In this thesis, we introduce a new cohomology theory associated to a Lie 2-algebras and a new cohomology theory associated to a Lie 2-group. These cohomology theories are shown to extend the classical cohomology theories of Lie algebras and Lie groups in that their second groups classify extensions. We use this fact to…
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48 results for “Lie group theory”
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…
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…
Symplectic Lie Groups I-IIImath.DG
We develop the structure theory of symplectic Lie groups based on the study of their isotropic normal subgroups. The article consists of three main parts. In the first part we show that every symplectic Lie group admits a sequence of subsequent symplectic reductions to a unique irreducible symplectic Lie group. The sec…
Paper develops Lie theory for Rota-Baxter operators on Lie algebras and groups.
problem Cohomology of relative Rota-Baxter operators on Lie algebras and groups.
method Cohomology construction, infinitesimal deformations study, differentiation and integration of Rota-Baxter operators.
result Integration of Rota-Baxter operators on Lie groups and Lie algebras.
Lecture notes on Lie groups and Chern-Simons theory for grad students.
problem Understanding Chern-Simons theory and its applications.
method Explains Lie groups and their relation to Chern-Simons theory.
result Motivated new fields in knot theory and topology.
Explores new perspectives in transverse index theory for Lie group actions.
problem Transverse index theory for compact Lie group actions.
method Kasparov's work on transverse index theory, connections to Berline-Vergne and Paradan-Vergne.
result Potential connections and new insights in transverse index theory.
We present a simple remark that assures that the invariant theory of certain real Lie groups coincides with that of the underlying affine, real algebraic groups. In particular, this result applies to the non-compact orthogonal or symplectic Lie groups.
Study index theory on Lie group homogeneous spaces using topological and analytic methods.
problem Index theory on homogeneous spaces of Lie groups.
method Topological and analytic approaches: Riemann-Roch formula and heat kernel methods.
result Local index formula representing higher indices of equivariant elliptic operators.
This paper is devoted to the development and applications of some (new) basic concepts in Lie theory, both from `computational" and "observability" viewpoint. We specify set of all "G-equivariant" maps from a given Lie group G to the underlying manifold M, namely -set, and also we introduce "conjugacy" in Lie group …
Discrete Lagrange problems solved with Lie group constraints.
problem Solving discrete Lagrange problems with Lie group constraints.
method Proving critical sections are solutions of unconstrained variational problems, applying Noether theory and multisymplectic forms.
result Critical sections of discrete Lagrange problems are solutions of unconstrained variational problems.
The paper extends Heintze-Kobayashi-Wolf theory to negatively curved homogeneous Finsler manifolds.
problem Understanding negatively curved homogeneous Finsler manifolds.
method Generalizing Heintze-Kobayashi-Wolf theory to homogeneous Finsler geometry, proving two main theorems.
result Negatively curved homogeneous Finsler manifolds are isometric to Lie groups with specific properties.
Motivated by the interesting and yet scattered developments in representation theory of Banach-Lie groups, we discuss several functional analytic issues which should underlie the notion of infinite-dimensional reductive Lie group: norm ideals, triangular integrals, operator factorizations, and amenability.
The book explores Lie groups and Carnot-Carathéodory spaces, highlighting their applications in metric geometry and geometric group theory.
problem Exploring non-smooth geometries on Lie groups and their applications.
method Study of left-invariant metrics on Lie groups, focusing on nilpotent and Carnot groups.
result Illustrates the role of metric Lie groups, particularly Carnot groups, in various mathematical contexts.
We introduce -positivity, a new notion of positivity in real semisimple Lie groups. The notion of -positivity generalizes at the same time Lusztig's total positivity in split real Lie groups as well as well known concepts of positivity in Lie groups of Hermitian type. We show that there are two other families of …
We compute the equivariant -theory for a simply connected Lie group (acting on itself by conjugation). We prove that is isomorphic to the algebra of Grothendieck differentials on the representation ring. We also study a special example of a non-simply connected Lie group , namely PSU(3),…
We generalize the Uhlenbeck-Segal theory for harmonic maps into compact semi-simple Lie groups to general Lie groups equipped with torsion free bi-invariant connection.
We survey the concept of multiplicativity from its initial appearance in the theory of Poisson-Lie groups to the far-reaching generalizations, for multivectors and differential forms in the geometry and the generalized geometry of Lie groupoids, as well as their infinitesimal counterparts in the theory of Lie algebroid…
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.
Survey on equivariant cohomology of Lie group actions.
problem Understanding Lie group actions on manifolds.
method Focus on equivariant formality and de Rham theory.
result Applications to ordinary cohomology and fixed points.
Lie groups applied to tech progress in economic growth.
problem Understanding the impact of technical progress on economic growth.
method Application of Lie theory and economic modeling.
result Estimation of GDP function for Viet Nam, highlighting tech progress impact.
Lecture notes on BGG complexes using Lie groups and algebras.
problem Constructing BGG complexes on open domains.
method Representation theory of semisimple Lie groups and Lie algebras.
result Introduction of BGG complexes with Lie group and algebra insights.
Proposes an auto-encoder for Gaussian distributions using Lie group theory.
problem Generative models for Gaussian distributions.
method Lie group auto-encoder with UTDATs, incorporating geometric properties.
result Eliminates matrix exponential operator and derives intrinsic loss.
Investigates connections in Lie group bundles, focusing on geometric reduction.
problem Geometric reduction of gauge field theories.
method Definition and analysis of equivariant connections in Lie group bundles.
result Provides conditions for the existence and properties of equivariant connections.
Lie Transformation Groups -- An Introduction to Symmetry Group Analysis of Differential Equationsmath.DG
These are lecture notes of a course on symmetry group analysis of differential equations, based mainly on P. J. Olver's book 'Applications of Lie Groups to Differential Equations'. The course starts out with an introduction to the theory of local transformation groups, based on the Stefan-Sussman theory on the integrab…
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…
Some simple examples from quantum physics and control theory are used to illustrate the application of the theory of Lie systems. We will show, in particular, that for certain physical models both of the corresponding classical and quantum problems can be treated in a similar way, may be up to the replacement of the in…
Investigates adjustments on Lie group crossed modules for gauge theory.
problem Existence and classification of adjustments on crossed modules of Lie groups.
method Differentiation/integration correspondence with infinitesimal adjustments; Lie algebra techniques.
result Infinitesimal adjustments exist if and only if the Kassel-Loday class lies in the image of the Chern-Weil homomorphism.
Chern-Weil and Chern-Simons theory extend to certain infinite-rank bundles that appear in mathematical physics. We discuss what is known of the invariant theory of the corresponding infinite-dimensional Lie groups. We use these techniques to detect cohomology classes for spaces of maps between manifolds and for diffeom…
Survey recent constructions of cyclic cocycles for Lie groups.
problem Higher index theory for proper cocompact -actions.
method Constructions of cyclic cocycles on Harish-Chandra Schwartz algebra.
result Applications to higher index theory.
We describe the reduction procedure for a symplectic Lie algebroid by a Lie subalgebroid and a symmetry Lie group. Moreover, given an invariant Hamiltonian function we obtain the corresponding reduced Hamiltonian dynamics. Several examples illustrate the generality of the theory.
Lie Calculusmath.GR
We explain that general differential calculus and Lie theory have a common foundation: Lie Calculus is differential calculus, seen from the point of view of Lie theory, by making use of the groupoid concept as link between them. Higher order theory naturally involves higher algebra (n-fold groupoids).(conceptual, topol…
Study 2D viscoelastic equations using Lie group theory.
problem Symmetry classification and reduction of 2D viscoelastic equations.
method Investigation through Lie group theory, including algebra of symmetries and optimal subalgebras.
result Classification of reductions of similarities related to Lie subalgebras.
In this article, we introduce a new cohomology theory associated to a Lie 2-algebras. This cohomology theory is shown to extend the classical cohomology theory of Lie algebras; in particular, we show that the second cohomology group classifies an appropriate type of extensions.
Study classifies Lie group representations with non-empty boundary orbit space.
problem Classifying representations of Lie groups with non-empty boundary orbit space.
method Detailed calculations based on previous work.
result Classification of Lie group representations with non-empty boundary orbit space.
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.
Study the spaces of flat connections for classical Lie groups using Chern-Weil theory.
problem Understanding the weak homotopy type of spaces of flat connections for classical Lie groups.
method Use Chern-Weil theory and relate to the functorial map involving continuous families of representations.
result Relate the spaces of flat connections to the weak homotopy type of the spaces of representations.
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.
Infinitesimal calculations link fundamental groups to Lie algebras.
problem Calculating logarithm maps in fundamental groups.
method Hopf invariants defined by Harrison cohomology of commutative cochains.
result Zeroth Harrison cohomology is a universal dual to Malcev Lie algebra.
Jet spaces on Carnot groupsmath.DG
Jet spaces on Carnot groups have a canonical Lie group structure.
problem Understanding jet spaces on Carnot groups.
method Constructing jet spaces over stratified Lie groups and showing they are stratified Lie groups.
result Every stratified Lie group of step can be embedded in a jet space over a stratified Lie group of step .
Using the relations between the theory of differentiable Bol loops and the theory of affine symmetric spaces we classify all connected differentiable Bol loops having an at most -dimensional semi-simple Lie group as the group topologically generated by their left translations. We show that all these Bol loops are is…
Quantum groups created from disk configuration space homologies.
problem Creating quantum groups from algebraic structures.
method Reconstructing quantum groups from homologies of configuration spaces of disks.
result New combinatorics and actual submanifolds of configuration spaces.
Curvature without connectionmath.DG
New theory connects geometry without relying on connections.
problem Developing geometric structures without connections.
method Proposed alternative approach to Lie groups and geometric structures.
result Geometric structures can be developed independently of connections.
The study examines fibrations of Lie groupoids and their classification.
problem Understanding and classifying fibrations of Lie groupoids.
method Introduced locally topological product Lie groupoid fibrations and used groupoid cohomology to obstruct and classify these fibrations.
result The existence of locally topological product Lie groupoid fibrations is obstructed by a specific groupoid cohomology.
Abstract sketches historical development of Lie brackets, crossed modules, and Lie-Rinehart algebras.
problem Characterizing and understanding the relationships between Lie brackets, crossed modules, and Lie-Rinehart algebras.
method Historical review and combinatorial group theory considerations.
result The mutual relationship between Lie-Rinehart algebras and Lie brackets, and the historical development of these concepts.
Course notes on Lie groups and Riemannian geometry, focusing on applications and low-dimensional examples.
problem Exploring Lie groups and their representations in Riemannian geometry.
method Review of well-known topics and recent advances in Riemannian geometry with symmetries.
result First construction of exceptional holonomy metrics.
New manifolds help understand group actions on complex chains.
problem Understanding compact Lie group actions on Morse and Floer chains.
method Introduced new manifolds called forest biassociahedra and bimultiplihedra.
result Derived algebraic structures like bialgebras and bimodules.
Every finite dimensional real representation of a compact real semisimple Lie algebra determines a metric 2-step nilpotent Lie algebra and a corresponding simply connected metric 2-step nilpotent Lie group N. We study the differential geometry of N using representation theory of the complexified complex semisimple Lie …