Study generalizes finiteness theorem using Lie theory.
problem Finiteness theorem in Hadwiger-Alesker theory.
method Generalization through Lie theory.
result Generalization of finiteness theorem.
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48 results for “Lie-theory”
Study compares thimbles to Morse theory on Lie theory models.
problem Exploring thimbles in Landau-Ginzburg models using Morse theory.
method Constructing real Lagrangian thimbles and comparing to gradient flow manifolds.
result Explicit construction and comparison of thimbles to gradient flow manifolds.
We briefly review our results on the Lie theory underlying vector bundles over Lie groupoids and Lie algebroids, pointing out the role of Poisson geometry in extending these results to double Lie algebroids and LA-groupoids.
This thesis bridges Lie theory and sketch theory using tangent categories.
problem Two diverging lines of research in Lie theory.
method Developing involution algebroids and using tangent categories to connect Lie algebroids and Weil algebras.
result The category of Lie algebroids is a functor category, and the Lie functor is a composition with a tangent categorical functor.
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…
We study VB-groupoids and VB-algebroids, which are vector bundles in the realm of Lie groupoids and Lie algebroids. Through a suitable reformulation of their definitions, we elucidate the Lie theory relating these objects, i.e., their relation via differentiation and integration. We also show how to extend our techniqu…
Fat Lie Theorymath.DG
Introduces fat Lie theory for Lie groupoids and algebroids.
problem Representation theory of Lie groupoids and algebroids.
method Introduces fat extensions and abstract 2-term representations up to homotopy (ruths). Establishes correspondences and equivalences.
result One-to-one correspondence between fat extensions and abstract 2-term representations up to homotopy.
Categorifies Jones polynomial using Lie theory.
problem Categorifying the coloured Jones polynomial.
method Lie theoretic categorification with Jones-Wenzl projectors.
result Constructs a categorification of the coloured Jones polynomial.
Poisson double structuresmath.SG
Introduces Poisson double algebroids and their relation to Lie 2-bialgebras.
problem Developing Lie theory for Poisson double structures.
method Introducing Poisson double algebroids and double Lie bialgebroids, and relating them through differentiation and integration.
result Revisits Lie 2-bialgebras using Poisson double structures.
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 …
Polytopes connect Lie theory to physics, integrating integrable systems.
problem Understanding connections between Lie theory and field theories.
method Using Coxeter Plane and integrable systems, a systematic mathematical treatment.
result Supports physical proposals linking polytopes to field theories.
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…
Summary of main work 1999-2012
The paper solves the dHYM equation on rational homogeneous varieties using Lie theory.
problem Solving the deformed Hermitian Yang-Mills equation on rational homogeneous varieties.
method Using Lie theory to describe the Lagrangian phase and characterize solutions.
result Characterization of all supercritical and hypercritical homogeneous solutions of the dHYM equation.
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.
Study of Riemannian geometry on quaternionic unit ball linked to Sp(1,1) group.
problem Understanding the geometry induced by slice Riemannian metric.
method Developed Lie theoretic study, computed isometry group, compared with quaternionic Poincaré geometry.
result Isometry group of slice Riemannian metric is built from symmetries of Sp(1,1) group.
These are notes for a very rapid introduction to the basics of exterior differential systems and their connection with what is now known as Lie theory, together with some typical and not-so-typical applications to illustrate their use.
The paper describes invariant twisted Kähler-Einstein metrics on flag varieties.
problem Existence and properties of invariant twisted Kähler-Einstein metrics on flag varieties.
method Invariant twisted Kähler-Einstein metrics on flag varieties, exploring applications and inequalities.
result Established inequalities related to optimal volume upper bounds for Kähler metrics.
Geometrically solves differentiating simplicial manifolds.
problem Differentiating simplicial manifolds.
method Establishes a normal form theorem, identifies a differentiating ideal, proves quotient semi-freeness, interprets as Chevalley-Eilenberg algebra of higher Lie algebroid.
result Introduces higher van Est map and proves van Est isomorphism theorem.
Lie theory for the integration of Lie algebroids to Lie groupoids, on the one hand, and of Poisson manifolds to symplectic groupoids, on the other, has undergone tremendous developements in the last decade, thanks to the work of Mackenzie-Xu, Moerdijk-Mrcun, Cattaneo-Felder and Crainic-Fernandes, among others. In this …
The paper describes spectra of operators on rational homogeneous varieties.
problem Spectrum of invariant -forms and Weitzenböck remainder.
method Detailed analysis using index theory and Lie-theoretic data.
result Explicit formula for smallest eigenvalue and new lower bounds.
Complete totally geodesic subsets of the complex hyperbolic plane: an elementary classificationmath.DG
Two new proofs classify complete totally geodesic subsets of complex hyperbolic plane.
problem Classify complete totally geodesic subsets of complex hyperbolic plane.
method Two new proofs: one algebraic and one geometric.
result Only complex geodesics and real planes are non-trivial complete totally geodesic subsets.
Semisimple (co)adjoint orbits through real hyperbolic elements are well-known to be symplectomorphic to cotangent bundles. We provide a new proof of this fact based on elementary results on both Lie theory and symplectic geometry. Our proof establishes a new connection between the Iwasawa horospherical projection and t…
Novel analysis of generalized Ricci solitons leads to stability results and deformations.
problem Stability and classification of generalized Ricci solitons.
method Group of generalized gauge transformations, novel connection, second variation formula.
result All Bismut flat manifolds are linearly stable and admit nontrivial deformations.
This paper classifies the symmetry groups of a specific Monge-Ampère equation.
problem Classifying the symmetry groups of a specific Monge-Ampère equation.
method Developed Lie theory for fully nonlinear PDEs in Convex Geometry.
result Completely classified the symmetric groups of the -Minkowski problem.
We study tensors on Lie groupoids suitably compatible with the groupoid structure, called {\em multiplicative}. Our main result gives a complete description of these objects only in terms of infinitesimal data. Special cases include the infinitesimal counterparts of multiplicative forms, multivector fields and holomorp…
Paper estimates neural network size needed for topology learning.
problem Estimating the smallest neural network size for topology learning.
method Using algebraic topology and Lie theory, the paper introduces a procedure based on persistent homology to determine the required dimension.
result The derived dimension is the smallest capable of capturing the topology of the data manifold.
We give a direct, explicit and self-contained construction of a local Lie groupoid integrating a given Lie algebroid which only depends on the choice of a spray vector field lifting the underlying anchor map. This construction leads to a complete account of local Lie theory and, in particular, to a finite-dimensional p…
New Einstein manifolds split into symmetric and compact parts.
problem Understanding Einstein manifolds with unimodular isometry groups.
method Theory of polar actions, Lie-theoretic arguments, and maximum principles.
result Negative Einstein manifolds split into symmetric and compact parts.
Using the symplectic geometry of certain manifolds which appear naturally in Lie theory, we define an invariant which assigns a graded abelian group to an oriented link. The relevant manifolds are transverse slices to certain nilpotent orbits inside sl_{2m}, and intersections of those with regular semisimple orbits. Th…
In the present paper we provide a description of complete Calabi-Yau metrics on the canonical bundle of generalized complex flag manifolds. By means of Lie theory we give an explicit description of complete Ricci-flat Kähler metrics obtained through the Calabi ansatz technique. We use this approach to provide several e…
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 …
Study non-integer power-law potentials for Schrödinger operators using Lie-Rinehart algebras.
problem Analyzing Schrödinger operators with non-integer power-law potentials.
method Using Lie-Rinehart algebras and microlocal analysis.
result Microlocal analysis can be applied to Schrödinger operators with non-integer power-law potentials.
New approach links 2D fluid dynamics to matrix theory.
problem Understanding swirling patterns in 2D fluids.
method Matrix hydrodynamics linking 2D fluid dynamics to matrix theory.
result Established connections between 2D hydrodynamics and matrix Lie theory.
Study finds conditions for Kähler-Einstein metrics on flag manifolds.
problem Characterizing Kähler-Einstein metrics on flag manifolds.
method Using Lie theoretic data, establish a sufficient and necessary condition for -extremality.
result Identifies criteria for a metric to be a critical point of the first eigenvalue functional.
This expository article is an introduction to the adjoint orbits of complex semisimple groups, primarily in the algebro-geometric and Lie-theoretic contexts, and with a pronounced emphasis on the properties of semisimple and nilpotent orbits. It is intended to build a foundation for more specialized settings in which a…
These are notes from lectures given at the Clay Institute Summer School on "Floer homology, gauge theory and low-dimensional topology" (Budapest, 2004). The first part describes as background some of the geometry of symplectic fibre bundles and their monodromy. The second part, overviewing joint work with Paul Seidel, …
We define involution algebroids which generalise Lie algebroids to the abstract setting of tangent categories. As a part of this generalisation the Jacobi identity which appears in classical Lie theory is replaced by an identity similar to the Yang-Baxter equation. Every classical Lie algebroid has the structure of an …
We prove a Frobenius theorem for Banach distributions on manifolds that are modelled over locally convex spaces. Moreover, we recall how Frobenius theorems can be applied to infinite-dimensional Lie groups and obtain, that given a Lie subalgebra of the Lie algebra of a Lie group that is modelled over a locally convex s…
Defines formal vertex laws related to Lie conformal algebras.
problem No specific problem stated; focuses on definitions and proofs.
method Definitions and proofs of vertex/conformal versions of classical Lie theory results.
result Proves vertex/conformal versions of important Lie theory results.
Singular subalgebroidsmath.DG
We introduce singular subalgebroids of an integrable Lie algebroid, extending the notion of Lie subalgebroid by dropping the constant rank requirement. We lay the bases of a Lie theory for singular subalgebroids: we construct the associated holonomy groupoids, adapting the procedure of Androulidakis-Skandalis for singu…
Relates quantum cohomology to tt*-Toda equations for minuscule flag manifolds.
problem Quantum cohomology of minuscule flag manifolds.
method Combining Lie-theoretic treatments of tt*-Toda equations and quantum cohomology.
result Relates quantum cohomology to tt*-Toda equations for minuscule flag manifolds.
The aim of this work is to lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed in recent work (collaboration with H. Gloeckner and K.-H. Neeb), without any restriction on the dimension or on the cha…
We classify all closed, aspherical Riemannian manifolds M whose universal cover has indiscrete isometry group. One sample application is the theorem that any such M with word-hyperbolic fundamental group must be isometric to a negatively curved, locally symmetric manifold. Another application is the classification of a…
We use Chen's iterated integrals to integrate representations up to homotopy. That is, we construct an A_infty functor from the representations up to homotopy of a Lie algebroid to those of its infinity groupoid. This construction extends the usual integration of representations in Lie theory. We discuss several exampl…
Develops Lie-theoretic perspective on Hitchin's equations for cyclic G-Higgs bundles.
problem Analytic and geometric properties of harmonic maps.
method Lie-theoretic perspective on Hitchin's equations for cyclic G-Higgs bundles.
result Proves Dai-Li's conjecture on the monotonicity of the energy density and negative curvature conjecture for Coxeter cyclic G-Higgs bundles.
Defines and classifies toric co-Higgs bundles on projective toric varieties.
problem Classifying co-Higgs bundles on toric varieties.
method Using Klyachko's fan filtration and studying the co-Higgs bundle fiber at a closed point.
result Provides a Lie-theoretic classification of toric co-Higgs bundles.
Geodesics on adjoint orbits of math.DG
Study geodesics on adjoint orbits.
problem Geodesics on adjoint orbits of .
method Translate problem into tangent bundle of -flag manifolds, use Sasaki metric and Lie Theory.
result Explicit description of geodesics families in .