Maps Lie 2-groups to Weil algebras, showing cohomology isomorphisms.
problem Cohomology of strict Lie 2-groups.
method Constructs van Est map using double complex and Weil algebra.
result Induces isomorphisms in cohomology under connectedness.
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35 results for “Lie-2-group”
Simple construction of Lie 2-groups from loop group extensions.
problem Constructing Lie 2-groups from loop group extensions.
method Using conjugation action of loop group on its central extension.
result Simple construction of string 2-group as a strict Fréchet Lie 2-group.
Cohomology of Lie 2-groupsmath.AT
In this paper we study the cohomology of (strict) Lie 2-groups. We obtain an explicit Bott-Shulman type map in the case of a Lie 2-group corresponding to the crossed module . The cohomology of the Lie 2-groups corresponding to the universal crossed modules $G\to \Aut(G)$ and $G\to \Aut^+(G)$ is the abutment of …
Study Lie 2-group actions on Riemannian groupoids, proving existence and developing geometric Killing vector fields.
problem Understanding isometric actions of Lie 2-groups on Riemannian groupoids.
method Exhibit properties, prove existence, construct bi-invariant metrics, provide infinitesimal description.
result Existence of 2-equivariant Slice Theorem and Equivariant Tubular Neighborhood Theorem.
We construct a model for the string group as an infinite-dimensional Lie group. In a second step we extend this model by a contractible Lie group to a Lie 2-group model. To this end we need to establish some facts on the homotopy theory of Lie 2-groups. Moreover, we provide an explicit comparison of string structures f…
Constructs 2-representations and 2-vector bundles for Lie 2-groups.
problem No strict model for string 2-group, so coherent model is constructed.
method Builds category of 2-representations and equivariant 2-vector bundles.
result Explicit formulas for 2-representations and 2-vector bundles.
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.
Poisson 2-groupsmath.DG
We prove a 2-categorical analogue of a classical result of Drinfeld: there is a one-to-one correspondence between connected, simply-connected Poisson Lie 2-groups and Lie 2-bialgebras. In fact, we also prove that there is a one-to-one correspondence between connected, simply connected quasi-Poisson 2-groups and quasi-L…
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.
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…
We show how to integrate a weak morphism of Lie algebra crossed-modules to a weak morphism of Lie 2-groups. To do so we develop a theory of butterflies for 2-term L_infty algebras. In particular, we obtain a new description of the bicategory of 2-term L_infty algebras. We use butterflies to give a functorial constructi…
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 …
Parallel transport defined for 2-bundles over Lie groupoids.
problem Defining parallel transport for 2-bundles over Lie groupoids.
method Using Lie 2-group torsors and pseudofunctors, extending principal 2-bundles to differentiable stacks.
result A smooth parallel transport functor defined for Haefliger paths.
New findings on how foliations and Lie group actions interact.
problem Understanding interactions between foliations and Lie group actions.
method Functorial properties of holonomy groupoids and quotient manifolds.
result Holonomy groupoids of quotients by Lie group actions exhibit similar properties.
We develop semistrict higher gauge theory from first principles. In particular, we describe the differential Deligne cohomology underlying semistrict principal 2-bundles with connective structures. Principal 2-bundles are obtained in terms of weak 2-functors from the Cech groupoid to weak Lie 2-groups. As is demonstrat…
Study of gauge theory and parallel transport in Lie 2-group bundles over Lie groupoids.
problem Classify and understand principal 2-bundles over Lie groupoids.
method Introduce principal Lie 2-group bundles, study connection structures, gauge transformations, and parallel transport.
result Extend classification of principal 2-bundles to differentiable stacks and establish connections between geometric and categorical parallel transport.
From Loop Groups to 2-Groupsmath.QA
We describe an interesting relation between Lie 2-algebras, the Kac-Moody central extensions of loop groups, and the group . A Lie 2-algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the "Jacobiator". Similarly, a Lie 2-group is a c…
Develops connections on principal 2-bundles using Lie 2-algebra-valued forms.
problem Defines connections on principal 2-bundles for strict Lie 2-groups.
method Introduces Lie 2-algebra-valued differential forms and connections on Lie groupoids.
result Provides a consistent global perspective to higher gauge theory.
This work introduces a unified approach to the reduction of Poisson manifolds using their description by graded symplectic manifolds. This yields a generalization of the classical Poisson reduction by distributions (Marsden-Ratiu reduction). Further it allows one to construct actions of strict Lie 2-groups and to descr…
We discuss various lifting and reduction problems for bundles and gerbes in the context of a strict Lie 2-group. We obtain a geometrical formulation (and a new proof) for the exactness of Breen's long exact sequence in non-abelian cohomology. We use our geometrical formulation in order to define a transgression map in …
We introduce an axiomatic framework for the parallel transport of connections on gerbes. It incorporates parallel transport along curves and along surfaces, and is formulated in terms of gluing axioms and smoothness conditions. The smoothness conditions are imposed with respect to a strict Lie 2-group, which plays the …
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.
The semidirect product of a Lie algebra and a 2-term representation up to homotopy is a Lie 2-algebra. Such Lie 2-algebras include many examples arising from the Courant algebroid appearing in generalized complex geometry. In this paper, we integrate such a Lie 2-algebra to a strict Lie 2-group in the finite dimensiona…
In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge '2-group'. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-gr…
The paper constructs a model for the string 2-group using the free loop group.
problem The string 2-group lacks a strict Lie-2-group model with circle group action.
method Using the free loop group (or ), the paper constructs a coherent model for the string 2-group with explicit formulas.
result There are obstructions for constructing a strict model for the string 2-group using the free loop group.
Lie's third theorem proven for Lie ∞-algebras.
problem Integrating finite-type Lie ∞-algebras to Lie ∞-groups.
method Local minimal models for Kan simplicial manifolds.
result Every finite-type Lie ∞-algebra integrates to a finite-dimensional Lie ∞-group.
This work constructs groupoids from flat bundles over surfaces.
problem Moduli spaces of flat bundles over surfaces.
method Using quasi-Hamiltonian structures and groupoid constructions.
result Constructs up to three distinct and compatible groupoid structures.
Leaves of Lie algebroids are Lie groupoids under certain conditions.
problem Understanding the structure of leaves in Lie algebroids.
method Analyzing cotangent Lie algebroids and Lie 2-group coadjoint orbits.
result Coadjoint orbits of Lie 2-groups are symplectic groupoids.
In this note we present a new construction of the string group that ends optionally in two different contexts: strict diffeological 2-groups or finite-dimensional Lie 2-groups. It is canonical in the sense that no choices are involved; all the data is written down and can be looked up (at least somewhere). The basis of…
Defines non-Abelian gerbes with connections for physical applications.
problem Tackles the definition and description of non-Abelian gerbes with connections.
method Provides a complete cocycle description for non-Abelian gerbes with connections using adjusted connections.
result Important generalization needed for physical applications, especially in supergravity.
In the first part of this paper, we work out a perturbative Lagrangian formulation of semistrict higher gauge theory, that avoids the subtleties of the relationship between Lie 2-groups and algebras by relying exclusively on the structure semistrict Lie 2-algebra v and its automorphism 2-group Aut(v). Gauge transformat…
The paper constructs automorphisms of Lie groupoids and applies them to symplectic reductions on orbifolds.
problem Formulating Hamiltonian actions of Lie 2-groups on orbifolds.
method Constructing automorphisms of Lie groupoids, using 2-group actions and Kan fibrations.
result Symplectic reductions of Lie 2-group actions on orbifolds are Lie 2-groupoids under certain conditions.
Higher Gauge Theorymath.DG
Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1-dimensional objects (e.g. strings) using 2-connections on 2-bundles. A 2-bundle is a categorified version of a bundle: that is, one where the fiber is not a ma…
PB-groupoids vs VB-groupoidsmath.DG
Paper defines PB-groupoids and their relation to VB-groupoids.
problem Establishing PB-groupoids as a Lie 2-groupoid counterpart.
method Defining PB-groupoids as Lie 2-groupoid actions on diagrams of Lie groupoids and principal bundles.
result PB-groupoids generalize VB-groupoids and provide a new correspondence between them and principal bundles.
Constructs 2-vector bundles and 2K-theory for Lie groupoids and 2-equivariant settings.
problem Developing a theory of 2-vector bundles and 2K-theory for Lie groupoids and their equivariant versions.
method Defines 2-vector bundles over Lie groupoids, constructs 2K-theory as Grothendieck completion, and proves classification theorems.
result Establishes an equivalence between homotopy categories of 2-vector bundles and simplicial maps, and computes 2-equivariant 2K-theories for specific Lie groups.