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 “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…
New approach to Lagrangian field theories using pro-finite structures and L-infinity algebras.
problem Formulating Lagrangian field theories with locality constraints.
method Using pro-finite structures and L-infinity algebras to define local observables and a pre-multisymplectic form.
result Definition of L-infinity algebra of local observables based on Lagrangian cohomology.
Establishes higher T-duality for super M-branes.
problem Generalizing T-duality for super p-branes.
method Super L-infinity-algebraic T-duality for super WZW-terms.
result Spherical T-duality of super M5-branes.
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…
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.…
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…
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.
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…
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.
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.
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.
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…
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.
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 …
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.
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…
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…
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 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.
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 …
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.
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.
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.
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 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.
-actions of Lie algebroidsmath.DG
Homotopy actions of Lie algebroids defined as -algebra morphisms.
problem Defining and studying homotopy actions of Lie algebroids.
method Constructing homological vector fields on the semi-direct product and proving bijection.
result The construction is a bijection between homotopy actions and homological vector fields.
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 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.
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.
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.
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.
This paper reinterprets Khovanov-Sano symmetries using BV formalism.
problem Understanding symmetries in equivariant Khovanov homology.
method Identifying Shumakovitch operator as a BV Laplacian and proving -algebra structure.
result Construction of an intrinsic -algebra on the Khovanov-Sano complex.
Higher curvature corrections to Bianchi identities conflict with exceptional generalised geometry.
problem Incompatibility of higher curvature corrections with exceptional generalised geometry.
method Analysis of higher derivative corrections to Bianchi identities in supergravity and M-theory.
result Higher curvature corrections imply an algebra for the gauge field, terminating at level ten.
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…
Study infinitesimal deformations of Lie algebroid pairs.
problem Infinitesimal deformations of Lie algebroid pairs.
method Investigate isomorphism classes of infinitesimal deformations of modulo automorphisms from exponentials of derivations of and those from the exponentials of inner derivations of .
result Find the associated governing -algebras in the sense of extended deformation theory.
The work of Oh and Park ([OP]) on the deformation problem of coisotropic submanifolds opened the possibility of studying a large and interesting class of foliations with some explicit geometric tools. These tools assemble into the structure of an L-infinity algebra on the shifted foliation complex (Ω^*[1](\fol), d_\fol…
New algebra governs deformations of Dirac-Jacobi structures.
problem Deformation theory of Dirac-Jacobi structures.
method Using higher derived brackets and split Courant-Jacobi algebroids, an algebra is associated with each Dirac-Jacobi structure.
result There is a one-to-one correspondence between MC elements of the algebra and small deformations of the Dirac-Jacobi structure.
Introduces new connections in higher geometry.
problem Defining connections in higher geometry.
method Develops formal differentiation and integration of maps to derived stacks.
result Establishes new -algebras of higher symmetries.
Develops homotopies for Lagrangian field theory using advanced algebraic structures.
problem Formulating a consistent framework for Lagrangian field theory.
method Introduces algebras and homotopies to enrich the Batalin-Vilkovisky framework.
result Provides an explicit lift of the Batalin-Vilkovisky framework to local forms.
The paper explores connections between dg manifolds and homotopy Lie algebras.
problem Understanding the relationship between dg manifolds and homotopy Lie algebras.
method Study of formal exponential maps, Atiyah classes, and Kapranov L-infinity algebras.
result Existence of formal exponential maps linked to vanishing of Atiyah classes.
We solve higher-order morphisms for twisted Courant algebras.
problem Construct canonical -morphisms for higher Courant algebroids.
method Develop a general framework for arbitrary .
result Affirmative answer to Zambon's question for higher degrees.
In the background effective field theory of heterotic string theory, the Green-Schwarz anomaly cancellation mechanism plays a key role. Here we reinterpret it and its magnetic dual version in terms of differential twisted String- and differential twisted Fivebrane-structures that generalize the notion of Spin-structure…
Extends Chern character to non-abelian cohomology, linking to physics.
problem Generalizing Chern character to non-abelian cohomology.
method Leveraging dg-algebraic rational homotopy theory and de Rham theorem.
result Generalizes Chern-Dold character, Chern-Weil homomorphism, and Cheeger-Simons homomorphism.
The paper studies deformations of Lie ideals in Lie algebras.
problem Understanding deformations of Lie ideals in Lie algebras.
method Develops deformation theory, compares cohomologies, enriches deformation complex.
result Deformation cohomology classes differentiate smooth deformations of ideals.
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.