The procedure "Lie group --> Lie algebra" has a generalization "simplicial manifold --> L_infinity algebra", or yet better, "presheaf on the category of surjective submersions --> L_infinity algebra". We describe this generalization, together with its higher-order extensions.
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48 results for “isotropy L-infinity algebra”
L-infinity algebra actionsmath.DG
We define the notion of action of an L-infinity algebra on a graded manifold , and show that such an action corresponds to a homological vector field on of a specific form. This generalizes the correspondence between Lie algebra actions on manifolds and transformation Lie algebroids. In particula…
Motivated by families of formal moduli problems, in this note we generalize the notion of L-infinity space by allowing sheaves of L-infinity algebras over any (reasonable) nilpotent dg manifold. We discuss various examples including those coming from Lie algebroids. Given a Lie algebroid, we show that there is an L-inf…
A universal Lie ∞-algebroid is constructed for singular foliations.
problem Describing the geometry and structure of singular foliations.
method Construction of a Lie ∞-algebroid for every resolution of a singular foliation.
result The universal Lie ∞-algebroid uniquely encodes the geometry of singular foliations.
The Deligne groupoid is a functor from nilpotent differential graded Lie algebras concentrated in positive degrees to groupoids; in the special case of Lie algebras over a field of characteristic zero, it gives the associated simply connected Lie group. We generalize the Deligne groupoid to a functor gamma from L-infin…
This paper upgrades Khovanov homology to an L-infinity module structure.
problem Exploring Khovanov homology with L-infinity algebra structures.
method Developed an L-infinity algebra structure on sl2(∧) and showed annular Khovanov homology is an L-infinity module over it.
result The annular Khovanov homology of a link L is an L-infinity module over sl2(∧) up to quasi-isomorphism.
Multisymplectic geometry admits an operation that has no counterpart in symplectic geometry, namely, taking the product of two multisymplectic manifolds endowed with the wedge product of the multisymplectic forms. We show that there is an L-infinity-embedding of the L-infinity-algebra of observables of the individual f…
Given an n-term L-infinity algebra L, we construct a Kan simplicial manifold which we think of as the 'Lie n-group' integrating L. This extends work of Getzler math.AT/0404003 . In the case of an ordinary Lie algebra, our construction gives the simplicial classifying space of the corresponding simply connect Lie group.…
New algebra structure for Legendrian knots preserves contact homology invariants.
problem Constructing an algebra for Legendrian knots.
method Combining rational Symplectic Field Theory and combinatorial methods.
result Invariant Poisson algebra of Legendrian links under isotopy.
Homotopy moment mapsmath.DG
Associated to any manifold equipped with a closed form of degree >1 is an `L-infinity algebra of observables' which acts as a higher/homotopy analog of the Poisson algebra of functions on a symplectic manifold. In order to study Lie group actions on these manifolds, we introduce a theory of homotopy moment maps. Such a…
Computes -algebroid for linear foliations on vector spaces.
problem Invariants of singular foliations on vector spaces induced by Lie subalgebras.
method Explicitly constructs projective resolutions and computes -algebroid structure.
result Provides invariants and constant-rank replacements of singular foliations.
Study equigeodesics on compact homogeneous spaces using Lie algebra properties.
problem Identifying equigeodesic vectors on compact homogeneous spaces.
method Formula for equigeodesic vectors based on isotropy representation and Lie algebra structure.
result Identification of equigeodesic vectors solely through Lie algebra properties.
In the first chapter, we give a precise and general description of gerbes valued in arbitrary crossed module and over an arbitrary differential stack. We do it using only Lie groupoids, hence ordinary differential geometry, by considering differential stacks as being Lie groupoids up to Morita equivalence. We prove the…
Explains how pre-symplectic structures can be changed.
problem Understanding how pre-symplectic structures can be deformed.
method Uses Dirac geometry to explain the geometric origin of -algebra controlling deformations.
result Discovers the geometric origin of the -algebra controlling deformations of pre-symplectic structures.
An -algebra is built on symplectic manifold homology.
problem No specific problem stated; focuses on construction of algebra.
method Construction of an -algebra on symplectic manifold homology.
result The constructed -algebra naturally projects to a Lie algebra extension.
We study the isotropy representation of real flag manifolds associated to simple Lie algebras that are split real forms of complex simple Lie algebras. For each Dynkin diagram the invariant irreducible subspaces for the compact part of the isotropy subgroup are described. Contrary to the complex flag manifolds the deco…
Study how algebraic conditions on isotropy group affect Lorentzian homogeneous space geometry.
problem Understand how algebraic conditions on isotropy group affect the geometry and curvature of Lorentzian homogeneous spaces.
method Prove that a Lorentzian locally homogeneous space is locally isometric to a plane wave if it admits an Ambrose--Singer connection with indecomposable, non-irreducible holonomy.
result Generalize existing results about Lorentzian homogeneous spaces with irreducible isotropy and prove results about Lorentzian connections with parallel torsion and 2-symmetric connections.
A manifold is multisymplectic, or more specifically n-plectic, if it is equipped with a closed nondegenerate differential form of degree n+1. In our previous work with Baez and Hoffnung, we described how the `higher analogs' of the algebraic and geometric structures found in symplectic geometry should naturally arise i…
Paper constructs observables using multisymplectic geometry and algebraic methods.
problem Building observables in multisymplectic geometry.
method Uses -algebras, Gerstenhaber algebras, BV-modules, and constraint triples.
result Reconstructs and explains recent geometric results.
Inverse function theorem and homotopy description for L-infinity bundles.
problem Inverse function theorem and homotopy description for L-infinity bundles.
method Local sections composed of elementary morphisms.
result Simple description of homotopy category of L-infinity bundles.
Homotopy operators help describe structures in equivariant deformation problems.
problem Equivariant deformation problems in algebraic structures.
method Use homotopy operators for an -algebra associated with the problem.
result Smooth parametrization of the space of structures around a given one.
Split Courant algebroids linked to special algebra structures.
problem Understanding the structure of split Courant algebroids.
method Established a correspondence with multiplicative curved -algebras.
result Split Courant algebroids correspond to multiplicative curved -algebras.
Field Theories in Physics can be formulated giving a local Lagrangian density. Locality is imposed using the infinite jet bundle. That bundle is viewed as a pro-finite dimensional smooth manifold and that point of view has been compared to different topological and Frechét structures on it. A category of local (insular…
We establish a higher generalization of super L-infinity-algebraic T-duality of super WZW-terms for super p-branes. In particular, we demonstrate spherical T-duality of super M5-branes propagating on exceptional-geometric 11d super spacetime.
We define and study the degeneration property for BV-infinity algebras and show that it implies that the underlying L-infinity algebras are homotopy abelian. The proof is based on a generalisation of the well-known identity Δ(e^x)=e^x(Δ(x)+[x,x]/2) which holds in all BV algebras. As an application we show that the high…
New algebraic structure derived from Hopf algebra and Drinfel'd twist.
problem Developing a new algebraic structure from existing mathematical concepts.
method Extending -algebra to a Hopf algebra, twisting with Drinfel'd twist, and identifying Hopf morphisms and braided morphisms.
result Braided -algebra is derived from the process.
Homotopy momentum map extends Noether's theorem in general relativity.
problem Extending Noether's theorem to spacetime vector fields.
method Using homotopy momentum map and -algebras.
result Extension of conserved currents to spacetime vector fields.
Study shows equivalence in foliations and pre-symplectic forms aligns with gauge equivalence.
problem Deformation theory of foliations and pre-symplectic forms.
method Proved geometric equivalence agrees with algebraic gauge equivalence using -algebras.
result Gauge equivalences for foliations and pre-symplectic structures are consistent.
Study on invariant Einstein metrics on specific flag manifolds.
problem Existence of invariant Einstein metrics on real flag manifolds.
method Analysis of isotropy representations and Riemannian metrics.
result Existence of non-diagonal Einstein metrics on real flag manifolds.
Develops Morse theory for commuting gradient-like vector fields.
problem Understanding the algebraic structure of the infrared data.
method Formal analogue of Morse theory for commuting vector fields, constructing L-infinity algebra and Maurer-Cartan elements.
result The algebraic formalism is similar to the algebra of the infrared of Gaiotto, Moore, and Witten.
The isotropy action on certain symmetric spaces is shown to be equivariantly formal.
problem Understanding the equivariant formality of isotropy actions on symmetric spaces.
method Developed a new approach to prove equivariant formality for -symmetric spaces.
result Symmetric spaces with -symmetry are equivariantly formal and formal in the Sullivan sense.
We give a generalization of the notion of a Cartan-Ehresmann connection from Lie algebras to L-infinity algebras and use it to study the obstruction theory of lifts through higher String-like extensions of Lie algebras. We find (generalized) Chern-Simons and BF-theory functionals this way and describe aspects of their …
The paper constructs -algebras from contact Courant algebroids and isotropic subbundles.
problem Understanding the structure of contact Courant algebroids and their associated -algebras.
method The construction of -algebras from -Courant algebroids and isotropic subbundles.
result A relationship between constructed -algebras is established by a morphism.
Study rational homotopy types of embedding spaces of manifolds.
problem Understanding the rational homotopy types of embedding spaces of manifolds.
method Express rational homotopy types through combinatorially defined L-infinity algebras of diagrams.
result Expressed the rational homotopy type of connected components of embedding spaces.
The paper studies deformations of Lagrangian submanifolds using algebraic tools.
problem Deformation theory of Lagrangian submanifolds in symplectic geometry.
method Graded versions of the Darboux Theorem and Weinstein's Lagrangian tubular neighbourhood Theorem, attaching an -algebra to each submanifold.
result Controls the deformation theory of Lagrangian -submanifolds using an -algebra.
We prove that a polar orthogonal representation of a real reductive algebraic group has the same closed orbits as the isotropy representation of a pseudo-Riemannian symmetric space. We also develop a partial structural theory of polar orthogonal representations of real reductive algebraic groups which slightly generali…
-structures are weak forms of multiplications on closed oriented manifolds. As shown by Hopf the rational cohomology algebras of manifolds admitting -structures are free over odd degree generators. We prove that this condition is also sufficient for the existence of -structures on manifolds which are nilpotent…
Higher Gauge Flow Models integrate higher geometry and symmetries into Generative Flow Models.
problem Improving generative models' performance.
method Integrates L-algebra into Generative Flow Models, leveraging higher geometry and symmetries.
result Substantial performance improvements on Gaussian Mixture Model datasets.
New method for flux quantization on phase space stacks.
problem Defining and constructing flux-quantized phase space stacks.
method Observation of flux densities and characterization of Cauchy data.
result Flux-quantized phase space stacks have classifying spaces with rational Whitehead L-infinity algebra.
Study of manifolds with special holonomy using Frölicher-Nijenhuis bracket.
problem Understanding manifolds with special holonomy.
method Use of Frölicher-Nijenhuis bracket to define cohomologies and -algebras.
result Definition and computation of Frölicher-Nijenhuis cohomology.
It is well known that the Einstein equation on a Riemannian flag manifold reduces to a algebraic system, if is a -invariant metric. In this paper we described this system for all flag manifolds of a classical Lie group. We also determined the number of isotropy summands for all of these spaces and prov…
The paper extends Chern-Weil-Lecomte map to -algebras.
problem Defining characteristic classes for -algebra extensions.
method Using the Chern-Weil-Lecomte map to define characteristic classes in an -algebra setting.
result Unified definition of several known cohomology classes.
We introduce the notions of Atiyah class and Todd class of a differential graded vector bundle with respect to a differential graded Lie algebroid. We prove that the space of vector fields on a dg-manifold with homological vector field admits a structure of L-infinity algebra with the Lie derivative as unary …
Uniform Lie algebras are combinatorially defined two-step nilpotent Lie algebras which can be used to define Einstein solvmanifolds. These Einstein spaces often have nontrivial isotropy groups. We derive basic properties of uniform Lie algebras and we classify uniform Lie algebras with five or fewer generators. We defi…
We develop further the approach to derived differential geometry introduced in Costello's work on the Witten genus. In particular, we introduce several new examples of L-infinity spaces, discuss vector bundles and shifted symplectic structures on L-infinity spaces, and examine in some detail the example of derived loop…
We give a soft geometric proof of the classical result due to Conn stating that a Poisson structure is linearizable around a singular point (zero) at which the isotropy Lie algebra is compact and semisimple.
Develops deformation theory for symplectic foliations using -algebras.
problem Deformation of symplectic foliations.
method Uses -algebras to control deformation problems.
result Establishes a correspondence between small deformations and Maurer-Cartan elements of -algebra.
I define higher codimensional versions of contact structures on manifolds as maximally non-integrable distributions. I call them multicontact structures. Cartan distributions on jet spaces provide canonical examples. More generally, I define higher codimensional versions of pre-contact structures as distributions on ma…