cohomology group classifies infinitesimal deformations of braided algebras.
New cohomology theory for Lie 2-algebras extends classical theory.
problem Classical cohomology theory limitations for Lie 2-algebras.
method Introduced a new cohomology theory for Lie 2-algebras.
result Second cohomology group classifies extensions of Lie 2-algebras.
We introduce a bicomplex which computes the triple cohomology of Lie--Rinehart algebras. We prove that the triple cohomology is isomorphic to the Rinehart cohomology \cite{Ri} provided the Lie--Rinehart algebra is projective over the corresponding commutative algebra. As an application we construct a canonical class in…
Introduces new cohomology theories for Lie 2-algebras and groups.
problem Classical cohomology theories do not extend to Lie 2-algebras and groups.
method Develops new cohomology theories and uses them to prove integrability.
result New cohomology theories classify extensions and prove integrability of Lie 2-algebras.
Analyses cohomology relations for moving frames and coframes.
problem Relating Hopf cyclic cohomology of moving frames and coframes.
method Uses van Est analogy for DG Hopf algebras.
result Establishes cohomology isomorphism for DG Hopf algebras.
New algebraic tools solve Poisson and Lie bialgebra problems.
problem Modular class and intrinsic biderivation in Poisson geometry.
method Algebraic tools from differential Gerstenhaber algebras and Batalin-Vilkobisky algebras.
result Applications to Lie bialgebra and Poisson cohomology.
Proves integrability of strict Lie 2-algebras using cohomological methods.
problem Integrability of strict Lie 2-algebras.
method Van Est theorems relating cohomologies of Lie 2-groups and algebras.
result Proves integrability of Lie 2-algebras.
Study on deformation cohomology for braided commutative structures.
problem Classifying and understanding deformations of braided commutative algebras.
method Extending Yang-Baxter Hochschild cohomology to braided commutative deformations.
result Classifies infinitesimal deformations of braided algebras that are braided commutative.
The first cohomology of Poisson algebras is described and conditions for its vanishing are established.
problem Understanding the first cohomology of Poisson algebras.
method Description and mapping of first cohomology to intrinsic cohomologies of Poisson submanifolds, formulation of vanishing conditions.
result Necessary and sufficient conditions for the vanishing of the first cohomology of infinitesimal Poisson algebras are derived.
Cohomology of Lie group quotient equals Lie algebra cohomology.
problem Cohomology of Lie group quotients.
method Diffeological de Rham cohomology and Lie algebra cohomology.
result Cohomology of G/H equals Lie algebra cohomology of g/h. Cohomology of 'book' Lie algebra Poisson structure computed.
problem Computing the cohomology of a specific Lie algebra structure.
method Direct computation of cohomology for the given Lie algebra structure.
result Explicit formula for Poisson cohomology of the 'book' Lie algebra.
Introduces Lie-Yamaguti algebra bundles and their cohomology.
problem Defining and studying Lie-Yamaguti algebra bundles.
method Defined Lie-Yamaguti algebra bundles and their cohomology groups.
result Lie-Yamaguti algebra bundles naturally arise from geometric considerations.
In this note we define a notion of Courant pair as a Courant algebra over the Lie algebra of linear derivations on an associative algebra. We study formal deformations of Courant pairs by constructing a cohomology bicomplex with coefficients in a module from the cochain complexes defining Hochschild cohomology and Leib…
Holomorphic Poisson cohomology on nilmanifolds identified and characterized.
problem Characterizing the cohomology of holomorphic Poisson structures on nilmanifolds.
method Construction of non-trivial holomorphic Poisson structures and identification of conditions for cohomology isomorphisms.
result Conditions for the cohomology of non-trivial holomorphic Poisson structures to be isomorphic to trivial ones.
This work is devoted to an intrinsic cohomology theory of Koszul-Vinberg algebras and their modules. Our results may be regarded as improvements of the attempt by Albert Nijenhuis in [NA]. The relationships between the cohomology theory developed here and some classical problems are pointed out, e.g. extensions of alge…
The paper studies deformations of Lie ideals in Lie algebras.
problem Understanding deformations of Lie ideals in Lie algebras.
method Develops deformation theory, compares cohomologies, enriches deformation complex.
result Deformation cohomology classes differentiate smooth deformations of ideals.
Study cohomology of Lie algebroids over algebraic spaces using derived functors and Čech cohomology.
problem Cohomology of Lie algebroids over algebraic spaces.
method Express hypercohomology as a derived functor, simplify via Čech cohomology, define Hochschild hypercohomology, present Hochschild-Kostant-Rosenberg theorem.
result Presented a version of Hochschild-Kostant-Rosenberg theorem for locally free Lie algebroids.
We construct a new equivariant cohomology theory for a certain class of differential vertex algebras, which we call the chiral equivariant cohomology. A principal example of a differential vertex algebra in this class is the chiral de Rham complex of Malikov-Schechtman-Vaintrob of a manifold with a group action. The ma…
Novel cohomology theories for operadic algebras and spaces.
problem Formulating cohomology theories for operadic algebras.
method Using cotangent complex formalism and spectral Hochschild cohomology.
result Controlled cohomologies of operads and their algebras.
After a self-contained introduction to Lie algebra cohomology, we present some recent applications in mathematics and in physics. Contents: 1. Preliminaries: L_X, i_X, d 2. Elementary differential geometry on Lie groups 3. Lie algebra cohomology: a brief introduction 4. Symmetric polynomials and higher order cocycles 5…
Researchers compute Hochschild cohomology of Grassmannians.
problem Computing Hochschild cohomology of Grassmannians.
method Explicit description of Gerstenhaber algebra structure, vanishing of higher cohomology.
result Decomposition of Hochschild cohomology concentrated in global sections for certain Grassmannians.
Researchers compute cohomology of Lie groups using Lie algebras.
problem Computing cohomology of left-invariant elliptic and hypocomplex structures on compact Lie groups.
method Used the Leray spectral sequence connecting Lie algebras to Dolbeault cohomology of homogeneous manifolds.
result Cohomology can be computed purely algebraically.
Classifies foliated complex manifolds using marked fans.
problem Understanding foliated complex manifolds similar to toric varieties.
method Classifies by marked fans and describes cohomology algebras.
result Basic cohomology and Dolbeault cohomology algebras described in terms of marked fans.
Develops a spectral sequence for Lie group actions on manifolds.
problem Understanding cohomology of manifolds with Lie group actions.
method Introduces a spectral sequence relating manifold cohomology to Lie algebra cohomology.
result Establishes a new description of de Rham cohomology for manifolds with Lie group actions.
This research classifies deformations of Yang-Baxter operators using cohomology of n-Lie algebras.
problem Classifying deformations of Yang-Baxter operators via cohomology of n-Lie algebras. method Introducing a cohomology theory for n-ary self-distributive objects, showing natural injections and isomorphisms, and constructing deformation theories. result The self-distributive deformations classify the Yang-Baxter operator deformations, with nontrivial examples provided.
In this paper we study the variability and rigidity of secondary characteristic classes which arise from flat connections on a manifold. Considering the connection as a Lie-algebra valued one-form, we study the characteristic map from Lie algebra cohomology to de Rham cohomology of the manifold, and prove that if the L…
The equivariant cohomology of a space with a group action is not only a ring but also an algebra over the cohomology ring of the classifying space of the acting group. We prove that toric manifolds (i.e. compact smooth toric varieties) are isomorphic as varieties if and only if their equivariant cohomology algebras are…
Formulae for Riemannian foliations cohomology.
problem Equivariant cohomology of Riemannian foliations.
method Localization and integration formulas for equivariant basic cohomology.
result Duistermaat-Heckman theorem for transversely symplectic foliations.
This is the second in a series of papers on a new equivariant cohomology that takes values in a vertex algebra. In an earlier paper, the first two authors gave a construction of the cohomology functor on the category of O(sg) algebras. The new cohomology theory can be viewed as a kind of "chiralization'' of the classic…
We define algebraic structures on graph cohomology and prove that they correspond to algebraic structures on the cohomology of the spaces of imbeddings of S^1 or R into R^n. As a corollary, we deduce the existence of an infinite number of nontrivial cohomology classes in Imb(S^1,R^n) when n is even and greater than 3. …
Topological Pontryagin classes are algebraically independent in high-dimensional spaces.
problem Algebraic independence of topological Pontryagin classes.
method Analyzing rationalised cohomology of BTop(d).
result Topological Pontryagin classes are algebraically independent.
Determines algebra structure of complex differential forms operators.
problem Identifying the algebra structure of differential operators on complex-valued differential forms.
method Shows it is the universal enveloping algebra of a graded Lie algebra and determines its cohomology.
result Determines the cohomology of the graded Lie algebra with respect to various inner differentials.
We associate to each infinite primitive Lie pseudogroup a Hopf algebra of `transverse symmetries', by refining a procedure due to Connes and the first author in the case of the general pseudogroup. The affiliated Hopf algebra can be viewed as a `quantum group' counterpart of the infinite-dimensional primitive Lie algeb…
An algebra A with identity (a∘b)∘c−a∘(b∘c)=(a∘c)∘b−a∘(c∘b), is called right-symmetric. Cohomology and deformation theory for right-symmetric algebras are developed. Cohomologies of gln and half-Witt algebras Wnrsym,p=0, Wnrsym(m),p>0, are calculated. In p…
The abstract discusses cohomologies and deformations of Rota-Baxter operators on Lie algebroids and Koszul-Vinberg structures.
problem Characterizing and studying deformations and cohomologies of relative Rota-Baxter operators.
method Constructing graded Lie algebras and studying their Maurer-Cartan elements, cohomology, and deformations.
result Homomorphisms between cohomology groups of relative Rota-Baxter operators and deformation cohomology groups of left-symmetric algebroids.
New algebraic structures for Hermitian geometry cohomologies.
problem Understanding cohomologies of Hermitian manifolds.
method Introducing BV-algebras and homotopy BV-algebras.
result Cohomologies of Hermitian manifolds are endowed with homotopy hypercommutative algebra structures.
The space of Lie algebra cohomology is usually described by the dimensions of components of certain degree even for the adjoint module as coefficients when the spaces of cochains and cohomology can be endowed with a Lie superalgebra structure. Such a description is rather imprecise: these dimensions may coincide for co…
Defines algebraic structures in Lagrangian Floer cohomology using differential forms.
problem Modeling algebraic structures in Lagrangian Floer cohomology.
method Defines two algebra structures using differential forms and a closed-open map, showing they coincide.
result Two algebra structures for the 2-dimensional Clifford torus coincide.
Simple construction of Rumin algebra for contact manifolds.
problem Computing the de Rham cohomology algebra of contact manifolds.
method Using Markl's Homotopy Transfer Theorem for a simple explicit construction.
result Recovery of the Rumin algebra as a contact invariant C∞-algebra. This paper constructs cohomological Hall algebras for 3-Calabi-Yau categories.
problem Mathematical definition of the algebra of BPS states.
method Construction of cohomological Hall algebras for 3-Calabi-Yau categories.
result Construction of cohomological Hall algebras and proof of Joyce's conjecture.
We construct a Hopf action, with an invariant trace, of a bicrossed product Hopf algebra $\cH=\big( \cU(\Fg_1) \acr \cR(G_2) \big)^{\cop}$ constructed from a matched pair of Lie groups G1 and G2, on a convolution algebra $\cA=C_c^{\ify}(G_1)\rtimes G_2^δ$. We give an explicit way to construct Hopf cyclic cohomolo…
We develop the theory of linear algebra over a (Z_2)^n-commutative algebra (n in N), which includes the well-known super linear algebra as a special case (n=1). Examples of such graded-commutative algebras are the Clifford algebras, in particular the quaternion algebra H. Following a cohomological approach, we introduc…
Let R be a commutative ring, and let A be a Poisson algebra over R. We construct an (R,A)-Lie algebra structure, in the sense of Rinehart, on the A-module of Kähler differentials of A depending naturally on A and the Poisson bracket. This gives rise to suitable algebraic notions of Poisson homology and cohomology for a…
Study cohomology rings of Grassmannians using Clifford algebras and symmetric spaces.
problem Understanding cohomology rings of Grassmannians over different fields.
method Explicit generators and relations for de Rham cohomology rings, filtered deformations related to Clifford algebras.
result Explicit generators and relations for the de Rham cohomology rings of Grassmannians.
We prove a cyclic cohomological analogue of Haefliger's van Est-type theorem for the groupoid of germs of diffeomorphisms of a manifold. The differentiable version of cyclic cohomology is associated to the algebra of transverse differential operators on that groupoid, which is shown to carry an intrinsic Hopf algebraic…
For a simply connected (non-nilpotent) solvable Lie group G with a lattice Γ the de Rham and Dolbeault cohomologies of the solvmanifold G/Γ are not in general isomorphic to the cohomologies of the Lie algebra g of G. In this paper we construct, up to a finite group, a new Lie algebra $\tilde{\mathfr…