arXiv research

Develops method to construct Lie algebra weight system kernel using Vogel algebra.

Unified approach to deform Lie-Hamilton systems using Poisson-Hopf algebra.

Automorphisms of Lie algebras and their root systems are fully lifted.

Study Poisson algebras for Hamiltonian systems linearization.

Investigates integrable systems with linear periodic integral for e(3) Lie algebra.

New weight systems derived from a specific Lie algebra for knot invariants.

New methods for solving hydrodynamic-type equations using quasi-rectifiable Lie algebras.

Researchers link vertex algebras to non-Kähler solutions of the Hull-Strominger system.

Researchers present and compare different representations of dissipative Hamiltonian DAE systems.

Novel duality theory for operator Frobenius algebras solves long-standing hydrodynamic integrable systems problem.

Paper constructs super integrable systems on color Lie algebra.

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.

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…

Paper defines and computes a new weight system for gl_N Lie algebra.

We define what it means for a proper continuous morphism between groupoids to be Haar system preserving, and show that such a morphism induces (via pullback) a *-morphism between the corresponding convolution algebras. We proceed to provide a plethora of examples of Haar system preserving morphisms and discuss connecti…

Abstract reviews symmetry and reduction in dynamical systems.

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…

Study Toda systems blowup masses linked to Weyl groups.

New algebraic approach classifies conformally superintegrable systems in arbitrary dimensions.

We prove that the set of non-degenerate second order maximally superintegrable systems in the complex Euclidean plane carries a natural structure of a projective variety, equipped with a linear isometry group action. This is done by deriving the corresponding system of homogeneous algebraic equations. We then solve the…

The study connects hypergraphs to strong homotopy Lie algebras.

Study uses geometric algebra to analyze credit cycles, revealing dangerous feedback loops.

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…

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 …

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 …

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…

The paper explores rigidity and proximality in dynamical systems, proving new results about $C^*$-algebras.

Abstract: Connections between Lie algebras and symplectic nilmanifolds explored.

Geometric models for Lie algebras from simple singularities.

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…

A Lie system is the non-autonomous system of differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot--Guldberg Lie algebra. This work pioneers the analysis of Lie systems admitting a Vessiot--Guldberg …

A collection $Δ$ of simple closed curves on an orientable surface is an algebraic $k$-system if the algebraic intersection number $\langle α,β\rangle$ is equal to $k$ in absolute value for every $α, β\in Δ$ distinct. Generalizing a theorem of [MRT14] we compute that the maximum size of an algebraic $k$-system of c…

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 $\frak{g}^*$ related to an algebraic Nijenhuis operator $N:\frak{g}\to \frak{g}$ on a finite-dimensional Lie algebra $\frak{g}$. As an application we get a series of examples of completely integrable systems on semisimple Lie algebras related t…

Symmetric spaces' connections form Lie admissible triple algebras.

New superintegrable systems derived from Frobenius structures.

We extend the Framization of the Temperley-Lieb algebra to Coxeter systems of type $\mathtt{B}$. We first define a natural extension of the classical Temperley-Lieb algebra to Coxeter systems of type $\mathtt{B}$ and prove that such an extension supports a unique linear Markov trace function. We then introduce the Fram…

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.

The study identifies exceptions to fiber-preserving symmetry in ODEs and systems.

In this paper we introduce a new dynamical system which we call Angular billiard. It acts on the exterior points of a convex curve in Euclidean plane. In a neighborhood of the boundary curve this system turns out to be dual to the Birkhoff billiard. Using this system we get new results on algebraic Birkhoff conjecture …

New integrable matrix PDEs derived from Frölicher-Nijenhuis brackets.

Second-order superintegrable systems in dimensions two and three are essentially classified. With increasing dimension, however, the non-linear partial differential equations employed in current methods become unmanageable. Here we propose a new, algebraic-geometric approach to the classification problem - based on a p…

Study heterotic G2-system on 2-step nilmanifolds with torus bundles.

Cluster algebras match for specific Lie algebras and surfaces.