Jordan-Kronecker invariants of finite-dimensional Lie algebras

For any finite-dimensional Lie algebra we introduce the notion of Jordan-Kronecker invariants, study their properties and discuss examples. These invariants naturally appear in the framework of the bi-Hamiltonian approach to integrable systems on Lie algebras and are closely related to Mischenko-Fomenko's argument shift method

Matrix Pencils and Entanglement Classification

In this paper, we study pure state entanglement in systems of dimension $2\otimes m\otimes n$. Two states are considered equivalent if they can be reversibly converted from one to the other with a nonzero probability using only local quantum resources and classical communication (SLOCC). We introduce a connection between entanglement manipulations in these systems and the well-studied theory of matrix pencils. All previous attempts to study general SLOCC equivalence in such systems have relied on somewhat contrived techniques which fail to reveal the elegant structure of the problem that can be seen from the matrix pencil approach. Based on this method, we report the first polynomial-time algorithm for deciding when two $2\otimes m\otimes n$ states are SLOCC equivalent. Besides recovering the previously known 26 distinct SLOCC equivalence classes in $2\otimes 3\otimes n$ systems, we also determine the hierarchy between these classes

Stratification theory of matrix pairs under equivalence and contragredient equivalence

We develop the theory of perturbations of matrix pencils basing on their miniversal deformations. Several applications of this theory are given. All possible Kronecker pencils that are canonical forms of pencils in an arbitrary small neighbourhood of a given pencil were described by A. Pokrzywa (Linear Algebra Appl., 1986). His proof is very abstract and unconstructive. Even more abstract proof of Pokrzywa’s theorem was given by K. Bongartz (Advances in Mathematics, 1996); he uses the representation theory of finite dimensional algebras. The main purpose of this thesis is to give a direct, constructive, and rather elementary proof of Pokrzywa’s theorem. We first show that it is sufficient to prove Pokrzywa’s theorem only for pencils that are direct sums of at most two indecomposable Kronecker pencils. Then we prove Pokrzywa’s theorem for such pencils. The latter problem is very simplified due to the following observation: it is sufficient to find Kronecker's canonical forms of only those pencils that are obtained by miniversal perturbations of a given pencil. We use miniversal deformations of matrix pencils that are given by M. I. García-Planas and V. V. Sergeichuk (Linear Algebra Appl., 1999) because their deformations have many zero entries unlike the miniversal deformations given by A. Edelman, E. Elmroth, and B. Kagstrom (SIAM J. Matrix Anal. Appl., 1997). Thus, we give not only all possible Kronecker’s canonical forms, but also the corresponding deformations of a given pencil, which is important for applications of this theory. P. Van Dooren (Linear Algebra Appl., 1979) constructed an algorithm for computing all singular summands of Kronecker’s canonical form of a matrix pencil. His algorithm uses only unitary transformations, which improves its numerical stability. We extend Van Dooren’s algorithm both to square complex matrices under consimilarity transformations and to pairs of complex matrices under mixed equivalence. We describe all pairs (A, B) of m-by-n and n-by-m complex matrices for which the product CD is a versal deformation of AB, in which (C, D) is the miniversal deformation of (A, B) under contragredient equivalence given by M. I. García-Planas and V. V. Sergeichuk (Linear Algebra Appl., 1999). We find all canonical matrix pairs (A, B) under contragredient equivalence, for which the first order induced perturbations are nonzero for all nonzero miniversal deformations of (A, B). This problem arises in the theory of differential matrix equations dx= ABx. A complex matrix pencil is called structurally stable if there exists its neighbourhood in which all pencils are strictly equivalent to it. We describe all complex matrix pencils that are structurally stable. We show that there are no pairs of complex matrices that are structurally stable with respect to contragredient equivalence.Es desenvolupa la teoria de pertorbacions de feixos de matrius a partir de les seves deformacions miniversals. Es donen diverses aplicacions d'aquesta teoria. A. Pokrzywa (Linear Algebra Appl., 1986) va descriure tots els possibles feixos en la seva forma de Kronecker que són formes canòniques dels feixos que es poden trobar en un petit entorn arbitrari d'un feix prèviament determinat. La demostració que presentava és molt abstracta i no constructiva. K. Bongartz (Advances in Mathematics, 1996) va donar una demostració encara més abstracta del teorema de Pokrzywa; utilitzant resultats de la teoria de representació d'àlgebres de dimensió finita. L’objectiu principal de aquesta tesi és presentar una demostració directa, constructiva i bastant elemental del teorema de Pokrzywa. Primer, es demostra que per a provar el teorema de Pokrzywa és suficient provar-lo solament per a feixos que són sumes directes de, com màxim, dos feixos de Kronecker indescomponibles. Per a continuació, provar el teorema de Pokrzywa per aquests feixos. L’últim problema es simplifica molt degut a la següent observació: és suficient per trobar les formes canòniques de Kronecker de només aquells feixox que s’obtenen de deformacions miniversals d’un feix determinat. Utilitzem les deformacions de feixos de matrius obtingudes per MI García-Planas i VV Sergeichuk (Linear Algebra Appl., 1999) perquè les seves deformacions tenen moltes entrades nul·les, a diferència de les deformacions miniversals obtingudes per A. Edelman, E. Elmroth i B. Kagstrom (SIAM J. Matrix Anal. Appl., 1997). Per tant, no solament donem totes les formes canòniques de Kronecker possibles, sinó també les deformacions corresponents a un feix prèviament fixat, la qual cosa és important per a les aplicacions d’aquesta teoria. P. Van Dooren (Linear Algebra Appl., 1979) va construir un algoritme per calcular tots els sumands singulars de la forma canònica de Kronecker, d’un feix de matrius. El seu algoritme utilitza solament transformacions unitàries, el que millora la seva estabilitat numèrica. Estenem l’algoritme de Van Dooren tant a matrius complexes quadrades respecte transformacions de cosimilaritat com a parells de matrius complexes respecte l’equivalència mixta. Descrivim tots els parells (A, B) de matrius complexes m per n i n per m, per les quals el producte CD és una deformació versal de AB, en la que (C, D) és la deformació miniversal de (A, B) respecte l’equivalència contragredient donada per MI García-Planas y VV Sergeichuk (Linear Algebra Appl., 1999). Descrivim tots los pares de matrius canòniques (A, B) respecte l’equivalència contragredient, per les quals les pertorbacions de primer ordre induïdes són diferents de cero para totes les deformacions miniversals no nul·les d¿(A, B). Aquest problema apareix en la teoria de les equacions matricials diferencials dx = ABx. Un feix de matrius complexes es diu estructuralment estable si existeix un entorn en el que tots els feixos són equivalents a ell respecte una relació d’equivalència considerada. Descrivim tots els feixos de matrius complexes que són estructuralment estables respecte la equivalència estricta. Mostrem que no hi ha parelles de matrius complexes que són estructuralment estables respecto l’equivalència contragredient.Postprint (published version

Finite-dimensional integrable systems: a collection of research problems

This article suggests a series of problems related to various algebraic and geometric aspects of integrability. They reflect some recent developments in the theory of finite-dimensional integrable systems such as bi-Poisson linear algebra, Jordan-Kronecker invariants of finite dimensional Lie algebras, the interplay between singularities of Lagrangian fibrations and compatible Poisson brackets, and new techniques in projective geometry

Open problems, questions, and challenges in finite-dimensional integrable systems

The paper surveys open problems and questions related to different aspects of integrable systems with finitely many degrees of freedom. Many of the open problems were suggested by the participants of the conference “Finite-dimensional Integrable Systems, FDIS 2017” held at CRM, Barcelona in July 2017.Postprint (updated version

Algebraic aspects of compatible poisson structures

This thesis consists of three chapters. In Chapter one, we introduce some notions and definitions for basic concepts of the theory of integrable bi-Hamiltonian systems. Brief statements of several open problems related to our main results are also mentioned in this part. In Chapter two, we applied the so-called Jordan–Kronecker decomposition theorem to study algebraic properties of the pencil P generated by two constant compatible Poisson structures on a vector space. In particular, we study the linear automorphism group GP that preserves P. In classical symplectic geometry, many fundamental results are based on the symplectic group, which preserves the symplectic structure. Therefore in the theory of bi-Hamiltonian structures, we hope GP also plays a fundamental role. In Chapter three, we study one of the famous Poisson pencils which is sometimes called “argument shift pencil”. This pencil is defined on the dual space g ∗ of an arbitrary Lie algebra g. This pencil is generated by the Lie-Poisson bracket { , } and constant bracket { , }a for a ∈ g ∗ . Thus we may apply the Jordan–Kronecker decomposition theorem to introduce the so-called Jordan–Kronecker invariants of a finite-dimensional Lie algebra g. These invariants can be understood as the algebraic type of the canonical Jordan–Kronecker form for the “argument shift pencil” at a generic point. Jordan–Kronecker invariants are found for all low-dimensional Lie algebras (dim g ≤ 5) and can be used to construct the families of polynomials in bi-involution. The results are found to be useful in the discussion of the existence of a complete family of polynomials in bi-involution w.r.t. these two brackets { , } and { , }a

Digraph based determination of Jordan block size structure of singular matrix pencils

AbstractThe generic Jordan block sizes corresponding to multiple characteristic roots at zero and at infinity of a singular matrix pencil will be determined graph-theoretically. An application of this technique to detect certain controllability properties of linear time-invariant differential algebraic equations is discussed