This paper associates homotopy Poisson-n algebras to higher symplectic structures.
problem Generalizing symplectic Poisson algebras to higher symplectic structures.
method Introducing a homotopy Poisson-n algebra associated with higher symplectic structures.
result The exterior product does not close on Poisson cotensors, but valid computations remain.
This paper shows the equivalence of the categories of N-manifolds of degree 2 with the category of double vector bundles endowed with a linear metric. Split Poisson N-manifolds of degree 2 are shown to be equivalent to self-dual representations up to homotopy. As a consequence, the equivalence above induces an …
The characterization of the Nambu-Poisson n-tensors as a subfamily of the Generalized-Poisson ones recently introduced (and here extended to the odd order case) is discussed. The homology and cohomology complexes of both structures are compared, and some physical considerations are made.
The study connects hypergraphs to strong homotopy Lie algebras.
problem Characterizing hypergraphs with a system of distinct representatives.
method Describing a procedure to attach nilpotent strong homotopy Lie algebras to hypergraphs.
result Isomorphic hypergraphs correspond to isomorphic strong homotopy Lie algebras.
Homotopy equivalence of cotangent bundles' function algebras is shown.
problem Understanding homotopy equivalence in cotangent bundles and their function algebras.
method Using shifted Poisson algebras and homotopy equivalence of bundles.
result Homotopy equivalent bundles have equivalent Poisson algebras.
Constructs a homotopy Loday algebra from symplectic 2-manifolds.
problem Tackles the construction of algebraic structures from symplectic 2-manifolds.
method Uses higher derived brackets and Voronov's technique to construct a homotopy Loday algebra.
result Constructs a homotopy Loday algebra with a specific structure accommodating the Dorfman bracket.
The settings for homotopical algebra---categories such as simplicial groups, simplicial rings, A∞ spaces, E∞ ring spectra, etc.---are often equivalent to categories of algebras over some monad or triple T. In such cases, T is acting on a nice simplicial model category in such a way that T descends…
This paper emphasizes the ubiquitous role of moduli spaces of algebraic curves in associative algebra and algebraic topology. The main results are: (1) the space of an operad with multiplication is a homotopy Gerstenhaber (i.e., homotopy graded Poisson) algebra; (2) the singular cochain complex is naturally an operad; …
We construct a functor from the derived category of homotopy Gerstenhaber algebras with finite-dimensional cohomology to the purely geometric category of so-called F∞-manifolds. The latter contains Frobenius manifolds as a subcategory (so that a pointed Frobenius manifold is itself a homotopy Gerstenhaber alg…
Study realizes symplectic algebras and homotopy types on manifolds.
problem Realizing symplectic algebras and homotopy types on manifolds.
method Addressing questions on realizability of symplectic algebras and rational homotopy types by closed symplectic manifolds.
result Realization of symplectic algebras and homotopy types in various dimensions.
Characterizes a specific type of Courant algebroid with a Calabi-Yau structure.
problem Understanding specific types of Courant algebroids with Calabi-Yau structures.
method Explains how a homotopy BV algebra with certain properties characterizes these algebroids.
result A Courant algebroid with a Calabi-Yau structure is a homotopy BV algebra with specific properties.
New algebraic structures for Hermitian geometry cohomologies.
problem Understanding cohomologies of Hermitian manifolds.
method Introducing BV-algebras and homotopy BV-algebras.
result Cohomologies of Hermitian manifolds are endowed with homotopy hypercommutative algebra structures.
We introduce higher Kirillov brackets and algebroids for supermanifolds.
problem Understanding higher structures on supermanifolds.
method Introducing homotopy Kirillov algebras and algebroids.
result Construction of homotopy versions of Kirillov's theorems.
Homotopy operators help describe structures in equivariant deformation problems.
problem Equivariant deformation problems in algebraic structures.
method Use homotopy operators for an L∞-algebra associated with the problem. result Smooth parametrization of the space of structures around a given one.
Relates field theory algebras to sigma model geometric structures.
problem Conformal field theory and sigma model geometric structures.
method Formulate conformal invariance conditions as generalized Maurer-Cartan equations.
result Einstein equations with extra fields in the quasi-classical limit.
Surveying A1-homotopy theory and contractible varieties.
problem Understanding the relationship between A1-homotopy theory and contractible varieties. method Exploring the interplay between A1-homotopy theory and affine algebraic geometry. result Highlighting the connection between A1-homotopy theory and contractible varieties. New invariant constructed using stable homotopy methods.
problem Constructing a new link invariant.
method Stable homotopy theory and Khovanov's method.
result A Khovanov slk-stable homotopy type constructed. Paper constructs characteristic classes using C-infinity algebras.
problem Characteristic classes of fiber bundles.
method Homotopy theory of C-infinity algebras.
result Chern-Weil-type construction of characteristic classes.
We study Maurer-Cartan elements on homotopy Poisson manifolds of degree n. They unify many twisted or homotopy structures in Poisson geometry and mathematical physics, such as twisted Poisson manifolds, quasi-Poisson $\g$-manifolds, and twisted Courant algebroids. Using the fact that the dual of an n-term $L_\infty…
Classifies two-dimensional extended homotopy field theories with aspherical targets.
problem Classifying two-dimensional extended homotopy field theories with aspherical targets.
method Defining and classifying E-HFTs with specific properties and using Frobenius algebras.
result Classifying E-HFTs taking values in symmetric monoidal bicategories of algebras and bimodules.
We study the semidirect product of a Lie algebra with a representation up to homotopy and provide various examples coming from Courant algebroids, string Lie 2-algebras, and omni-Lie algebroids. In the end, we study the semidirect product of a Lie group with a representation up to homotopy and use it to give an integra…
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.
Symmetries of Poisson manifolds are in general quantized just to symmetries up to homotopy of the quantized algebra of functions. It is therefore interesting to study symmetries up to homotopy of Poisson manifolds. We notice that they are equivalent to Poisson principal bundles and describe their quantization to symmet…
An involutive distribution C on a smooth manifold M is a Lie-algebroid acting on sections of the normal bundle TM/C. It is known that the Chevalley-Eilenberg complex associated to this representation of C possesses the structure X of a strong homotopy Lie-Rinehart algebra. It is natural to interpret …
Two approaches study the homotopy of blow ups in algebraic and symplectic geometry.
problem Investigate the homotopy of blow ups in algebraic and symplectic geometry.
method Develops fibrewise surgery theory and a purely homotopy theoretic approach.
result Obtained homotopy decompositions of the based loop space on blow ups.
Simple construction of Rumin algebra for contact manifolds.
problem Computing the de Rham cohomology algebra of contact manifolds.
method Using Markl's Homotopy Transfer Theorem for a simple explicit construction.
result Recovery of the Rumin algebra as a contact invariant C∞-algebra. We give a construction of homotopy algebras based on ``higher derived brackets''. More precisely, the data include a Lie superalgebra with a projector on an Abelian subalgebra satisfying a certain axiom, and an odd element Δ. Given this, we introduce an infinite sequence of higher brackets on the image of the project…
It is shown that any compact Kähler manifold M gives canonically rise to two strongly homotopy algebras, the first one being associated with the Hodge theory of the de Rham complex and the second one with the Hodge theory of the Dolbeault complex. In these algebras the product of two harmonic differential forms is ag…
This Master Thesis is devoted to the study of n-plectic manifolds and the Strongly Homotopy Lie algebras, also called L∞-algebras, that can be associated to them. Since multisymplectic geometry and L∞-algebras are relevant in Theoretical Physics, and in particular in String Theory, we introduce th…
New homotopy refinements for tangle invariants.
problem Stable homotopy refinements for tangle invariants.
method Refined Khovanov and Chen-Khovanov spectra.
result Induces refinements of platform algebras and invariants.
Lie algebroids and curved Lie algebras are equivalent categories.
problem Understanding the relationship between Lie algebroids and curved Lie algebras.
method Developed a method to study the ∞-category of curved Lie algebras using homotopy theory of algebras over a complete operad. result Equivalence of ∞-categories between Lie algebroids and certain kinds of curved Lie algebras. 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 L∞-algebras. result Extension of conserved currents to spacetime vector fields.
New algebra models refine complex manifold homotopy groups.
problem Understanding complex manifold homotopy groups better.
method Free, bigraded bidifferential algebra models with quasi-isomorphism.
result Obtained minimal models unique up to isomorphism.
Abstract: Homotopy Poisson algebra models for reduced spaces derived from Poisson structures.
problem Homotopy Poisson algebra models for reduced spaces.
method Cattaneo-Zambon compatibility and regularity conditions, equivariant map, homotopy Poisson algebra.
result Derivation of homotopy Poisson algebra generalizing classical BFV algebra.
Homotopy actions of Lie algebroids defined as L∞-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.
New homotopy refinements for tangle invariants defined.
problem Stable homotopy refinements for tangle invariants.
method Defined stable homotopy refinements of Khovanov's arc algebras and tangle invariants.
result Stable homotopy refinements of Khovanov's arc algebras and tangle invariants defined.
New invariants define the rational and real homotopy types of closed manifolds.
problem Defining invariants for the rational and real homotopy types of closed manifolds.
method Introducing isotopy modulo k and minimal unital cyclic C-infinity-algebras.
result A complete set of invariants uniquely defines the rational and real homotopy types of closed simply connected manifolds.
Study sheaves of Lie-Rinehart algebras and their morphisms, generalizing Lie algebroid concepts.
problem Understanding sheaves of Lie-Rinehart algebras and their morphisms.
method Introduced morphisms and comorphisms, proved factorization theorems, and defined higher homotopy groups and groupoids.
result Sheaves of Lie-Rinehart algebras over smooth manifolds induce partitions into orbits of the fundamental groupoid.
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.
Higher homotopy generalizations of Lie-Rinehart algebras, Gerstenhaber-, and Batalin-Vilkovisky algebras are explored. These are defined in terms of various antisymmetric bilinear operations satisfying weakened versions of the Jacobi identity, as well as in terms of operations involving more than two variables of the L…
New method for moment maps in multisymplectic geometry using Lie 2-algebras.
problem Existence and construction of moment maps in multisymplectic geometry.
method Introducing homotopy moment maps defined on a Lie 2-algebra.
result Existence criteria and construction of homotopy moment maps.
Scalable spaces are simply connected manifolds with nice cohomology properties.
problem Understanding the limitations of formality in higher homotopy groups.
method Analyzing the embedding of cohomology algebras into differential forms.
result Spaces that are formal but not scalable provide counterexamples to Gromov's conjecture.
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…
Study homotopy types of free racks and quandles, proving analogs of Milnor's theorem.
problem Understanding the homotopy types of free racks and quandles.
method Proved analogs of Milnor's theorem for racks and quandles and their pointed variants.
result Identified the homotopy types of free racks and quandles on spaces of generators.
New examples of manifolds that are homotopy but not simple homotopy equivalent.
problem Characterizing simple homotopy types of even dimensional manifolds.
method Using algebraic K-theory, surgery obstruction map, and homotopy automorphisms.
result Construction of infinite families of manifolds that are homotopy equivalent but not simple homotopy equivalent.
Study shows weak homotopy equivalences for complete minimal surfaces.
problem Understanding complete minimal surfaces and their properties.
method Analyzes algebraic null immersions and conformal minimal immersions.
result Inclusion and differential mappings are weak homotopy equivalences.
Algebra Situs is a branch of mathematics which has its roots in Jones' construction of his polynomial invariant of links and Drinfeld's work on quantum groups. It encompasses the theory of quantum invariants of knots and 3-manifolds, algebraic topology based on knots, operads, planar algebras, q-deformations, quantum g…
Study ideals in the Goldman Lie algebra S by mapping to a simpler algebra.
problem Identify and classify ideals in the Goldman Lie algebra S. method Construct an algebra homomorphism to a simpler structure and classify ideals.
result Found infinite classes of ideals in both Z-module and Q-module cases.