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.
We develop a systematic approach to contact and Jacobi structures on graded supermanifolds. In this framework, contact structures are interpreted as symplectic principal GL(1,R)-bundles. Gradings compatible with the GL(1,R)-action lead to the concept of a graded contact manifold, in particular a linear (more generally,…
We show that every Lie algebra is equipped with a natural (1,1)-variant tensor field, the "canonical endomorphism field", naturally determined by the Lie structure, and satisfying a certain Nijenhuis bracket condition. This observation may be considered as complementary to the Kirillov-Kostant-Souriau theorem on symp…
Jacobi algebroids, that is graded Lie brackets on the Grassmann algebra associated with a vector bundle which satisfy a property similar to that of the Jacobi brackets, are introduced. They turn out to be equivalent to generalized Lie algebroids in the sense of Iglesias and Marrero and can be viewed also as odd Jacobi …
It is proven that a local Lie algebra in the sense of A. A. Kirillov determines the base manifold up to a diffeomorphism provided the anchor map is nowhere-vanishing. In particular, the Lie algebras of nowhere-vanishing Poisson or Jacobi brackets determine manifolds. This result has been proven for different types of d…
We consider coefficient bodies Mn for univalent functions. Based on the Löwner-Kufarev parametric representation we get a partially integrable Hamiltonian system in which the first integrals are Kirillov's operators for a representation of the Virasoro algebra. Then Mn are defined as sub-Riemann…
Binary operations on algebras of observables are studied in the quantum as well as in the classical case. It is shown that certain natural compatibility conditions with the associative product imply the properties which usually are additionally required. In particular, it is proved that locality of a Loday bracket on s…
Axioms of Lie algebroid are discussed in order to review some known aspects for non-experts. In particular, it is shown that a Lie QD-algebroid (i.e. a Lie algebra bracket on the Functions(M)-module F of sections of a vector bundle E over a manifold M which satisfies [X,fY]=f[X,Y]+A(X,f)Y for all X,Y from F, all f from…
The paper discusses reducing Hamiltonian systems by scaling and standard symmetries, leading to Kirillov Hamiltonian systems.
problem Reduction of symplectic Hamiltonian systems by scaling and standard symmetries.
method Proof of Kirillov Hamiltonian systems and equivalence of reductions.
result Equivalent Kirillov Hamiltonian systems from different reduction orders.
Introduces LRY skein algebras generalizing Kauffman bracket and Roger-Yang skein algebras.
problem Generalizing skein algebras for surfaces with arbitrary ground rings.
method Constructs LRY skein algebras, quantum traces, and Dehn-Thurston coordinates.
result LRY skein algebras are domains, have degenerations to monomial subalgebras of quantum tori, and are orderly finitely generated.
The paper constructs bundles and recovers Kirillov character formula.
problem Constructing smooth vector bundles over deformation to the normal cone.
method Rescaling of vector bundles and equivariant constructions.
result Recovery of Kirillov character formula for equivariant index.
Method calculates polytope volume using graph combinatorics and Kirillov-Reshetikhin invariants.
problem Computing the volume of hyperbolic polyhedra.
method Combining combinatorial graph reductions and geometric splitting into tetrahedra.
result Volume of polytope can be expressed through critical values of a potential function.
New proof connects Kashiwara-Vergne equations to Goldman-Turaev Lie bialgebra.
problem Proving Kashiwara-Vergne equations from isomorphism of Lie bialgebras.
method Novel characterization of conjugacy classes in free Lie algebra via cyclic words.
result Automorphisms inducing isomorphisms in Goldman-Turaev Lie bialgebra satisfy Kashiwara-Vergne equations.
Simplified approach to Jacobi and contact geometry.
problem Complexity in presenting Jacobi and contact geometry.
method Kirillov manifolds and linear Kirillov structures, relating homogeneity to principal GL(1,R)-bundle structure.
result Simplified and clearer understanding of Jacobi and contact geometry.
Geometric models for Lie--Hamilton systems on \(\mathbb{R}^2\) are described.
problem Analyzing Lie--Hamilton systems on \(\mathbb{R}^2\).
method Two geometric models: 1) restriction to symplectic leaves, 2) projection onto quotient space.
result Natural framework for Lie--Hamilton systems on \(\mathbb{R}^2\).
The study examines differential smoothness in specific Artin-Schelter regular algebras of dimension 5.
problem Investigating the differential smoothness of Artin-Schelter regular algebras of dimension 5.
method Analyzing the relationship between the number of generators and Gelfand-Kirillov dimension to identify structural obstructions.
result Certain two- and four-generator AS-regular algebras of global dimension five fail to admit a differential calculus, while a five-generator graded Clifford algebra provides a positive example.
We study actions of Lie supergroups, in particular, the hitherto elusive notion of orbits through odd (or more general) points. Following categorical principles, we derive a conceptual framework for their treatment and therein prove general existence theorems for the isotropy (or stabiliser) supergroups and orbits thro…
Leaves of Lie algebroids are Lie groupoids under certain conditions.
problem Understanding the structure of leaves in Lie algebroids.
method Analyzing cotangent Lie algebroids and Lie 2-group coadjoint orbits.
result Coadjoint orbits of Lie 2-groups are symplectic groupoids.
The paper studies geodesics in Teichmüller spaces and their smoothness or blowup.
problem Analyzing geodesics in Teichmüller spaces and their behavior over time.
method Proof of global existence for smooth solutions and blowup for initial smooth solutions in two Teichmüller spaces.
result Global existence for smooth solutions in one Teichmüller space and blowup for initial smooth solutions in another.
Quantum traces embed into quantum tori for surface skein algebras.
problem Embedding stated skein algebras into quantum tori.
method Two different embeddings using quantum trace maps and lambda length coordinates.
result Quantum cluster algebra of Muller equals reduced stated skein algebra.
Researchers analyze hypoelliptic heat kernels on nilpotent Lie groups.
problem Analyzing hypoelliptic heat kernels on nilpotent Lie groups.
method Using generalized Fourier transform and Kirillov's orbit method to describe unitary irreducible representations and write hypoelliptic heat kernels.
result Explicit formula for hypoelliptic heat kernel on Gn. Geometrically realises restricted tempered representations of Lie groups.
problem Realising the restriction of tempered representations to maximal compact subgroups.
method Using Dirac operators on homogeneous spaces identified with coadjoint orbits.
result Explicit geometric expression for multiplicities of K-types. We construct a Poisson isomorphism between the formal Poisson manifolds g^* and G^*, where g is a finite dimensional quasitriangular Lie bialgebra. Here g^* is equipped with its Lie-Poisson (or Kostant-Kirillov-Souriau) structure, and G^* with its Poisson-Lie structure. We also quantize Poisson-Lie dynamical r-matrices…
New bracket unifies nonholonomic dynamics and Hamilton-Jacobi theory.
problem Unified description of nonholonomic dynamics and Hamilton-Jacobi theory.
method Defined and proved coincidence of three nonholonomic brackets.
result Three nonholonomic brackets coincide.
Geometrically calculates multiplicities of K-types in tempered representations.
problem Calculating multiplicities of K-types in tempered representations of Lie groups.
method Geometric formula based on Kirillov's orbit method and quantisation commutes with reduction.
result Geometric expression for multiplicities of K-types in tempered representations.
New bracket theory connects three nonholonomic dynamics models.
problem Nonholonomic dynamics and their bracket formulations.
method Definition and proof of equivalence of three nonholonomic brackets.
result Three nonholonomic brackets are equivalent.
This note proves equivalence between Dorfman brackets and lifts, showing universality of the Courant-Dorfman bracket.
problem Characterizing twistings and symmetries of transitive Dorfman brackets.
method Proving equivalence between Dorfman brackets and lifts, intertwining with Courant-Dorfman bracket.
result Universality of the Courant-Dorfman bracket and characterization of Dorfman brackets via lifts.
Flat connections derived from Poisson brackets on loop spaces.
problem Understanding the structure of Poisson brackets on loop spaces.
method Defined connections by explicit linear combinations of standard connections associated with the Poisson bracket.
result Connections are shown to be flat.
It is shown that two definitions for an exterior differential in superspace, giving the same exterior calculus, yet lead to different results when applied to the Poisson bracket. A prescription for the transition with the help of these exterior differentials from the given Poisson bracket of definite Grassmann parity t…
The paper constructs compatible Poisson brackets on gl(N).
problem Constructing compatible Poisson brackets on gl(N).
method Using constant tensors and Schouten brackets, the paper explicitly constructs quadratic Poisson brackets compatible with the standard Lie-Poisson bracket.
result Explicit construction of quadratic Poisson brackets compatible with the standard Lie-Poisson bracket on gl(N).
We survey the many instances of derived bracket construction in differential geometry, Lie algebroid and Courant algebroid theories, and their properties. We recall and compare the constructions of Buttin and Vinogradov, and we prove that the Vinogradov bracket is the skew-symmetrization of a derived bracket. Odd (resp…
Researchers attempt to categorify biquandle brackets using Khovanov homology methods.
problem Categorify biquandle brackets using Khovanov homology methods.
method Outline a Khovanov homology-style construction for biquandle brackets.
result A canonical biquandle 2-cocycle is defined, but not a true categorification of biquandle brackets.
New formulas for equivariant indices of non-product Dirac operators near boundaries.
problem Equivariant indices of non-product Dirac operators near boundaries.
method Generalized APS boundary problem, Kirillov formula, explicit formulas.
result Explicit formulas for the equivariant signature of local systems over manifolds with boundary.
Extends Nambu-Poisson bracket to superspace R^{n|m}.
problem No new problem introduced.
method Constructs Nambu-Poisson algebras of even degree functions using superdeterminant.
result Proves the n-ary bracket satisfies conditions for Nambu-Poisson bracket in R^{n|1} and R^{n|2}.
Explains biquandle brackets and quivers for a topology talk.
problem None explicitly stated; focuses on background information.
method Review of biquandle concepts and related structures.
result Clarifies understanding of biquandle bracket quivers.
We prove that the groupoid of transformations of rigid structures on surfaces has a finite presentation as a 2-groupoid establishing a result first conjectured by G.Moore and N.Seiberg. An alternative proof was given by B.Bakalov and A.Kirillov Jr. We present some applications to TQFTs. This is also related to recent w…
New method calculates knot and link biquandle brackets using trace diagrams.
problem Computing biquandle brackets of knots and links efficiently.
method Using trace diagrams to compute biquandle brackets of oriented knots and links.
result Identified algebraic conditions for strand moves and stop conditions.
A new framework describes dissipation using a metriplectic 4-bracket.
problem Describing dissipation in a way that preserves energy and entropy.
method Using a metriplectic 4-bracket, a quantity like the Poisson bracket with symmetries motivated by Riemannian curvature.
result The metriplectic 4-bracket dynamics includes all known previous binary bracket theories for dissipation.
New biquandle bracket invariants are linked to biquandle 2-cocycles.
problem Quantum enhancements and biquandle colored links.
method Proving biquandle bracket invariants are pointwise products of other invariants and biquandle 2-cocycles.
result New biquandle bracket invariants are equivalent to the Jones polynomial on knots.
New geometric definition of Lie bracket for undirected curves.
problem Understanding the Lie bracket of undirected curves on a surface.
method Local geometric definition and proof of three results.
result The TWG bracket counts intersection and suggests disjoint representatives.
New examples show non-trivial parity-biquandle bracket.
problem Constructing non-trivial parity-biquandle bracket examples.
method Slightly changed notation and constructed examples of knots and links.
result Minimality theorem: graphs appear as link invariants.
Abstract: Poisson bracket on shear coordinates relates to Fenchel-Nielsen bracket on gluing parameters.
problem Relationship between Fenchel-Nielsen coordinates and shear coordinates on Riemann surfaces.
method Explicitly showed the Poisson bracket on shear coordinates induces the Fenchel-Nielsen bracket on gluing parameters.
result Poisson bracket on shear coordinates relates to Fenchel-Nielsen bracket on gluing parameters.
Kishino's knot is not detected by the fundamental group or the bracket polynomial; these invariants cannot differentiate between Kishino's knot and the unknot. However, we can show that Kishino's knot is not equivalent to unknot by applying either the 3-strand bracket polynomial or the surface bracket polynomial. In th…
Enhances quantum invariants using tribracket brackets.
problem Quantum invariants of tribracket-colored knots and links.
method Introduces tribracket brackets as skein invariants.
result Provides new quantum invariants and examples.
Defines a new Poisson bracket on differential forms for symplectic and pseudo-Riemannian metrics.
problem No specific problem stated; defining a new mathematical structure.
method Defined a non-degenerate even Poisson bracket on the algebra of differential forms.
result Established properties and compared with the Koszul-Schouten bracket.
The paper defines symmetric brackets for skew-symmetric algebroids with totally skew-symmetric torsion.
problem Defining symmetric brackets for skew-symmetric algebroids.
method Using connections with totally skew-symmetric torsion and pseudo-Riemannian metrics.
result Explicit formula for the Levi-Civita connection and symmetric brackets on almost Hermitian manifolds.
The goal of this diploma thesis is to give a detailed description of Kirillov's Orbit Method for the case of compact connected Lie groups. The theory of Kirillov aims at finding all irreducible unitary representations of a given Lie group G. The first chapter is intended to recall some facts about Lie groups. The mos…
Positive basis of Kauffman bracket skein algebras proven using Chebyshev polynomials.
problem Positivity of Kauffman bracket skein algebras of surfaces.
method Using Chebyshev polynomials as a basic block.
result Chebyshev polynomials form a positive basis.