Matrix Lie Algebra Research Articles

A combined Kaup–Newell type integrable hierarchy with four potentials and its bi-Hamiltonian formulation

This paper aims to introduce a Kaup–Newell type matrix eigenvalue problem with four potentials, based on a specific matrix Lie algebra, and construct its associated Liouville integrable Hamiltonian hierarchy, through the zero curvature formulation. The Liouville integrability of the resulting hierarchy is shown by determining its recursion operator and bi-Hamiltonian formulation. An illustrative example of combined derivative nonlinear Schrödinger equations with two arbitrary constants is explicitly presented.

A generic quantum Wielandt's inequality

Quantum Wielandt's inequality gives an optimal upper bound on the minimal length k such that length-k products of elements in a generating system span Mn(C). It is conjectured that k should be of order O(n2) in general. In this paper, we give an overview of how the question has been studied in the literature so far and its relation to a classical question in linear algebra, namely the length of the algebra Mn(C). We provide a generic version of quantum Wielandt's inequality, which gives the optimal length with probability one. More specifically, we prove based on [KS16] that k generically is of order Θ(log⁡n), as opposed to the general case, in which the best bound to date is O(n2log⁡n). Our result implies a new bound on the primitivity index of a random quantum channel. Furthermore, we shed new light on a long-standing open problem for Projected Entangled Pair State, by concluding that almost any translation-invariant PEPS (in particular, Matrix Product State) with periodic boundary conditions on a grid with side length of order Ω(log⁡n) is the unique ground state of a local Hamiltonian. We observe similar characteristics for matrix Lie algebras and provide numerical results for random Lie-generating systems.

Identities of matrix powers of a matrix and matrix power of matrix on lie groups

This paper introduces fundamental definitions pertaining to Matrix Lie Groups and Lie Algebra. We illustrate the concepts of the exponential of a matrix and the logarithm of a matrix as integral components of our discourse. We introduce novel identities related to the matrix powers of a matrix. To prove these identities, we employ the results of the matrix powers of a matrix. Finally, we derive an expression for the matrix powers of a matrix within the context of a connected matrix lie group.

Chaos in Stochastic 2d Galerkin-Navier–Stokes

We prove that all Galerkin truncations of the 2d stochastic Navier–Stokes equations in vorticity form on any rectangular torus subjected to hypoelliptic, additive stochastic forcing are chaotic at sufficiently small viscosity, provided the frequency truncation satisfies N≥392\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\ge 392$$\\end{document}. By “chaotic” we mean having a strictly positive Lyapunov exponent, i.e. almost-sure asymptotic exponential growth of the derivative with respect to generic initial conditions. A sufficient condition for such results was derived in previous joint work with Alex Blumenthal which reduces the question to the non-degeneracy of a matrix Lie algebra implying Hörmander’s condition for the Markov process lifted to the sphere bundle (projective hypoellipticity). The purpose of this work is to reformulate this condition to be more amenable for Galerkin truncations of PDEs and then to verify this condition using (a) a reduction to genericity properties of a diagonal sub-algebra inspired by the root space decomposition of semi-simple Lie algebras and (b) computational algebraic geometry executed by Maple in exact rational arithmetic. Note that even though we use a computer assisted proof, the result is valid for all aspect ratios and all sufficiently high dimensional truncations; in fact, certain steps simplify in the formal infinite dimensional limit.

Transposed Poisson structures on the Lie algebra of upper triangular matrices

We describe transposed Poisson structures on the upper triangular matrix Lie algebra T_{n}(F) , n>1 , over a field F of characteristic zero. We prove that, for n>2 , any such structure is either of Poisson type or the orthogonal sum of a fixed non-Poisson structure with a structure of Poisson type, and for n=2 , there is one more class of transposed Poisson structures on T_{n}(F) . We also show that, up to isomorphism, the full matrix Lie algebra M_{n}(F) admits only one non-trivial transposed Poisson structure, and it is of Poisson type.

Structure and Approximation Properties of Laplacian-Like Matrices

Many of today’s problems require techniques that involve the solution of arbitrarily large systems Atextbf{x}=textbf{b}. A popular numerical approach is the so-called Greedy Rank-One Update Algorithm, based on a particular tensor decomposition. The numerical experiments support the fact that this algorithm converges especially fast when the matrix of the linear system is Laplacian-Like. These matrices that follow the tensor structure of the Laplacian operator are formed by sums of Kronecker product of matrices following a particular pattern. Moreover, this set of matrices is not only a linear subspace it is a Lie sub-algebra of a matrix Lie Algebra. In this paper, we characterize and give the main properties of this particular class of matrices. Moreover, the above results allow us to propose an algorithm to explicitly compute the orthogonal projection onto this subspace of a given square matrix A in mathbb {R}^{Ntimes N}.

Nonlocal reduced integrable mKdV-type equations from a vector integrable hierarchy

This paper aims to present two hierarchies of nonlocal reduced integrable mKdV-type equations from a vector integrable hierarchy associated with a matrix Lie algebra, not being A type. The key point is to make similarity transformations for the spectral matrix, which keep the associated zero curvature equations invariant and then there follow reduced nonlocal integrable mKdV-type equations. The success lies in determining a Laurent series solution to the corresponding reduced stationary zero curvature equation, which generates temporal matrix spectral problems in the zero curvature formation.

An efficient geometric method for incompressible hydrodynamics on the sphere

We present an efficient and highly scalable geometric numerical method for two-dimensional ideal fluid dynamics on the sphere. The starting point is Zeitlin's finite-dimensional model of hydrodynamics. The efficiency stems from exploiting a tridiagonal splitting of the discrete spherical Laplacian combined with highly optimized, scalable numerical algorithms. For time-stepping, we adopt a recently developed isospectral integrator able to preserve the geometric structure of Euler's equations, in particular conservation of the Casimir functions. To overcome previous computational bottlenecks, we formulate the matrix Lie algebra basis through a sequence of tridiagonal eigenvalue problems, efficiently solved by well-established linear algebra libraries. The same tridiagonal splitting allows for computation of the stream matrix, involving the inverse Laplacian, for which we design an efficient parallel implementation on distributed memory systems. The resulting overall computational complexity is O(N3) per time-step for N2 spatial degrees of freedom. The dominating computational cost is matrix-matrix multiplication, carried out via the parallel library ScaLAPACK. Scaling tests show approximately linear scaling up to around 2500 cores for the matrix size N=4096 with a computational time per time-step of about 0.55 seconds. These results allow for long-time simulations and the gathering of statistical quantities while simultaneously conserving the Casimir functions. We illustrate the developed algorithm for Euler's equations at the resolution N=2048.

A Lie algebra representation for efficient 2D shape classification

Riemannian manifold plays a vital role as a powerful mathematical tool in computer vision, with important applications in curved shape analysis and classification. Significant progress has recently been made by Riemannian framework based methods that achieved state-of-the-art classification accuracy and robustness. However, these Riemannian manifold and Lie group methods require a very high computational complexity and do not include a description of the shape regions. This paper presents a novel mathematical tool, called Block Diagonal Symmetric Positive Definite Matrix Lie Algebra (BDSPDMLA) to represent curves, which extends the existing Lie group representations to a compact yet informative Lie algebra representation. The proposed Lie algebra based method addresses the computational bottleneck problem of the Riemannian framework based methods. In addition, it allows the natural fusion of various regions information with curved shape features for a more discriminative shape description. Here the region information is represented by values of distance maps, local binary patterns (LBP) and image intensity. Extensive experiments on five publicly available databases demonstrate that the proposed Lie algebra based method can achieve a speed of over ten thousand times faster than the Riemannian manifold and Lie group based baseline methods, while obtaining comparable accuracies for 2D shape classification.

Several Isospectral and Non-Isospectral Integrable Hierarchies of Evolution Equations

By introducing a 3×3 matrix Lie algebra and employing the generalized Tu scheme, a AKNS isospectral–nonisospectral integrable hierarchy is generated by using a third-order matrix Lie algebra. Through a matrix transformation, we turn the 3×3 matrix Lie algebra into a 2×2 matrix case for which we conveniently enlarge it into two various expanding Lie algebras in order to obtain two different expanding integrable models of the isospectral–nonisospectral AKNS hierarchy by employing the integrable coupling theory. Specially, we propose a method for generating nonlinear integrable couplings for the first time, and produce a generalized KdV-Schrödinger integrable system and a nonlocal nonlinear Schrödinger equation, which indicates that we unite the KdV equation and the nonlinear Schrödinger equation as an integrable model by our method. This method presented in the paper could apply to investigate other integrable systems.

Regular Hom–Lie structures on strictly upper triangular matrix Lie algebras

A Hom-structure on a Lie algebra [Formula: see text] is a linear map [Formula: see text] which satisfies the Hom–Jacobi identity [Formula: see text] for all [Formula: see text]. A Hom-structure is called regular if [Formula: see text] is also a Lie algebra isomorphism. Let [Formula: see text] be the Lie algebra consisting of all strictly upper triangular [Formula: see text] matrices over a field [Formula: see text]. In this paper, we prove that if [Formula: see text], any regular Hom-structure [Formula: see text] on [Formula: see text] is a product of a special inner automorphism, an extremal inner automorphism and a central automorphism of [Formula: see text]. As its application, the set of all regular Hom-structures on [Formula: see text] forms a normal subgroup of the automorphism group of [Formula: see text].

The impracticalities of multiplicatively-closed codon models: a retreat to linear alternatives.

A matrix Lie algebra is a linear space of matrices closed under the operation [Formula: see text]. The "Lie closure" of a set of matrices is the smallest matrix Lie algebra which contains the set. In the context of Markov chain theory, if a set of rate matrices form a Lie algebra, their corresponding Markov matrices are closed under matrix multiplication; this has been found to be a useful property in phylogenetics. Inspired by previous research involving Lie closures of DNA models, it was hypothesised that finding the Lie closure of a codon model could help to solve the problem of mis-estimation of the non-synonymous/synonymous rate ratio, [Formula: see text]. We propose two different methods of finding a linear space from a model: the first is the linear closure which is the smallest linear space which contains the model, and the second is the linear version which changes multiplicative constraints in the model to additive ones. For each of these linear spaces we then find the Lie closures of them. Under both methods, it was found that closed codon models would require thousands of parameters, and that any partial solution to this problem that was of a reasonable size violated stochasticity. Investigation of toy models indicated that finding the Lie closure of matrix linear spaces which deviated only slightly from a simple model resulted in a Lie closure that was close to having the maximum number of parameters possible. Given that Lie closures are not practical, we propose further consideration of the two variants of linearly closed models.

The index of nilpotent Lie poset algebras

We establish combinatorial formulas for the index of a class of matrix Lie algebras whose matrix forms are encoded by strict partial orderings. Our methods are linear algebraic.

Two integrable couplings of a generalized D-Kaup–Newell hierarchy and their Hamiltonian and bi-Hamiltonian structures

We enlarge the spectral problem of a generalized D-Kaup–Newell (D-KN) spectral problem. Solving the enlarged zero-curvature equations, we produce integrable couplings. A reduction of the spectral matrix leads to a second integrable coupling system. Next, bilinear forms that are symmetric, ad-invariant, and non-degenerate on the given non-semisimple matrix Lie algebra are computed to employ the variational identity. The variational identity is then applied to the original enlarged spectral problem of a generalized D-KN hierarchy and the reduced problem. Hamiltonian structures are presented, as well as a bi-Hamiltonian formulation of the reduced problem. Both hierarchies have infinitely many commuting high-order symmetries and conserved densities, i.e., are Liouville integrable.

Strong commutativity preserving maps of strictly triangular matrix Lie algebras

Let [Formula: see text] be the Lie algebra consisting of all strictly upper triangular [Formula: see text] matrices over a field [Formula: see text]. An invertible linear map [Formula: see text] on [Formula: see text] is called to be strong commutativity preserving (simply denoted by SCP) if [Formula: see text] for any [Formula: see text]. We show that for [Formula: see text], an invertible linear map [Formula: see text] preserves strong commutativity if and only if there exist [Formula: see text] and a linear function [Formula: see text] with [Formula: see text] such that [Formula: see text], where [Formula: see text] is an inner SCP, [Formula: see text] are extremal SCPs, [Formula: see text] is a central SCP.

Algorithm to compute minimal matrix representation of nilpotent lie algebras

ABSTRACTAs it is well-known there exist matrix representations of any given finite-dimensional complex Lie algebra. More concretely, such representations can be obtained by means of an isomorphic matrix Lie algebra consisting of upper-triangular square matrices. However, there is no general information about the minimal order for the matrices involved in such representations. In this way, our main goal is to revisit, debug and implement an algorithm which provides the minimal order for matrix representations of any finite-dimensional nilpotent Lie algebra from its law, as well as returning a matrix representative of such an algebra by using the minimal order previously computed. In order to show the applicability of this procedure, we have computed minimal representative for each nilpotent Lie algebra of dimensions 6 and 7 and we have also obtained the representation of some families with an arbitrary dimension.

A new approach for finding standard heat equation and a special Newell-whitehead equation

Under a frame of 2 ? 2 matrix Lie algebras, Tu and Meng [9] once established a united integrable model of the Ablowitz-Kaup-Newel-Segur (AKNS) hierarchy, the D-AKNS hierarchy, the Levi hierarchy and the TD hierarchy. Based on this idea, we introduce two block-matrix Lie algebras to present an isospectral problem, whose compatibility condition gives rise to a type of integrable hierarchy which can be reduced to the Levi hierarchy and the AKNS hierarchy, and so on. A united integrable model obtained by us in the paper is different from that given by Tu and Meng. Specially, the main result in the paper can be reduced to two new various integrable couplings of the Levi hierarchy, from which we again obtain the standard heat equation and a special Newell-Whitehead equation.

Nonlinear anti-commuting maps of strictly triangular matrix Lie algebras

Nonlinear anti-commuting maps of strictly triangular matrix Lie algebras

Some novel soliton solution, breather solution and Darboux transformation for a generalized coupled Toda soliton hierarchy

A few of discrete integrable coupling systems(DICSs) of previous papers are linear discrete integrable couplings(LDICS). We take a special matrix Lie algebra system(non-semisimple) to construct the Lax pairs, and establish a method for deriving the nonlinear discrete integrable coupling systems(NDICS). From the Lax pairs of the generalized Toda(G-Toda) spectral problem, we can derive a novel NDICS, which is a real NDICS. For the obtained lattice integrable coupling equation, we establish a Darboux transformation (DT) with 4 × 4 Lax pairs, and apply the gauge transformation to a specific equation, then the explicit solutions of the lattice integrable coupling equation are given, which contains discrete soliton solution, breather solution and rogue wave solution. Furthermore, we can derive the discrete explicit solutions with free parameters to depict their dynamic behaviors.

Travelling waves and conservation laws of a (2+1)-dimensional coupling system with Korteweg-de Vries equation

Abstract In this paper we study a (2+1)-dimensional coupling system with the Korteweg-de Vries equation, which is associated with non-semisimple matrix Lie algebras. Its Lax-pair and bi-Hamiltonian formulation were obtained and presented in the literature. We utilize Lie symmetry analysis along with the (G′/G)–expansion method to obtain travelling wave solutions of this system. Furthermore, conservation laws are constructed using the multiplier method.