The paper defines and studies the first Pontryagin class for quadratic Lie 2-algebroids.
problem Defining and studying the first Pontryagin class for quadratic Lie 2-algebroids.
method Detailed study of transitive Lie 2-algebroids, introduction of quadratic Lie 2-algebroids, definition of first Pontryagin class, construction of quadratic Lie 2-algebroids.
result The first Pontryagin class is the obstruction class for the existence of a CLWX-extension and trivial for certain quadratic Lie 2-algebroids.
The paper categorifies Lie and Courant algebroids, establishing correspondences and new constructions.
problem Categorification of Lie and Courant algebroids to better understand geometric structures.
method Introducing and studying new algebraic structures like VB-Lie 2-algebroids and VB-LWX 2-algebroids.
result Established correspondences and new constructions between Lie and Courant algebroids.
New algebraic structures for Lie 2-algebroids and their connections.
problem Characterizing and understanding Lie 2-algebroids and their structures.
method Construction of homotopy Poisson algebra and introduction of Dirac structures.
result One-to-one correspondence between Manin triples and Lie 2-bialgebroids.
Simplified definition of LA-Courant algebroids and Poisson Lie 2-algebroids.
problem Defining and characterizing LA-Courant algebroids and Poisson Lie 2-algebroids.
method Using split Lie 2-algebroids and self-dual 2-representations to define LA-Courant algebroids, and studying geometric examples and induced structures.
result New examples of Poisson Lie 2-algebroids and a new construction of Courant algebroids.
In this paper, we give the notion of a CLWX 2-algebroid and show that a QP-structure of degree 3 gives rise to a CLWX 2-algebroid. This is the higher analogue of the result that a QP-structure of degree 2 gives rise to a Courant algebroid. A CLWX 2-algebroid can also be viewed as a categorified Courant algebroid. We sh…
Li-Bland's correspondence between linear Courant algebroids and Lie 2-algebroids is explained and shown to be an equivalence of categories. Decomposed VB-Courant algebroids are shown to be equivalent to split Lie 2-algebroids in the same manner as decomposed VB-algebroids are equivalent to 2-term representations up t…
This paper shows the equivalence of the categories of N-manifolds of degree 2 with the category of double vector bundles endowed with a linear metric. Split Poisson N-manifolds of degree 2 are shown to be equivalent to self-dual representations up to homotopy. As a consequence, the equivalence above induces an …
Defines Lie and Courant algebroids over Lie groupoids using homological vector fields.
problem Provides a Morita invariant definition of Lie and Courant algebroids over Lie groupoids.
method Views vector fields as Maurer-Cartan elements in a differential graded Lie algebra and as functors and natural transformations.
result Obtains a unifying conceptual framework for studying various algebraic structures.
The paper studies Lie n-algebroids and their representations up to homotopy.
problem Understanding Lie n-algebroids and their representations.
method Analyzes differential graded modules and representations up to homotopy, describes adjoint and coadjoint modules, and computes the Weil algebra.
result Alternative characterisation of non-degeneracy of higher Poisson structures.
Pre-Courant algebroids are `Courant algebroids' without the Jacobi identity for the Courant-Dorfman bracket. In this paper we examine the corresponding supermanifold description of pre-Courant algebroids and some direct consequences thereof - such as the definition of (sub-)Dirac structures and the notion of the naive …
Study quantum aspects of 1-form symmetries using BV-BRST cohomology.
problem Quantum aspects of gauging continuous 1-form global symmetries.
method BV-BRST quantization of a U(1) 2-form gauge field, Lie 2-algebroid construction, Čech-de Rham bicomplex.
result Anomaly descent for U(1) 1-form symmetries is naturally set up in the Čech-de Rham bicomplex.
Study quantum aspects of 1-form symmetries using BV-BRST cohomology and gerbes.
problem Quantum aspects of gauging continuous 1-form global symmetries.
method BV-BRST quantization of a U(1) 2-form gauge field, Lie 2-algebroid construction, Čech-de Rham bicomplex.
result Anomaly descent for U(1) 1-form symmetries is naturally set up in the Čech-de Rham bicomplex.
Starting with minimal requirements from the physical experience with higher gauge theories, i.e. gauge theories for a tower of differential forms of different form degrees, we discover that all the structural identities governing such theories can be concisely recombined into a so-called Q-structure or, equivalently, a…
We investigate geometric aspects of double field theory (DFT) and its formulation as a doubled membrane sigma-model. Starting from the standard Courant algebroid over the phase space of an open membrane, we determine a splitting and a projection to a subbundle that sends the Courant algebroid operations to the correspo…
Lie groups of automorphisms of cotangent bundles of Lie groups are completely characterized and interesting results are obtained. We give prominence to the fact that the Lie groups of automorphisms of cotangent bundles of Lie groups are super symmetric Lie groups. In the cases of orthogonal Lie lgebras, semi-simple Lie…
Characterizes Lie groups with specific structures and finds a correspondence between carrollian and galilean Lie algebras.
problem Understanding Lie groups with specific structures.
method Using structure theory of metric Lie algebras and defining new Lie algebras with skew-symmetric derivations.
result A canonical correspondence between carrollian and galilean Lie algebras mediated by bargmannian Lie algebras.
The article proves Lie's third theorem for Lie algebroids via singular Lie groupoids.
problem Lie's third theorem does not hold for Lie groupoids and Lie algebroids.
method Introducing a subcategory of diffeological spaces called quasi-etale, constructing a functor mapping singular Lie groupoids to Lie algebroids.
result Lie's third theorem is valid for Lie algebroids within the context of singular Lie groupoids.
Introduces Hom-Lie groups and their integrability, defining Hexp map and adjoint representation.
problem Integrability of Hom-Lie algebras and associated Hom-Lie groups.
method Definition of Hom-Lie groups and algebras, integration of Hom-Lie algebras, Hexp map definition.
result Every regular Hom-Lie algebra is integrable, Hexp map is universal.
Lie's Third Theorem, asserting that each finite-dimensional Lie algebra is the Lie algebra of a Lie group, fails in infinite dimensions. The modern account on this phenomenon is the integration problem for central extensions of infinite-dimensional Lie algebras, which in turn is phrased in terms of an integration proce…
The paper integrates Lie-Leibniz triples into Lie group-rack triples.
problem Integrating Lie-Leibniz triples into Lie group structures.
method Defining Lie group-rack triples and integrating finite-dimensional Lie-Leibniz triples.
result Any finite-dimensional Lie-Leibniz triple can be integrated to a local Lie group-rack triple.
Lie algebroids are like infinitesimal Lie groupoids.
problem None explicitly stated in the abstract.
method Overview article.
result Lie algebroids are infinitesimal counterparts of Lie groupoids.
The paper connects Lie bialgebras, Rota-Baxter Lie algebras, and their properties.
problem Exploring connections between Lie bialgebras and Rota-Baxter Lie algebras.
method Introducing quadratic Rota-Baxter Lie algebras, matched pairs, bialgebras, and Manin triples.
result Established a correspondence between factorizable Lie bialgebras and quadratic Rota-Baxter Lie algebras.
A Lie 2-group's left-invariant vector fields are isomorphic to its Lie 2-algebra.
problem Understanding the relationship between Lie 2-groups and their associated Lie 2-algebras.
method Analyzing the Lie 2-group structure to derive left-invariant vector fields and comparing them to the Lie 2-algebra.
result The Lie 2-algebra of a Lie 2-group is isomorphic to the Lie 2-algebra of its left-invariant vector fields.
The paper integrates Rota-Baxter Lie algebras into Lie group structures and geometries.
problem Integrating Rota-Baxter operators into Lie group structures and geometries.
method Introducing Rota-Baxter operators on Lie groups, Lie algebroids, and groupoids.
result Geometrization of Rota-Baxter Lie algebras and groups.
Following Sullivan's spacial realization of a differential algebra, we construct a universal integrating Lie 2-groupoid for every Lie algebroid. Then We show that unlike Lie algebras which one-to-one correspond to simply connected Lie groups, Lie algebroids (integrable or not) one-to-one correspond to a sort of etale L…
New Lie group structure on vertical bisections of Lie groupoids.
problem Constructing Lie group structure on vertical bisections of Lie groupoids.
method Construct Lie group structure on the group of vertical bisections of a regular Lie groupoid, identify Lie algebra, discuss regularity properties.
result Established Lie theoretic properties of vertical bisections of Lie groupoids over non-compact bases.
Constructing 3-Lie algebroids via connections
problem Constructing Lie algebroids and 3-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 3-Lie algebroid structures Develops a bialgebra theory for post-Lie algebras using geometric interpretations and bilinear forms.
problem Characterizing and understanding post-Lie algebras and their associated structures.
method Utilizes Manin triples and generalized Hessian Lie groups to define and characterize post-Lie algebras with nondegenerate symmetric invariant bilinear forms.
result Establishes a bialgebra theory for post-Lie algebras via the Manin triple approach, including new algebraic structures like pp-post-Lie algebras.
The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of étale Lie 2-groups. In finite dimensions, central extensions of Lie algebras integrate to central extensions of Lie groups, a fact which is due to the vanishing of π_2 for each …
A symplectic Lie group is a Lie group with a left-invariant symplectic form. Its Lie algebra structure is that of a quasi-Frobenius Lie algebra. In this note, we identify the groupoid analogue of a symplectic Lie group. We call the aforementioned structure a \textit{t-symplectic Lie groupoid}; the "t" is motivated …
In this paper, we present a study on the prolongations of representations of Lie algebras. We show that a tangent bundle of a given Lie algebra attains a Lie algebra structure. Then, we prove that this tangent bundle is algebraically isomorphic to the Lie algebra of a tangent bundle of a Lie group. Using these, we defi…
We define a general notion of abstract double Lie algebroid. We show (1) that the double Lie algebroid of a double Lie groupoid is a double Lie algebroid in this sense; (2) that the double cotangent constructed from Lie algebroid structures on a vector bundle A and its dual A* is a double Lie algebroid if and only if (…
In this work we deal with coverings and actions of Lie group- groupoids being a sort of the structured Lie groupoids. Firstly, we define an action of a Lie group-groupoid on some Lie group and the smooth coverings of Lie group-groupoids. Later, we show the equivalence of the category of smooth actions of Lie group-grou…
Investigate local Lie group structure of bisections over compact manifolds
problem Study the local Lie group structure associated with the space of admissible bisections of a local Lie groupoid over a compact manifold.
method Investigate the relation of this local Lie group to the Lie algebra of sections of the associated Lie algebroid.
result Prove that the globalizability of a local Lie groupoid implies the globalizability of its associated local Lie group of bisections.
Current Lie groupoids generalize mappings to Lie groupoids.
problem Generalizing Lie group properties to mappings into Lie groupoids.
method Study of superposition operators and Lie groupoid properties.
result Current Lie groupoids inherit properties from underlying maps.
Lie algebroids and curved Lie algebras are equivalent categories.
problem Understanding the relationship between Lie algebroids and curved Lie algebras.
method Developed a method to study the ∞-category of curved Lie algebras using homotopy theory of algebras over a complete operad. result Equivalence of ∞-categories between Lie algebroids and certain kinds of curved Lie algebras. The paper studies prolongations of Lie algebras associated with pseudo H-type Lie algebras.
problem Investigating prolongations of Lie algebras associated with pseudo H-type Lie algebras. method Analyzing prolongations of associated fundamental graded Lie algebra and associated conformal pseudo-subriemannian fundamental graded Lie algebra.
result The prolongation of the associated conformal pseudo-subriemannian fundamental graded Lie algebra coincides with that of the associated fundamental graded Lie algebra under certain conditions.
Study on simplicity of Lie skew braces, proving new results for compact cases.
problem Simplicity of Lie skew braces, focusing on compact connected cases.
method Reviewing correspondence, investigating ideals and rigidity, proving main result for compact Lie skew braces.
result Compact connected simple Lie skew braces are either trivial or have simple underlying Lie groups.
Generalizes Lie supergroups to Lie superheaps.
problem No specific problem stated; generalization of Lie supergroups.
method Uses functor of points to show isomorphism between categories of pointed Lie superheaps and Lie supergroups.
result There is an isomorphism between the categories of pointed Lie superheaps and Lie supergroups.
In this survey, we report on the state of the art of some of the fundamental problems in the Lie theory of Lie groups modeled on locally convex spaces, such as integrability of Lie algebras, integrability of Lie subalgebras to Lie subgroups, and integrability of Lie algebra extensions to Lie group extensions. We furthe…
A Lie version of Turaev's G-Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a \textit{g-quasi-Frobenius Lie algebra} for g a finite dimensional Lie algebra. The latter consists of a quasi-Frobenius…
Decomposes Goldman-Turaev Lie bialgebra via cutting a surface.
problem Decomposing the Goldman-Turaev Lie bialgebra of a surface.
method Algebraic construction of double Lie bimodules and their combination.
result Decomposes the Goldman-Turaev Lie bialgebra along a simple separating curve.
Study first-order locally convex Lie algebroids in Bastiani calculus.
problem Define and study first-order locally convex Lie algebroids.
method Define sheaves of Lie algebroid forms and morphisms, prove category structure, study representations and cohomology.
result First-order locally convex Lie algebroids form a category and have applications in Lie II theorems.
We introduce post-Lie algebra structures on pairs of Lie algebras $(\Lg,\Ln)$ defined on a fixed vector space V. Special cases are LR-structures and pre-Lie algebra structures on Lie algebras. We show that post-Lie algebra structures naturally arise in the study of NIL-affine actions on nilpotent Lie groups. We obtai…
Integrates transitive Lie algebroids to Lie groupoids, explaining obstructions.
problem Integrating transitive Lie algebroids to Lie groupoids.
method Geometric explanation and explicit construction of integration, with obstructions considered.
result Obstructions explained and integration constructed when they vanish.
Local description of solvable Lie algebras of vector fields.
problem Understanding solvable Lie algebras of vector fields.
method Local and constructive differential geometric description.
result Implication of Lie's conjecture for solvable Lie algebras.
We study generalized Lie bialgebroids over a single point, that is, generalized Lie bialgebras. Lie bialgebras are examples of generalized Lie bialgebras. Moreover, we prove that the last ones can be considered as the infinitesimal invariants of Lie groups endowed with a certain type of Jacobi structures. We also propo…
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.