Study Hom-Lie algebroid connections on complex manifolds.
problem Irreducible connections on Hom-Lie algebroids.
method Proved moduli space structure using H-gauge theory.
result Moduli space has a Hausdorff Hilbert manifold structure.
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48 results for “Hom-Lie algebroid connections”
In this paper, first we modify the definition of a Hom-Lie algebroid introduced by Laurent-Gengoux and Teles and give its equivalent dual description. Many results that parallel to Lie algebroids are given. In particular, we give the notion of a Hom-Poisson manifold and show that there is a Hom-Lie algebroid structure …
A hom-Lie algebroid is a vector bundle together with a Lie algebroid like structure which is twisted by a homomorphism. In this paper we use the idea of representations up to homotopy of Lie algebroids to construct a same structure for hom-Lie algebroids and we will explain how representations up to homotopy of length …
In this paper, we develop the theory of Hom-Lie algebroids, Hom-Lie bialgebroids and Hom-Courant algebroids introduced by Cai, Liu and Sheng. Specifically, we introduce the notions of Hom-Poisson, Hom-Nijenhuis and Hom-Poisson-Nijenhuis structures on a Hom-Lie algebroid and the notion of Hom-Dirac structures on a Hom-C…
Hom-Lie algebroidsmath.DG
We define hom-Lie algebroids, a definition that may seem cumbersome at first, but which is justified, first, by a one-to-one corespondence with hom-Gerstenhaber algebras, a notion that we also introduce, and several examples, including hom-Poisson structures.
Hom-Lie-Rinehart Algebrasmath.KT
We introduce hom-Lie-Rinehart algebras as an algebraic analogue of hom-Lie algebroids, and systematically describe a cohomology complex by considering coefficient modules. We define the notion of extensions for hom-Lie-Rinehart algebras. In the sequel, we deduce a characterisation of low dimensional cohomology spaces i…
In this paper, we introduce the notion of a (regular) Hom-Lie group. We associate a Hom-Lie algebra to a Hom-Lie group and show that every regular Hom-Lie algebra is integrable. Then, we define a Hom-exponential (Hexp) map from the Hom-Lie algebra of a Hom-Lie group to the Hom-Lie group and discuss the universality of …
The study examines conditions for completeness and simplicity in hom-Lie superalgebras.
problem Conditions for completeness and simplicity in hom-Lie superalgebras.
method Equivalent conditions and derivations analysis.
result Conditions for completeness and simplicity in hom-Lie superalgebras.
The paper defines Z-graded hom-Lie superalgebras and explores their properties.
problem Understanding the structure and properties of Z-graded hom-Lie superalgebras.
method Definition and exploration of Z-graded hom-Lie superalgebras, invariant bilinear forms, and simplicity conditions.
result Maximal and minimal Z-graded hom-Lie superalgebras for local hom-Lie superalgebras are identified, and conditions for simplicity are checked.
The paper explores geometric structures on Hom-Lie groups and algebras.
problem Exploring Kähler-Norden structures on Hom-Lie groups and algebras.
method Analyzing the relationship between holomorphic Norden structures and Kähler-Norden structures on Hom-Lie groups.
result Left-invariant holomorphic Hom-Lie groups with abelian complex structures are flat.
Establishing Hom-versions of Bochner theorems in pseudo-Riemannian Hom-Lie algebras
problem Killing vectors and Bochner-type theorems in pseudo-Riemannian Hom-Lie algebras
method Hom-versions of Bochner theorems
result Space of Killing vectors forms a totally geodesic Hom-Lie subalgebra
Para-Kahler hom-Lie algebrasmath.DG
In this paper, we introduce the notions of pseudo-Riemannian, para-Hermitian and para- Kahler structures on hom-Lie algebras. In addition, we present the characterization of these structures. Also, we provide an example including these structures. We then introduce the phase space of a hom-Lie algebra and using the hom…
Complex and Hermitian structures on hom-Lie algebras are introduced and some examples of these structures are presented. Also, it is shown that there not exists a proper complex (Hermitian) home-Lie algebra of dimension two. Then using a hom-left symmetric algebra, a phase space is provided and then a complex structure…
Recently, some concepts such as Hom-algebras, Hom-Lie algebras, Hom-Lie admissible algebras, Hom-coalgebras are studied and some of classical properties of algebras and some geometric objects are extended on them. In this paper by recall the concept of Hom--commutative algebras, we intend to develop some of the most…
Constructing -Lie algebroids via connections
problem Constructing Lie algebroids and -Lie algebroids
method Using connections generated by finite families of differential operators and dual sections
result Providing sufficient conditions for generating families to determine Lie algebroid and -Lie algebroid structures
Defines connections on parabolic vector bundles for Lie algebroids.
problem Characterizing parabolic vector bundles with Lie algebroid connections.
method Constructs Lie algebroid connections on parabolic vector bundles, uses Atiyah exact sequence.
result Characterizes stable Lie algebroid vector bundles with connections.
Holomorphic Lie algebroid connections on Riemann surfaces are characterized.
problem Characterizing holomorphic Lie algebroid connections on Riemann surfaces.
method Analyzes conditions for holomorphic vector bundles to admit Lie algebroid connections based on Lie algebroid properties.
result Conditions for holomorphic vector bundles to admit holomorphic Lie algebroid connections are determined.
Paper develops a unified framework for Lie algebroid connections on various bundles.
problem Unified framework for Lie algebroid connections on vector and principal bundles.
method Generalized Atiyah algebroid structure and its short exact sequence.
result Explicit constructions of Atiyah classes for Lie algebroid connections.
Criterion found for Lie algebroid connections on parabolic bundles.
problem Conditions for parabolic vector bundles to have Lie algebroid connections.
method Necessary and sufficient condition based on Lie algebroid structure.
result Found criterion for existence of parabolic Lie algebroid connections.
A Lie algebroid classifies G-structures with connections.
problem Classifying G-structures with connections.
method Associate a Lie algebroid to G-structures with connections.
result The Lie algebroid encapsulates all equivalence information.
New algebroids allow studying various geometries simultaneously.
problem Constructing geometries on a broad class of algebroids.
method Introduced `anti-commutable` pre-Leibniz algebroids and admissible connections.
result Proved properties of admissible connections on various algebroids.
Criterion for Lie algebroid connections on compact Riemann surfaces.
problem Finding conditions for Lie algebroid connections on compact Riemann surfaces.
method Analyzing stable holomorphic vector bundles and their connections.
result Necessary and sufficient condition for Lie algebroid connections on compact Riemann surfaces.
Characterizes Filippov n-algebroids using connections and formulas.
problem Generalizing Lie algebroids to Filippov n-algebroids.
method Introducing Filippov connections and transforming the Jacobi identity into the Bianchi-Filippov identity.
result Expressed the n-ary bracket using a torsion-free formula.
Study flat connections on Courant algebroids using Lie groups.
problem Flatness conditions on Courant algebroids.
method Metric generalized connections, flatness condition analysis.
result Existence of compact simple Lie groups as building blocks for flat transitive Courant algebroids.
Introduces new construction for Courant algebroids and curved structures.
problem Understanding and classifying Courant algebroids and their lifts.
method Introduces Courant algebroid lift and curved Courant algebroids, establishing connections to various geometric structures.
result Established a classification of exact curved Courant algebroids and related connections to various geometric structures.
The paper introduces statistical and geometric structures on anti-commutable pre-Leibniz algebroids.
problem Generalizing differential geometric structures to algebroids.
method Introducing statistical, conjugate connection, and Hessian structures on anti-commutable pre-Leibniz algebroids.
result Statistical and conjugate connection structures are equivalent for admissible connections.
Constructs a triple on an Atiyah algebroid with connection.
problem Dynamics of systems on principal bundles and Atiyah algebroids.
method Constructs a Tulczyjew triple on a principal bundle with connection, then reduces to the Atiyah algebroid.
result Dynamics of systems on principal bundles and Atiyah algebroids are discussed and applied.
Develops theory of para-holomorphic algebroids with para-complex connections.
problem Defines para-holomorphic algebroids and para-complex connections.
method Invokes Lie bialgebroids and almost para-complex structures.
result Generalizes para-Kähler geometry and Poisson-Lie groups.
Study Lie algebroid connections on principal bundles over complex projective varieties.
problem Existence and properties of Lie algebroid connections on principal bundles.
method Definition and study of Lie algebroid valued connections on holomorphic principal G-bundles, investigation of existence criteria.
result Investigation of criteria for existence of Lie algebroid connections on principal G-bundles over smooth complex projective curves.
A generalized notion of a Lie algebroid is presented. Using this, the Lie algebroid generalized tangent bundle is obtained. A new point of view over (linear) connections theory on a fiber bundle is presented. These connections are characterized by o horizontal distribution of the Lie algebroid generalized tangent bundl…
The Chern-Simons forms for R-linear connections on Lie algebroids are considered. A generalized Chern-Simons formula for such R-linear connections is obtained. We it apply to define Chern character and secondary characteristic classes for R-linear connections of Lie algebroids.
A linear connection in a Lie algebroid is said to be metrizable if there exists a Riemannian metric in the Lie algebroid such that . Conditions for the linear connection to be metrizable are investigated.
Study integrability of specific geometric structures on odd Courant algebroids.
problem Characterize integrability of B_n-generalized structures on odd exact Courant algebroids.
method Characterize integrability in terms of existence of adapted generalized connections.
result Describe affine spaces of adapted generalized connections for integrable structures.
We extend the notion of connection in order to be able to study singular geometric structures, namely, we consider a notion of connection on a Lie algebroid which is a natural extension of the usual concept of connection. Using connections, we are able to define holonomy of the orbit foliation of a Lie algebroid and pr…
Study families of Lie algebroids on complex spaces, introducing unfoldings.
problem Investigate singular holomorphic Lie algebroids on complex analytic spaces.
method Introduce and study unfoldings of Lie algebroids, showing a correspondence with holomorphic flat connections.
result Existence of a one-to-one correspondence between transversal unfoldings and holomorphic flat connections.
Study of real and quaternionic Lie algebroid connections on manifolds.
problem Understanding moduli spaces of connections in real and quaternionic geometry.
method Proved the moduli space has a Hausdorff Hilbert manifold structure.
result Generalized results from complex vector bundles to real and quaternionic settings.
Extends Lie algebroids by Lie algebroids with specific conditions.
problem Generalizing Lie algebroid extensions with curvature.
method Using strict covariant adjustments and Cartan connections.
result Provides an obstruction theory for certain Cartan connections.
Courant-Dorfman algebras of differential operators and Dorfman connections of Courant algebroidsmath.DG
Study of multidifferential operators and Dorfman connections on Courant algebroids.
problem Exploring multidifferential operators and Dorfman connections on Courant algebroids.
method Construction of an algebra and complex of multidifferential operators, study of Dorfman connections.
result Cartan calculus, curvatures of induced connections and basic differential geometric identities make sense in the constructed algebra.
We express any Courant algebroid bracket by means of a metric connection, and construct a Courant algebroid structure on any orthogonal Whitney sum where E is a given Courant algebroid and C is a flat, pseudo- Euclidean vector bundle. Then, we establish the general expression of the bracket of a transitive …
Holomorphic Lie algebroid connections on holomorphic principal bundles on compact Riemann surfacesmath.AG
Study connections on complex Riemann surfaces for Lie algebroid structures.
problem Investigating connections on holomorphic Lie algebroid structures on Riemann surfaces.
method Analyzing equivariant holomorphic Lie algebroid connections on holomorphic principal bundles over compact Riemann surfaces.
result Every holomorphic principal G-bundle admits an equivariant holomorphic Lie algebroid connection under certain conditions.
Study shows connection-preserving vector fields are equivalent to certain algebroid structures.
problem Equivalence between vector fields and algebroid structures.
method Quasi-isomorphism between Lie 2-algebras of sections of algebroids and vector fields.
result Lie 2-algebra of sections of -twisted Courant algebroid is quasi-isomorphic to Lie 2-algebra of connection-preserving vector fields.
We introduce and study a class of Lie algebroids associated to faithful modules which is motivated by the notion of cotangent Lie algebroids of Poisson manifolds. We also give a classification of transitive Lie algebroids and describe Poisson algebras by using the notions of algebroid and Lie connections.
In this paper we show how connections and their generalizations on transitive Lie algebroids are related to the notion of connections in the framework of the derivation-based noncommutative geometry. In order to compare the two constructions, we emphasize the algebraic approach of connections on Lie algebroids, using a…
Para-associative Algebroidsmath.DG
New algebraic structure for vector bundles with special properties.
problem Developing new algebraic structures for vector bundles.
method Introducing para-associative algebroids and showing local triviality conditions.
result Existence of a differential connection is necessary and sufficient for local triviality.
The authors define some secondary characteristic homomorphism for the triple (A,B,\bigtriangledown), in which B\subset A is a pair of regular Lie algebroids over the same foliated manifold and \bigtriangledown:L\rightarrow A is a homomorphism of Lie algebroids (i.e. a flat L-connection in A) where L is an arbitrary (no…
Just like Atiyah Lie algebroids encode the infinitesimal symmetries of principal bundles, exact Courant algebroids are believed to encode the infinitesimal symmetries of -gerbes. At the same time, transitive Courant algebroids may be viewed as the higher analogue of Atiyah Lie algebroids, and the non-commutative a…
Novel ternary structures reveal new interpretations of linear connections.
problem Examining the ternary structure of Lie algebroid connections.
method Study of endomorphisms and explicit presentation of the endomorphism truss.
result Explicitly presented endomorphism truss of linear connections.
Complex Finsler vector bundles have been studied mainly by T. Aikou, who defined complex Finsler structures on holomorphic vector bundles. In this paper, we consider the more general case of a holomorphic Lie algebroid E and we introduce Finsler structures, partial and Chern-Finsler connections on it. First, we recall …