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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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

CM-type projective varieties X of complex dimension n are characterized by their CM-type rational Hodge structures on the cohomology groups. One may impose such a condition in a weakest form when the canonical bundle of X is trivial; the rational Hodge structure on the level- n subspace of H n (X ; Q) is required to be of CM-type. This brief note addresses the question whether this weak condition implies that the Hodge structure on the entire H ⁎ (X ; Q) is of CM-type. We study in particular abelian varieties when the dimension of the level- n subspace is two or four, and K3 × T 2. It turns out that the answer is affirmative. Moreover, such an abelian variety is always isogenous to a product of CM-type elliptic curves or abelian surfaces. This extends a result of Shioda and Mitani in 1974. [ABSTRACT FROM AUTHOR]

The stable envelopes of Okounkov et al. realize some representations of quantum algebras associated to quivers, using geometry. We relate these geometric considerations to quantum field theory. The main ingredients are the supersymmetric interfaces in gauge theories with four supercharges, relation of supersymmetric vacua to generalized cohomology theories, and Berry connections. We mainly consider softly broken compactified three dimensional N = 4 theories. The companion papers will discuss applications of this construction to symplectic duality, Bethe/gauge correspondence, generalizations to higher dimensional theories, and other topics. [ABSTRACT FROM AUTHOR]

Algebraic structures relative to a semigroup Ω (also called family algebraic structures) first appeared in the work of renormalization in quantum field theory. In this paper, we first consider Ω-diassociative algebras and diassociative family algebras as family analogues of diassociative algebras. We define the cohomologies of these algebras generalizing the cohomology of diassociative algebras introduced by Frabetti. We also introduce (relative) averaging family operators as the family analogue of averaging operators and show that they induce diassociative family algebras and Ω-diassociative algebras. Next, we define the cohomology of a (relative) averaging family operator as the cohomology of the induced Ω-diassociative algebra with coefficients in a suitable representation. Finally, as an application of our cohomology, we study formal one-parameter deformations of relative averaging family operators. [ABSTRACT FROM AUTHOR]

Affine superspace C 1 | n has a single bosonic coordinate z and n fermionic coordinates θ 1 , ... , θ n. Let M be the supertorus obtained by quotienting C 1 | n by the abelian group generated by the maps S : (z , θ 1 , ... , θ n) ↦ (z + 1 , θ 1 , ... , θ n) and T : (z , θ 1 , ... , θ n) ↦ (z + t , θ 1 + α 1 , ... , θ n + α n) where t ∈ C has positive imaginary part and α 1 , ... , α n are independent fermionic parameters. We compute the zeroth and first cohomology groups of the structure sheaf O of M as doubly graded S n -modules, exhibiting an instance of Serre duality between these groups. We use skein relations and noncrossing matchings to give a combinatorial presentation of H 0 (M , O) in terms of generators and relations. [ABSTRACT FROM AUTHOR]

Consider the holomorphic Hamiltonian action of a compact Lie group K on a compact Kähler manifold M with a moment map Φ : M → k ∗ . Assume that 0 is a regular value of the moment map. Weitsman raised the question of what we can say about the cohomology of the Kähler quotient M 0 ≔ Φ − 1 (0) ∕ K if all the ordinary cohomology of M is of type (p , p). In this paper, using the Cartan–Chern–Weil theory we show that in the above context there is a natural surjective Kirwan map from an equivariant version of the Dolbeault cohomology of M onto the Dolbeault cohomology of the Kähler quotient M 0. As an immediate consequence, this result provides an answer to the question posed by Weitsman. [ABSTRACT FROM AUTHOR]

In this paper, we study non-abelian extensions of strict Lie 2-algebras via the cohomology theory. A non-abelian extension of a strict Lie 2-algebra g by h gives rise to a strict homomorphism from g to SOut (h). Conversely, we prove that the obstruction of existence of non-abelian extensions of strict Lie 2-algebras associated to a strict Lie 2-algebra homomorphism from g to SOut (h) is given by an element in the third cohomology group. We further prove that the isomorphism classes of non-abelian extensions of strict Lie 2-algebras are classified by the second cohomology group. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We compute the Poisson cohomology associated with several three dimensional Lie algebras. Together with existing results and the classification of three dimensional Lie algebras, this provides the Poisson cohomology of all linear Poisson structures in dimension 3. [ABSTRACT FROM AUTHOR]

Abstract Starting from the Kähler moduli space of the rigid orbifold Z = E 3 ∕ Z 3 one would expect for the cohomology of the generalized mirror to be a Hodge structure of Calabi–Yau type (1 , 9 , 9 , 1). We show that such a structure arises in a natural way from rational Hodge structures on Λ 3 Q [ ω ] 6 , where ω is a primitive third root of unity. We do not try to identify an underlying mirror geometry, but we show how special geometry arises in our abstract construction. We also show how such Hodge structure can be recovered as a polarized substructure of a bigger Hodge structure given by the third cohomology group of a six-dimensional abelian variety of Weil-type. Moreover, we recover a result of Zheng Zhang on the associates variation of Hodge structure. [ABSTRACT FROM AUTHOR]

Abstract We consider the V ect (R) -module structure on the spaces of bilinear differential operators acting on the spaces of weighted densities. We compute the first differential cohomology of the vector fields Lie algebra V ect (R) with coefficients in space D λ , ν ; μ of bilinear differential operators acting on weighted densities. [ABSTRACT FROM AUTHOR]

Abstract Each sub-Riemannian geometry with bracket generating distribution enjoys a background structure determined by the distribution itself. At the same time, those geometries with constant sub-Riemannian symbols determine a unique Cartan connection leading to their principal invariants. We provide cohomological description of the structure of these curvature invariants in the cases where the background structure is one of the parabolic geometries. As an illustration, constant curvature models are discussed for certain sub-Riemannian geometries. [ABSTRACT FROM AUTHOR]

Abstract In this note, we examine the bundle picture of the pullback construction of Lie algebroids. The notion of submersions by Lie algebroids is introduced, which leads to a new proof of the local normal form for Lie algebroid transversals of Bursztyn et al. (2017) and which we use to deduce that Lie algebroids transversals concentrate all local cohomology. The locally trivial version of submersions by Lie algebroids S is then discussed, and we show that this notion is equivalent to the existence of a complete Ehresmann connection for S , extending the main result in del Hoyo (2016). Finally, we show that locally trivial version of submersions by Lie algebroids gives rise to a system of local coefficients, which is an integral part of a version of the homotopy invariance of de Rham cohomology in the context of Lie algebroids, and we apply such local systems to extend the localization theorem of Chen and Liu (2006). [ABSTRACT FROM AUTHOR]

Abstract We introduce multiplicative differential forms on Lie groupoids with values in VB-groupoids. Our main result gives a complete description of these objects in terms of infinitesimal data. By considering split VB-groupoids, we are able to present a Lie theory for differential forms on Lie groupoids with values in 2-term representations up to homotopy. We also define a differential complex whose 1-cocycles are exactly the multiplicative forms with values in VB-groupoids and study the Morita invariance of its cohomology. [ABSTRACT FROM AUTHOR]

Abstract Let W A be the vector field Lie algebra with values in the Frobneius algebra A of rank N. We show that the second continuous cohomology group is H 2 (W A , K) = ⨁ k = 1 N K ω k , K = ℂ o r A , where ω 1 , ... , ω N are ℂ -valued basic Gelfand–Fuchs cocycles. As an application, we present a generalized Virasoro algebra vir A [ α 1 , ... , α k ] . We also consider another kind of generalization and propose a multi-component Frobenius–Virasoro algebra vir A [ n ] and study its related Euler-type equations. [ABSTRACT FROM AUTHOR]

Abstract Suppose the ground field 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. [ABSTRACT FROM AUTHOR]

Abstract We define the notion of hom-Batalin–Vilkovisky algebras and strong differential hom-Gerstenhaber algebras as a special class of hom-Gerstenhaber algebras and provide canonical examples associated to some well-known hom-structures. Representations of a hom-Lie algebroid on a hom-bundle are defined and a cohomology of a regular hom-Lie algebroid with coefficients in a representation is studied. We discuss about relationship between these classes of hom-Gerstenhaber algebras and geometric structures on a vector bundle. As an application, we associate a homology to a regular hom-Lie algebroid and then define a hom-Poisson homology associated to a hom-Poisson manifold. [ABSTRACT FROM AUTHOR]

Abstract Let ( X m + 1 , g ) be an ( m + 1 ) -dimensional globally hyperbolic spacetime with Cauchy surface M m , and let M ˜ m be the universal cover of the Cauchy surface. Let N X be the contact manifold of all future directed unparameterized light rays in X that we identify with the spherical cotangent bundle S T ∗ M. Jointly with Stefan Nemirovski we showed that when M ˜ m is not a compact manifold, then two points x , y ∈ X are causally related if and only if the Legendrian spheres S x , S y of all light rays through x and y are linked in N X. In this short note we use the contact Bott–Samelson theorem of Frauenfelder, Labrousse and Schlenk to show that the same statement is true for all X for which the integral cohomology ring of a closed M ˜ is not the one of the CROSS (compact rank one symmetric space). If M admits a Riemann metric g ¯ , a point x and a number ℓ > 0 such that all unit speed geodesics starting from x return back to x in time ℓ , then ( M , g ¯ ) is called a Y ℓ x manifold. Jointly with Stefan Nemirovski we observed that causality in ( M × R , g ¯ ⊕ − t 2 ) is not equivalent to Legendrian linking. Every Y ℓ x -Riemann manifold has compact universal cover and its integral cohomology ring is the one of a CROSS. So we conjecture that Legendrian linking is equivalent to causality if and only if one can not put a Y ℓ x Riemann metric on a Cauchy surface M. [ABSTRACT FROM AUTHOR]

The study of n -Lie algebras which are natural generalization of Lie algebras is motivated by Nambu Mechanics and recent developments in String Theory and M-branes. The purpose of this paper is to define cohomology complexes and study deformation theory of n -Lie algebra morphisms. We discuss infinitesimal deformations, equivalent deformations and obstructions. Moreover, we provide various examples. [ABSTRACT FROM AUTHOR]

Starting with Lie’s classification of finite-dimensional transitive Lie algebras of vector fields on C 2 we construct transitive Lie algebras of vector fields on the bundle C 2 × C by lifting the Lie algebras from the base. There are essentially three types of transitive lifts and we compute all of them for the Lie algebras from Lie’s classification. The simplest type of lift is encoded by Lie algebra cohomology. [ABSTRACT FROM AUTHOR]

We show that the classical results on the existence and uniqueness of moment maps in symplectic geometry generalize directly to weak homotopy moment maps in multisymplectic geometry. In particular, we show that their existence and uniqueness is governed by a Lie algebra cohomology complex which reduces to the Chevalley–Eilenberg complex in the symplectic setup. [ABSTRACT FROM AUTHOR]

Let V be a complete discrete valuation ring with residue field k of positive characteristic and with fraction field K of characteristic 0. We clarify the analysis behind the Monsky–Washnitzer completion of a commutative V -algebra using completions of bornological V -algebras. This leads us to a functorial chain complex for a finitely generated commutative algebra over the residue field k that computes its rigid cohomology in the sense of Berthelot. [ABSTRACT FROM AUTHOR]

The important unsolved problem in theory of integrable systems is to find conditions guaranteeing existence of a Lax representation for a given pde . The exotic cohomology of the symmetry algebras opens a way to formulate such conditions in internal terms of the pde s under the study. In this paper we consider certain examples of infinite-dimensional Lie algebras with nontrivial second exotic cohomology groups and show that the Maurer–Cartan forms of the associated extensions of these Lie algebras generate Lax representations for integrable systems, both known and new ones. [ABSTRACT FROM AUTHOR]

In this paper we generalize the definition of the FKMM-invariant introduced in De Nittis and Gomi (2015) for the case of “Quaternionic” vector bundles over involutive base spaces endowed with free involution or with a non-finite fixed-point set. In De Nittis and Gomi (2015) it has already be shown how the FKMM-invariant provides a cohomological description of the Fu–Kane–Mele index used to classify topological insulators in class AII. It follows that the FKMM-invariant described in this paper provides a cohomological generalization of the Fu–Kane–Mele index which is applicable to the classification of protected phases for other type of topological quantum systems (TQS) which are not necessarily related to models for topological insulators (e.g. the two-dimensional models of adiabatically perturbed systems discussed in Gat and Robbins, 2017). As a byproduct we provide the complete classification of “Quaternionic” vector bundles over a big class of low dimensional involutive spheres and tori. [ABSTRACT FROM AUTHOR]

We define the β -function of a perturbative quantum field theory in the mathematical framework introduced by Costello – combining perturbative renormalization and the BV formalism – as the cohomology class of a certain functional measuring scale dependence of the effective interaction. We show that the one-loop β -function is a well-defined element of the obstruction–deformation complex for translation-invariant and classically scale-invariant theories, and furthermore that it is locally constant as a function on the space of classical interactions and computable as a rescaling anomaly, or as the logarithmic one-loop counterterm. We compute the one-loop β -function in first-order Yang–Mills theory, recovering the famous asymptotic freedom for Yang–Mills in a mathematical context. [ABSTRACT FROM AUTHOR]

We consider the a f f ( n | 1 ) − module structure on the spaces of differential bilinear operators acting on the superspaces of weighted densities. We classify a f f ( n | 1 ) − invariant binary differential operators acting on the spaces of weighted densities. This result allows us to compute the first a f f ( n | 1 ) − relative differential cohomology of K ( n ) with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities. [ABSTRACT FROM AUTHOR]

In the present paper we indicate some Leibniz algebras whose closures of orbits under the natural action of GL n form an irreducible component of the variety of complex ( n + 1 ) -dimensional Leibniz algebras. Moreover, for these algebras we calculate the bases of their second cohomology groups. [ABSTRACT FROM AUTHOR]

We introduce a notion of crystalline cohomology for superschemes and show that it is isomorphic to the usual crystalline cohomology of the underlying commutative scheme if 2 is invertible on the base. [ABSTRACT FROM AUTHOR]

We study type one generalized complex and generalized Calabi–Yau manifolds. We introduce a cohomology class that obstructs the existence of a globally defined, closed 2-form which agrees with the symplectic form on the leaves of the generalized complex structure, the twisting class . We prove that in a compact, type one, 4 n -dimensional generalized complex manifold the Euler characteristic must be even and equal to the signature modulo four. The generalized Calabi–Yau condition places much stronger constraints: a compact type one generalized Calabi–Yau fibers over the 2-torus and if the structure has one compact leaf, then this fibration can be chosen to be the fibration by the symplectic leaves of the generalized complex structure. If the twisting class vanishes, one can always deform the structure so that it has a compact leaf. Finally we prove that every symplectic fibration over the 2-torus admits a type one generalized Calabi–Yau structure. [ABSTRACT FROM AUTHOR]

The classification of Seifert manifolds was given in terms of numeric data by Seifert (1933), and then generalized by Raymond (1968) and Orlik and Raymond (1968) to circle actions on closed 3d manifolds. In this paper, we further generalize the classification to circle actions on 3d manifolds with boundaries by adding a numeric parameter and a graph of cycles. Then, we describe the rational equivariant cohomology of 3d manifolds with circle actions in terms of ring, module and vector-space structures. We also compute equivariant Betti numbers and Poincaré series for these manifolds and discuss the equivariant formality. [ABSTRACT FROM AUTHOR]

We study rank two locally-free Fourier–Mukai transforms on K3 surfaces and show that they come in two distinct types according to whether the determinant of a suitable twist of the kernel is positive or not. We show that a necessary and sufficient condition on the existence of Fourier–Mukai transforms of rank 2 between the derived categories of K3 surfaces X and Y with negative twisted determinant is that Y is isomorphic to X and there must exist a line bundle with no cohomology. We use these results to prove that all reflexive K3 surfaces (including the degenerate ones) admit Fourier–Mukai transforms. [ABSTRACT FROM AUTHOR]

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 suitable condition. Finally, we show the equivalence between r -matrix-Nijenhuis structures and Rota-Baxter-Nijenhuis structures on Lie-Yamaguti algebras. [ABSTRACT FROM AUTHOR]

Over the (1 , 2) -dimensional real superspace R 1 | 2. We compute the second aff (2 | 1) -relative cohomology space of K (2) with coefficients in the module of λ -densities F λ 2 on R 1 | 2 , where K (2) is the Lie superalgebra of contact vector fields on R 1 | 2 and aff (2 | 1) is the affine Lie superalgebra. We give explicitly 2-cocycles spanning these cohomology spaces. This result is based specially on the classification of aff (2 | 1) -invariant bilinear differential operators acting on the superspaces of weighted densities given in [2]. [ABSTRACT FROM AUTHOR]

We show that the C ∞ -pure-and-full condition is not stable in the almost Kähler category. [ABSTRACT FROM AUTHOR]

We show that supercocycles on super L ∞ -algebras capture, at the rational level, the twisted cohomological charge structure of the fields of M -theory and of type IIA string theory. We show that rational 4-sphere-valued supercocycles for M-branes in M -theory descend to supercocycles in type IIA string theory with coefficients in the free loop space of the 4-sphere, to yield the Ramond–Ramond fields in the rational image of twisted K -theory, with the twist given by the B -field. In particular, we derive the M2/M5 ↔ F1/Dp/NS5 correspondence via dimensional reduction of sphere-valued super- L ∞ -cocycles. [ABSTRACT FROM AUTHOR]

On a projective complex manifold, the Abelian group of divisors maps surjectively onto that of holomorphic line bundles (the Picard group). On a G 2 -manifold we use coassociative submanifolds to define an analogue of the divisors, and a gauge theoretical equation for a connection on a gerbe to define an analogue of the Picard group. Then, we construct a map from the former to the later. We also prove that the canonical map from our analogue of the Picard group to the third cohomology group with integer coefficients is surjective. As a side remark we make an observation relating the topological type of coassociative submanifolds and the cohomology classes they represent. [ABSTRACT FROM AUTHOR]

We compute the Poisson cohomology of a scalar Poisson bracket of Dubrovin–Novikov type with D independent variables. We find that the second and third cohomology groups are generically non-vanishing in D > 1 . Hence, in contrast with the D = 1 case, the deformation theory in the multivariable case is non-trivial. [ABSTRACT FROM AUTHOR]

The work establishes a relation between deformed cohomologies of symmetry pseudo-groups and coverings of differential equations. Examples include the potential Khokhlov–Zabolotskaya equation and the Boyer–Finley equation. [ABSTRACT FROM AUTHOR]

The Dolbeault cohomology plays an important role in the study of some ∂ ¯ -invariants of complex and holomorphic vector bundles as ∂ ¯ -Chern classes and Atiyah classes. In this paper we generalize similar invariants and their properties in the tangential Dolbeault cohomology. More exactly, we introduce and we study tangential Atiyah classes for F -holomorphic vector bundles and ∂ ¯ -tangential Chern classes for tangentially smooth vector bundles over a manifold M endowed with a complex foliation F . Also, ∂ ¯ -tangential secondary invariants are studied following similar constructions for Lie algebroids. The notions are introduced by a global formalism that is used in the tangential theory of foliated spaces. [ABSTRACT FROM AUTHOR]

A cohomology theory of Rota-Baxter 3-Lie algebras of arbitrary weights is introduced. Formal deformations, abelian extensions, skeletal Rota-Baxter 3-Lie 2-algebras and crossed modules of Rota-Baxter 3-Lie algebras are interpreted by using lower degree cohomology groups. [ABSTRACT FROM AUTHOR]

We study the connection between N = 2 supersymmetric gauge theories, quantum cohomology and quantum integrable systems of hydrodynamic type. We consider gauge theories on ALE spaces of A and D -type and discuss how they describe the quantum cohomology of the corresponding Nakajima’s quiver varieties. We also discuss how the exact evaluation of local BPS observables in the gauge theory can be used to calculate the spectrum of quantum Hamiltonians of spin Calogero integrable systems and spin Intermediate Long Wave hydrodynamics. This is explicitly obtained by a Bethe Ansatz Equation provided by the quiver gauge theory in terms of its adjacency matrix. [ABSTRACT FROM AUTHOR]

The topological classification of gerbes, as principal bundles with the structure group the projective unitary group of a complex Hilbert space, over a topological space H is given by the third cohomology H 3 ( H , Z ) . When H is a topological group the integral cohomology is often related to a locally continuous (or in the case of a Lie group, locally smooth) third group cohomology of H . We shall study in more detail this relation in the case of a group extension 1 → N → G → H → 1 when the gerbe is defined by an abelian extension 1 → A → N ˆ → N → 1 of N . In particular, when H s 1 ( N , A ) vanishes we shall construct a transgression map H s 2 ( N , A ) → H s 3 ( H , A N ) , where A N is the subgroup of N -invariants in A and the subscript s denotes the locally smooth cohomology. Examples of this relation appear in gauge theory which are discussed in the paper. [ABSTRACT FROM AUTHOR]

We investigate the second cohomology space associated with the embedding of the orthosymplectic Lie superalgebra o s p ( n | 2 ) on the ( 1 , n ) -dimensional superspace R 1 | n in the Lie superalgebra S Ψ D O ( n ) of superpseudodifferential operators with smooth coefficients, where n = 0 , 1 , 2 . We show that this space is purely even and we give explicit expressions of the basis cocycles. [ABSTRACT FROM AUTHOR]

The k -Cauchy–Fueter operator D 0 ( k ) on one dimensional quaternionic space H is the Euclidean version of spin k / 2 massless field operator on the Minkowski space in physics. The k -Cauchy–Fueter equation for k ≥ 2 is overdetermined and its compatibility condition is given by the k -Cauchy–Fueter complex. In quaternionic analysis, these complexes play the role of Dolbeault complex in several complex variables. We prove that a natural boundary value problem associated to this complex is regular. Then by using the theory of regular boundary value problems, we show the Hodge-type orthogonal decomposition, and the fact that the non-homogeneous k -Cauchy–Fueter equation D 0 ( k ) u = f on a smooth domain Ω in H is solvable if and only if f satisfies the compatibility condition and is orthogonal to the set ℋ ( k ) 1 ( Ω ) of Hodge-type elements. This set is isomorphic to the first cohomology group of the k -Cauchy–Fueter complex over Ω , which is finite dimensional, while the second cohomology group is always trivial. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of a LeibtsDer pair, i.e., a Leibniz triple system with a derivation. We define a representation of a LeibtsDer pair and the corresponding cohomology theory. We prove that a LeibtsDer pair is rigid if the H D 3 (L , L) = 0 , and a deformation of order n is extensible if its obstruction class (Ob (μ t , D t) i 3 , Ob (μ t , D t) 1) = ∂ i 3 (μ n + 1 , D n + 1) (i = 1 , 2). We also show that the central extensions of a LeibtsDer pair can be classified by the third cohomology group. [ABSTRACT FROM AUTHOR]

A homological spark complex ( F U ∗ , E U ∗ , I U ∗ ) is constructed for any good atlas U of an effective orbifold X = ( X , [ U ] ) . It is proved that all these homological spark complexes are quasi-isomorphic for X . Furthermore, smooth Deligne-Beilinson cohomology H D p ( X , Z ( q ) ) of X is investigated and it is shown that the ring H ˆ ∗ ( X ) of spark classes of X is isomorphic to the subring H D ∗ ( X , Z ( ∗ ) ) of the orbifold Deligne-Beilinson cohomology ring of X . [ABSTRACT FROM AUTHOR]

We formulate and prove an extension of Connes’s reconstruction theorem for commutative spectral triples to so-called Connes–Landi or isospectral deformations of commutative spectral triples along the action of a compact Abelian Lie group G , also known as toric noncommutative manifolds. In particular, we propose an abstract definition for such spectral triples, where noncommutativity is entirely governed by a deformation parameter sitting in the second group cohomology of the Pontryagin dual of G , and then show that such spectral triples are well-behaved under further Connes–Landi deformation, thereby allowing for both quantisation from and dequantisation to G -equivariant abstract commutative spectral triples. We then use a refinement of the Connes–Dubois-Violette splitting homomorphism to conclude that suitable Connes–Landi deformations of commutative spectral triples by a rational deformation parameter are almost-commutative in the general, topologically non-trivial sense. [ABSTRACT FROM AUTHOR]

We study properties concerning decomposition in cohomology by means of generalized-complex structures. This notion includes the C ∞ -pure-and-fullness introduced by Li and Zhang in the complex case and the Hard Lefschetz Condition in the symplectic case. Explicit examples on the moduli space of the Iwasawa manifold are investigated. [ABSTRACT FROM AUTHOR]

Let M be an exact symplectic manifold with c 1 ( M ) = 0 . Denote by Fuk ( M ) the Fukaya category of M . We show that the dual space of the bar construction of Fuk ( M ) has a differential graded noncommutative Poisson structure. As a corollary we get a Lie algebra structure on the cyclic cohomology HC • ( Fuk ( M ) ) , which is analogous to the ones discovered by Kontsevich in noncommutative symplectic geometry and by Chas and Sullivan in string topology. [ABSTRACT FROM AUTHOR]