Develops method to construct Lie algebra weight system kernel using Vogel algebra.
problem Detecting correlators and distinguishing knots in 3D Chern-Simons theory.
method Uses Vogel's Λ algebra and Jacobi diagrams.
result Explicitly provides Jacobi diagrams in the kernel of sl_N weight system.
Unified approach to deform Lie-Hamilton systems using Poisson-Hopf algebra.
problem Deforming Lie systems with quantum algebras.
method Poisson-Hopf algebra deformations applied to Lie-Hamilton systems.
result Unified approach to deformations of Lie-Hamilton systems on the real plane.
Automorphisms of Lie algebras and their root systems are fully lifted.
problem Understanding automorphisms of real semisimple Lie algebras and their root systems.
method Proving every automorphism of the restricted root system can be lifted to a Lie algebra automorphism.
result Automorphisms of restricted root systems can be fully lifted to Lie algebras.
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.
Investigates integrable systems with linear periodic integral for e(3) Lie algebra.
problem Analyzes singularities and topological properties of integrable systems.
method Examines singularities of Liouville foliation, bifurcation diagram, transformations of Liouville tori, and isoenergy surfaces.
result Discovers topological properties of integrable systems with linear periodic integral.
The paper defines Haar system preserving morphisms and applies them to groupoid C∗-algebras.
problem Understanding and constructing inverse systems of groupoids.
method Defining Haar system preserving morphisms and using them to induce *-morphisms between convolution algebras.
result Inverse systems of groupoids with Haar system preserving bonding maps have limits, and corresponding direct systems of groupoid C∗-algebras. We classify superintegrable systems in the Euclidean plane using algebraic geometry.
problem Classifying superintegrable systems in the Euclidean plane.
method Derived and solved a system of algebraic equations to classify systems.
result Associated a unique line triple arrangement to each superintegrable system.
The maximum size of algebraic k-systems on surfaces is determined.
problem Finding the maximum size of algebraic k-systems of curves on surfaces.
method Generalizing a theorem from [MRT14], the maximum size is computed based on the surface's genus and the parity of k.
result The maximum size of an algebraic k-system of curves on a surface of genus g is 2g+1 when g≥3 or k is odd, and 2g otherwise.
New weight systems derived from a specific Lie algebra for knot invariants.
problem Constructing universal weight systems for knot invariants.
method Using a minimal Z22-graded Lie algebra to create weight systems. result Weight system derived from A1ε shows hybrid properties of sl(2) and gl(1∣1). New methods for solving hydrodynamic-type equations using quasi-rectifiable Lie algebras.
problem Solving systems of hydrodynamic-type equations.
method Introducing and analyzing quasi-rectifiable Lie algebras and vector fields.
result New methods for solving hydrodynamic-type equations.
Extends Framization to Coxeter system of type B.
problem Extending Framization to Coxeter systems of type B.
method Define a natural extension of the classical Temperley-Lieb algebra, prove the existence of a unique linear Markov trace function, introduce Framization as a quotient of the Yokonuma-Hecke algebra, and provide conditions for the Markov trace to pass to the quotient.
result Construct invariants for framed and classical links inside the solid torus.
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.
Researchers present and compare different representations of dissipative Hamiltonian DAE systems.
problem Understanding and transforming dissipative Hamiltonian DAE systems.
method Global geometric and algebraic points of view, translations between representations, characterizations, and numerical methods for computing structural information.
result A general DAE system can be transformed into a dissipative Hamiltonian or port-Hamiltonian DAE system.
Novel duality theory for operator Frobenius algebras solves long-standing hydrodynamic integrable systems problem.
problem Long-standing Eisenhart-Stäckel problem for non-degenerate integrable systems.
method Introduce duality for operator Frobenius algebras and use mutual symmetry assumption.
result Construct new infinite-dimensional integrable systems of hydrodynamic type.
New invariant for virtual links using multi-switches and algebraic systems.
problem Creating invariants for virtual links.
method Introducing a general approach to construct invariant algebraic systems from multi-switches and virtual links.
result Introduces a new quandle invariant for virtual links.
This paper develops an algebraic structure for systems of systems and networks.
problem Formalizing interactions between complex systems and networks.
method Developing a monoidal double category of surjective submersions to encompass systems and maps between them.
result Recovering results on fibrations of networks of manifolds as a special case.
Paper constructs super integrable systems on color Lie algebra.
problem Super integrable systems on color Lie algebra.
method Using non-isospectral problems with matrices from color Lie algebra sp1(6), constructing (1+1)- and (2+1)-dimensional systems. result Super integrable systems and their Hamiltonian structures constructed on color Lie algebra sp1(6). A new category of Lie algebras, called generalized Lie algebras, is presented such that classical Lie algebras and Lie-Rinehart algebras are objects of this new category. A new philosophy over generalized Lie algebroids theory is presented using the notion of generalized Lie algebra and examples of objects of the categ…
Proves properties of complex algebraic varieties and local systems.
problem Properties of complex algebraic varieties and local systems.
method Analytic Zariski open subsets and algebraic maps.
result Trivializing covering spaces of complex algebraic varieties.
Integrable Pfaffian systems invariant under Lie group actions have their cohomology equal to Lie algebra cohomology.
problem Detecting obstructions to the existence of finite or infinitesimal actions leaving a given system invariant.
method Using Lie algebra cohomology to compute and detect obstructions in the variational cohomology of invariant Pfaffian systems.
result The vertical variational cohomology of an invariant Pfaffian system is equal to the Lie algebra cohomology of the Lie algebra g. New algebraic-geometric method classifies superintegrable systems in any dimension.
problem Classifying superintegrable systems in arbitrary dimensions is challenging.
method Algebraic-geometric approach based on quasi-projective varieties.
result Established foundations for classification in arbitrary dimensions.
Paper defines and computes a new weight system for gl_N Lie algebra.
problem Understanding the weight system of Lie algebra gl_N.
method Two approaches: Kazarian's invariant and Harish-Chandra isomorphism.
result Computes the gl_N weight system on chord diagrams.
We construct a natural framed weight system on chord diagrams from the curvature tensor of any pseudo-Riemannian symmetric space. These weight systems are of Lie algebra type and realized by the action of the holonomy Lie algebra on a tangent space. Among the Lie algebra weight systems, they are exactly characterized b…
We show that the algebraic dimension of a twistor space over n#CP^2 cannot be two if n>4 and the fundamental system (i.e. the linear system associated to the half-anti-canonical bundle, which is available on any twistor space) is a pencil. This means that if the algebraic dimension of a twistor space on n#CP^2, n>4, is…
Abstract reviews symmetry and reduction in dynamical systems.
problem Understanding symmetries and reductions in dynamical systems.
method Algebraic formulation for dynamics of physical systems.
result Describes a reduction procedure for classical and quantum evolutions.
Study Toda systems blowup masses linked to Weyl groups.
problem Understanding blowup phenomena in Toda systems.
method Concrete examples of Toda systems solutions and blowup masses.
result Blowup masses correspond to Weyl groups.
New algebraic approach classifies conformally superintegrable systems in arbitrary dimensions.
problem Classifying conformally superintegrable systems in arbitrary dimensions.
method Algebraic geometric approach extended to conformally superintegrable systems.
result An algebraic equation governs the classification under conformal equivalence for a prolific class of second order conformally superintegrable systems.
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.
Study uses geometric algebra to analyze credit cycles, revealing dangerous feedback loops.
problem Understanding and predicting dangerous feedback loops in credit cycles.
method Represent economic states as multi-vectors in Clifford algebra, focusing on bivector elements for rotational coupling.
result Geometric relationship between unemployment and credit contraction shifts from simple correlation to dangerous rotational dynamics during crises.
The notion of integrability will often extend from systems with scalar-valued fields to systems with algebra-valued fields. In such extensions the properties of, and structures on, the algebra play a central role in ensuring integrability is preserved. In this paper a new theory of Frobenius-algebra valued integrable s…
Since its original publication in 1916 under the title "The Algebraic Theory of Modular Systems", the book by F. S. Macaulay has attracted a lot of scientists with a view towards pure mathematics (D. Eisenbud,...) or applications to control theory (U. Oberst,...).However, a carefull examination of the quotations clearl…
The main object of our study is a four dimensional Lie algebra which describes the symmetry properties of a nonlinear Black-Scholes model. This model implements a feedback effect which is typical for an illiquid market. The structure of the Lie algebra depends on one parameter, i.e. we have to do with a one-parametric …
The paper explores rigidity and proximality in dynamical systems, proving new results about C∗-algebras.
problem Understanding rigidity and proximality in dynamical systems and their algebraic counterparts.
method Analyzing crossed products of dynamical systems and their C∗-algebras, focusing on uniform rigidity and proximality. result Uniformly rigid systems are almost reflecting, and certain crossed products are reflecting.
An algebraic system is proposed that represent surface cobordisms in thickened surfaces. Module and comodule structures over Frobenius algebras are used for representing essential curves. The proposed structure gives a unified algebraic view of states of categorified Jones polynomials in thickened surfaces and virtual …
Abstract: Connections between Lie algebras and symplectic nilmanifolds explored.
problem Understanding the relationship between Lie algebras and symplectic nilmanifolds.
method Central extensions of Lie algebras, dynamical systems, and symplectic nilmanifolds constructed.
result Covering Lie groups for symplectic nilmanifolds can have any rank as solvable Lie groups.
We give an account of the construction of exterior differential systems based on the notion of tableaux over Lie algebras as developed in [Comm. Anal. Geom 14 (2006), 475-496; math.DG/0412169]. The definition of a tableau over a Lie algebra is revisited and extended in the light of the formalism of the Spencer cohomolo…
Geometric models for Lie algebras from simple singularities.
problem Classifying simply-laced simple Lie algebras.
method Using polygonal wheels derived from Milnor fibers of simple singularities.
result Geometric root systems are isomorphic to Lie algebras.
Let S be a path-connected, locally-compact CW-complex, and let M be a subcomplex with finitely-many components. A `decorated SL_2(C)-local system' is an SL_2(C)-local system on S, together with a choice of `decoration' at each component of M (a section of the stalk of an associated vector bundle). We study the (decorat…
We compute symmetry algebras of a system of two equations y^(k)=z^(l)=0, where 2<=k<l. It appears that there are many ways to convert such system of ODEs to an exterior differential system. They lead to different series of finite-dimensional symmetry algebras. For example, for (k,l)=(2,3) we get two non-isomorphic symm…
We give a criterion of (micro-)kroneckerity of the linear Poisson pencil on g∗ related to an algebraic Nijenhuis operator N:g→g on a finite-dimensional Lie algebra g. As an application we get a series of examples of completely integrable systems on semisimple Lie algebras related t…
New geometric methods find constants and tensors for Lie systems.
problem Finding constants and tensors for Lie systems.
method Geometric methods to attach multisymplectic Lie systems to tensor coalgebras.
result Derivation of constants of motion and invariant tensor fields.
Symmetric spaces' connections form Lie admissible triple algebras.
problem Understanding the algebraic structure of symmetric spaces' connections.
method Analyzing the connection as a binary operator on tangent bundle sections, identifying Lie admissibility constraints.
result Connection algebra of symmetric spaces is a Lie admissible triple algebra.
New Einstein manifolds from symplectic systems.
problem Creating Einstein manifolds from symplectic structures.
method Using symplectic triple systems to derive reductive pairs and prove Einstein properties.
result Simple symplectic triple systems yield Einstein manifolds.
New superintegrable systems derived from Frobenius structures.
problem Constructing second-order superintegrable systems.
method Using conification and direct product construction, applying to semi-simple and nilpotent algebras.
result Explicitly constructed second-order superintegrable systems in three dimensions.
The purpose of this paper is to connect two subjects: the theory of quantum integrable systems (complete commutative rings of differential operators), and differential Galois theory. We define quantum completely integrable systems (QCIS), algebraically integrable QCIS, the differential Galois group of a QCIS. We show t…
A Lie system is a system of first-order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of vector fields: a so-called Vessiot-Guldberg Lie algebra. We suggest the definition of a particular class of Lie systems, the $k…
The paper explores weight systems and their applications to graph and embedded graph invariants.
problem Developing weight systems for graphs and embedded graphs.
method Construction of weight systems from graph invariants and metrized Lie algebras, and extending to arbitrary embedded graphs.
result Explicit forms of generating functions and recurrence relations for weight systems on chord diagrams and embedded graphs.
The study identifies exceptions to fiber-preserving symmetry in ODEs and systems.
problem Identifying exceptions to fiber-preserving symmetry in ODEs and systems.
method Lie's classification of Lie algebras of vector fields, absolute and relative scalar differential invariants, conditional and vector-valued relative invariants, prolongations of actions.
result Examples of scalar ODEs and systems with symmetry groups not fiber-preserving.