Study classifies Hamiltonian operators with skew-symmetric constraints.
problem Classifying Hamiltonian operators with skew-symmetric constraints.
method Using Poisson vertex algebras and differential-geometric constraints.
result Complete classification results for 2-component and 3-component cases.
The theory of Poisson Vertex Algebras (PVAs) is a good framework to treat Hamiltonian partial differential equations. A PVA consists of a pair (A,{⋅λ⋅}) of a differential algebra A and a bilinear operation called the λ-bracket. We extend the definition to the class of algebras $\mat…
The paper studies deformations of Poisson brackets in two dimensions, finding non-trivial cohomology groups.
problem Deformations of multidimensional Poisson brackets of hydrodynamic type.
method Cohomology computation of PVAs associated with Poisson brackets at third differential degree.
result Non-trivial third cohomology group indicates non-equivalent infinitesimal deformations.
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.
Let M be a Riemannian manifold. For p∈M, the tensor algebra of the negative part of the (complex) affinization of the tangent space of M at p has a natural structure of a meromorphic open-string vertex algebra. These meromorphic open-string vertex algebras form a vector bundle over M with a connection. We …
Researchers link vertex algebras to non-Kähler solutions of the Hull-Strominger system.
problem Constructing representations of vertex algebras from non-Kähler solutions of the Hull-Strominger system.
method Embedding the N=2 superconformal vertex algebra in the chiral de Rham complex of a string Courant algebroid, with a condition on the Hermitian-Yang-Mills connection.
result Any solution of the Hull-Strominger system satisfying the Hermitian-Yang-Mills condition has an associated N=2 embedding.
Graph Lie algebras have infinite prolongation if they have a vertex of degree one.
problem Determining when the prolongation of a graph Lie algebra is infinite-dimensional.
method Analyzing labeled direct graphs and their associated Lie algebras.
result Graph Lie algebras are infinite-dimensional if and only if they have a vertex of degree one.
Mathematical construction of vertex algebra representations from integrable G2 structures.
problem Constructing representations of a specific vertex algebra from geometric input.
method Integrable G2 structures with closed torsion on group manifolds, embedding into superaffine vertex algebra and chiral de Rham complex.
result Embeddings of deformed Shatashvili-Vafa vertex algebra in the chiral algebra of heterotic G2 backgrounds.
Find first (0,2) mirror symmetry examples on Hopf surfaces.
problem Find (0,2) mirror symmetry on compact non-Kähler manifolds.
method Use Borisov's approach with vertex algebras and chiral de Rham complex. Study Killing spinors on quadratic Lie algebras and embeddings of superconformal vertex algebras.
result Construct first (0,2) mirror pairs of Hopf surfaces.
Study Poisson algebras for Hamiltonian systems linearization.
problem Linearize dynamics along Poisson submanifolds.
method Use contravariant derivative to characterize Poisson algebras.
result Infinitesimal Poisson algebras provide a framework for Hamiltonization.
We construct a new equivariant cohomology theory for a certain class of differential vertex algebras, which we call the chiral equivariant cohomology. A principal example of a differential vertex algebra in this class is the chiral de Rham complex of Malikov-Schechtman-Vaintrob of a manifold with a group action. The ma…
The Bergman space and conformally flat 2-disk operads are linked to vertex operator algebras.
problem Understanding invariants of 2D Riemannian manifolds using algebraic structures.
method Introducing a suboperad and showing algebraic structures, using conformally flat factorization homology.
result The Bergman space is identified with the ind-Hilbert space completion of the affine Heisenberg vertex operator algebra.
New algebraic tools solve Poisson and Lie bialgebra problems.
problem Modular class and intrinsic biderivation in Poisson geometry.
method Algebraic tools from differential Gerstenhaber algebras and Batalin-Vilkobisky algebras.
result Applications to Lie bialgebra and Poisson cohomology.
Abstract studies 3-manifolds and vertex algebras, expanding known connections.
problem Understanding the relationship between 3-manifolds and vertex algebras.
method Developed a dictionary between 3-manifolds and vertex algebras using q-series. result Generalized known entries to higher rank Lie groups and 3-manifolds with toral boundaries.
We introduce two K-theories, one for vector bundles whose fibers are modules of vertex operator algebras, another for vector bundles whose fibers are modules of associative algebras. We verify the cohomological properties of these K-theories, and construct a natural homomorphism from the VOA K-theory to the associa…
We give a vertex algebra proof of the Berglund-Hübsch duality of nondegenerate invertible potentials. We suggest a way to unify it with the Batyrev-Borisov duality of reflexive Gorenstein cones.
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.
Study meromorphic open-string vertex algebras and modules over Riemannian manifolds.
problem Characterize meromorphic open-string vertex algebras and their modules over Riemannian manifolds.
method Explicitly determine bases for meromorphic open-string vertex algebras and their modules, using parallel tensors and eigenfunctions of the Laplace-Beltrami operator.
result Every irreducible module of a specific type is completely reducible if every composition factor is generated by eigenfunctions of eigenvalue p(p−1)K for some p∈Z+. We construct bundles of modules of vertex operator algebras, and prove the rigidity and vanishing theorem for the Dirac operator on loop space twisted by such bundles. This result generalizes many previous results.
Abstract proposes a new categorical approach to quantization of Poisson algebras.
problem Quantization of Poisson algebras.
method Defining quantization categories as subcategories of R-module categories with classical limits.
result Categories of strict deformation quantization, prequantization, and matrix regularization are equivalent, while Poisson enveloping algebra is not.
New Poisson structures on algebras linked to derivatives.
problem Understanding Poisson structures on trivial extension algebras.
method Introducing a correspondence between Poisson structures and derivatives, and formulating results in terms of Lie algebroids.
result One-to-one correspondence between Poisson structures and data involving derivatives.
Injective map found between Poisson algebras.
problem Mapping between Poisson algebras on Grassmannian and swapping algebra.
method Injective Poisson homomorphism from Grassmannian to swapping algebra.
result Found an injective Poisson homomorphism.
The first cohomology of Poisson algebras is described and conditions for its vanishing are established.
problem Understanding the first cohomology of Poisson algebras.
method Description and mapping of first cohomology to intrinsic cohomologies of Poisson submanifolds, formulation of vanishing conditions.
result Necessary and sufficient conditions for the vanishing of the first cohomology of infinitesimal Poisson algebras are derived.
Poisson algebra is usually defined to be a commutative algebra together with a Lie bracket, and these operations are required to satisfy the Leibniz rule. We describe Poisson structures in terms of a single bilinear operation. This enables us to explore Poisson algebras in the realm of non-associative algebras. We stud…
New bialgebra structures for relative Poisson algebras are introduced.
problem Extending bialgebra structures from commutative differential algebras to relative Poisson algebras.
method Introducing new bialgebra structures (relative PCA bialgebras) and using commutative 2-cocycles.
result New bialgebra structures (relative PCA bialgebras) are equivalent to certain Manin triples.
Cohomology of 'book' Lie algebra Poisson structure computed.
problem Computing the cohomology of a specific Lie algebra structure.
method Direct computation of cohomology for the given Lie algebra structure.
result Explicit formula for Poisson cohomology of the 'book' Lie algebra.
Expands on graded Poisson algebras, their properties, and applications.
problem None explicitly stated; focuses on overview and properties.
method Overview and discussion of properties and applications.
result Provides detailed overview of graded Poisson algebras and their contexts.
We associate a homotopy Poisson-n algebra to any higher symplectic structure, which generalizes the common symplectic Poisson algebra of smooth functions. This provides robust n-plectic prequantum data for most approaches to quantization. UPDATE: It has been brought to my attention that the exterior product does not cl…
In this thesis, we study deformations of compact holomorphic Poisson manifolds and algebraic Poisson schemes in the framework of Kodaira-Spencer's analytic deformation theory and Grothendieck's algebraic deformation theory.
Method computes centers of Poisson and skein algebras for loops on surfaces.
problem Computing centers of Poisson and skein algebras associated to loops on surfaces.
method Systematic method using Goldman and Wolpert's Poisson algebras and Turaev's skein algebras.
result Computed centers of various Poisson and skein algebras for finite type hyperbolic surfaces.
The purpose of this paper is to establish an explicit correspondence between various geometric structures on a vector bundle with some well-known algebraic structures such as Gerstenhaber algebras and BV-algebras. Some applications are discussed. In particular, we found an explicit connection between the Koszul-Brylins…
Study of Batalin-Vilkovisky algebra on Poisson manifolds with diagonalizable modular symmetry.
problem Exploring Batalin-Vilkovisky algebra structures on Poisson manifolds with specific symmetry conditions.
method Analysis of twisted Poincaré duality and mixed complex structure, combined with Kontsevich's deformation quantization and Koszul duality.
result Generalization of Batalin-Vilkovisky algebra structure to Poisson manifolds with diagonalizable modular symmetry.
New 4-manifold invariants derived from vertex algebras.
problem Computing 4-manifold invariants, including new and old ones.
method Using chiral correlation functions in half-twisted 2d N=(0,2) theories from compactified fivebranes. result Prediction of structural properties of multi-monopole invariants and non-abelian generalizations.
Symmetric Poisson structures linked to geodesic foliations and Jordan algebras.
problem Understanding geometric structures related to geodesic foliations and dynamics.
method Introducing symmetric Poisson structures, proving correspondences with geodesic foliations and Jordan algebras.
result Symmetric Poisson structures correspond to totally geodesic foliations and Jacobi-Jordan algebras.
Study quantum groups and vertex algebras, linking tangle invariants and asymptotic dimensions.
problem Relationships between quantum groups and singlet vertex algebras.
method Use deformable families of modules to compute tangle invariants and relate to asymptotic dimensions.
result Regularized asymptotic dimensions of characters of singlet vertex algebras match modified traces of open Hopf link invariants.
Let R be a commutative ring, and let A be a Poisson algebra over R. We construct an (R,A)-Lie algebra structure, in the sense of Rinehart, on the A-module of Kähler differentials of A depending naturally on A and the Poisson bracket. This gives rise to suitable algebraic notions of Poisson homology and cohomology for a…
Derived Poisson structures from Lie pairs are studied and their algebraic properties are explored.
problem Exploring derived Poisson structures from Lie pairs.
method Algebraic and homotopy transfer theorems for derived Poisson algebras.
result Derived Poisson algebra structure on totΩA∙(Λ∙(L/A)) is unique up to isomorphism. Study examines linear Poisson structures tied to W*-algebras.
problem Understanding fiber-wise linear Poisson structures related to W*-algebras.
method Investigates structures defined by W*-algebra structure and shows their arrangement in a short exact sequence of VB-groupoids.
result Fiber-wise linear Poisson structures are arranged in a short exact sequence of VB-groupoids.
We study noncommutative generalizations of such notions of the classical symplectic geometry as degenerate Poisson structure, Poisson submanifold and quotient manifold, symplectic foliation and symplectic leaf for associative Poisson algebras. We consider these structures for the case of the endomorphism algebra of a v…
Let P be a Poisson algebra, E a vector space and π:E→P an epimorphism of vector spaces with V=Ker(π). The global extension problem asks for the classification of all Poisson algebra structures that can be defined on E such that π:E→P becomes a morphism of Poisson algebras. From a geometri…
Develops Poisson algebras for field theories using synthetic differential geometry.
problem Constructing Poisson algebras for non-linear field theories.
method Synthetic differential geometry and Cahiers topos model.
result Formulates a Poisson algebra for field theories, showing it forms a family of observables.
Study of Riemann-Poisson Lie groups with compatibility conditions.
problem Characterizing and constructing Riemann-Poisson Lie groups.
method Left-invariant metrics and Poisson tensors compatible in Lie groups.
result Characterization and construction of Lie algebras up to dimension 5.
The notion of Poisson manifold with compatible pseudo-metric was introduced by the author in [1]. In this paper, we introduce a new class of Lie algebras which we call a pseudo-Rieamannian Lie algebras. The two notions are strongly related: we prove that a linear Poisson structure on the dual of a Lie algebra has a com…
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…
A Riemann-Lie algebra is a Lie algebra G such that its dual G∗ carries a Riemannian metric compatible (in the sense introduced by th author in C. R. Acad. Paris, t. 333, Série I, (2001) 763-768) with the canonical linear Poisson sructure of G∗. The notion of Riemann-Lie algebra has its origin…
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.
Jacobi/Poisson algebras are algebraic counterparts of Jacobi/Poisson manifolds. We introduce representations of a Jacobi algebra A and Frobenius Jacobi algebras as symmetric objects in the category. A characterization theorem for Frobenius Jacobi algebras is given in terms of integrals on Jacobi algebras. For a vecto…
The paper studies quadratic Poisson structures on Lie algebras, finding a 10-parametric family.
problem Compatibility of quadratic Poisson structures with linear structures on Lie algebras.
method Developed general theory and studied families of functions in involution.
result Found a 10-parametric family of quadratic Poisson structures on $\gl(3)^*$.