Lie Algebra Research Articles

Lax representation and Bäcklund–Darboux transformation for coupled BKP hierarchy

By using vertex operator representation of polynomial Lie algebra b∞(n), a theoretical interpretation of coupled BKP hierarchy is given, where corresponding Hirota bilinear equations are derived. Then based upon this, a wave function matrix is introduced, so that we can construct the corresponding dressing operator matrix and obtain the coupled BKP constraints and Sato equation, which allows us to define the Lax operator matrix and obtain the Lax equation. It is found that the coupled BKP hierarchy is a special case of matrix KP theory. Lastly, we present the Bäcklund–Darboux transformations for the coupled BKP equations from both fermionic and bosonic pictures.

WITHDRAWN: Construction of a new [formula omitted]-dimensional KdV equation and its closed-form solutions with solitary wave behaviour and conserved vectors

This paper discusses the construction of a new (3+1)-dimensional Korteweg–de Vries (KdV) equation. By employing the KdV’s recursion operator, we extract two equations, and with elemental computation steps, the obtained result is 3uxyt+3uxzt−(ut−6uux+uxxx)yz−2ux∂x−1uyxz−2ux∂x−1uzxy=0. We then transform the new equation to a simpler one to avoid the appearance of the integral in the equation. Thereafter, we apply the Lie symmetry technique and gain a 7-dimensional Lie algebra L7 of point symmetries. The one-dimensional optimal system of Lie subalgebras is then computed and used in the reduction process to achieve seven exact solutions. These obtained solutions are graphically illustrated as 3D and 2D plots that show different propagations of solitary wave solutions such as breather, periodic, bell shape, and others. Finally, the conserved vectors are computed by invoking Ibragimov’s method.

Notes on emergent conformal symmetry for black holes

We examine the motion of the massless scalar field and nearly bound null geodesics in the near-ring region of a black hole, which may possess either acceleration or a gravitomagnetic mass. Around such black holes, the photon ring deviates from the equatorial plane. In the large angular momentum limit, we demonstrate that the massless scalar field exhibits an emergent conformal symmetry in this near-ring region. Additionally, in the nearly bound limit, we observe the emergence of a conformal symmetry for the null geodesics that constitute the photon ring in the black hole image. These findings suggest that the hidden conformal symmetry, associated with the Lie algebra 𝔰𝔩(2, ℝ), persists even for black holes lacking north-south reflection symmetry, thereby broadening the foundation of photon ring holography. Finally, we show that the conformal symmetry also emerges for nearly bound timelike geodesics and scalar fields in proximity to the particle ring, and with specific mass around a Schwarzschild black hole.

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.

Whittaker vectors in singular Whittaker modules

Let g be a complex semisimple Lie algebra with Borel subalgebra b and corresponding nilradical n . We show that singular Whittaker modules M are simple if and only if the space Wh M of Whittaker vectors is 1-dimensional. For arbitrary locally n -finite g -modules V, an immediate corollary is that the dimension of Wh V is bounded by the composition length of V.

Set inversion and box contraction on Lie groups using interval analysis

Set inversion and box contraction consist in finding guaranteed solutions to an equation where variables belong to bounded sets. These problems are classic in Euclidean spaces and can be solved using interval analysis. However, they quickly become complex when dealing with non-Euclidean manifolds. This paper investigates set inversion and box contraction problems when variables belong to generic Lie groups (e.g. the set of rotation matrices SO(3)). The concept of Lie group boxes is introduced and discussed. The core idea is to treat Lie group subsets via interval analysis in the Euclidean space of the Lie algebra. The concept of inclusion function on a Lie group is defined as an extension of classic inclusion functions in Euclidean interval analysis. It is shown that set inversion problems on Lie groups can be solved via the Lie algebra using Euclidean interval analysis and that the solution set has a finite volume in the group. On the same principle, the box contraction problem is extended to Lie groups via a Euclidean contractor in the Lie algebra. Numerical simulations demonstrate the high interest of using Lie group boxes instead of classic Euclidean parametrization for nonlinear problems (such as the Euler angles for rotations) both in terms of computational efficiency and accuracy. The proposed approach is of practical interest for state estimation and navigation in robotics since it makes it possible to solve highly nonlinear problems with a limited complexity and a high accuracy, in particular for attitude and pose estimation.

2-Local derivations of Heisenberg–Virasoro type Lie algebras

Let [Formula: see text] with [Formula: see text] be the Heisenberg–Virasoro type Lie algebras, and [Formula: see text] be a thin Lie algebra with only one diamond. In this paper, we study [Formula: see text]-local derivations on [Formula: see text] and [Formula: see text]. For [Formula: see text], we show that every [Formula: see text]-local derivation is a derivation for almost all parameter pairs [Formula: see text], generalizing some previous results, while for [Formula: see text], we show that it admits infinite many [Formula: see text]-local derivations which are not derivations. Along the way, we also prove that the space of outer derivations of [Formula: see text] is infinite-dimensional.

SKdV, SmKdV flows and their supersymmetric gauge-Miura transformations

The construction of Integrable Hierarchies in terms of zero curvature representation provides a systematic construction for a series of integrable non-linear evolution equations (flows) which shares a common affine Lie algebraic structure. The integrable hierarchies are then classified in terms of a decomposition of the underlying affine Lie algebra $\hat {\cal{G}} $ into graded subspaces defined by a grading operator $Q$. In this paper we shall discuss explicitly the simplest case of the affine $\hat {sl}(2)$ Kac-Moody algebra within the principal gradation given rise to the KdV and mKdV hierarchies and extend to supersymmetric models. It is known that the positive mKdV sub-hierachy is associated to some positive odd graded abelian subalgebra with elements denoted by $E^{(2n+1)}$. Each of these elements in turn, defines a time evolution equation according to time $t=t_{2n+1}$. An interesting observation is that for negative grades, the zero curvature representation allows both, even or odd sub-hierarchies. In both cases, the flows are non-local leading to integro-differential equations. Whilst positive and negative odd sub-hierarchies admit zero vacuum solutions, the negative even admits strictly non-zero vacuum solutions. Soliton solutions can be constructed by gauge transforming the zero curvature from the vacuum into a non trivial configuration (dressing method). Inspired by the dressing transformation method, we have constructed a gauge-Miura transformation mapping mKdV into KdV flows. Interesting new results concerns the negative grade sector of the mKdV hierarchy in which a double degeneracy of flows (odd and its consecutive even) of mKdV are mapped into a single odd KdV flow. These results are extended to supersymmetric hierarchies based upon the affine $\hat {sl}(2,1)$ super-algebra.

(Non-Symmetric) Yetter–Drinfel’d Module Category and Invariant Coinvariant Jacobians

In this paper, we generalize the homomorphisms of modules over groups and Lie algebras as being morphisms in the category of (non-symmetric) Yetter–Drinfel’d modules. These module homomorphisms play a key role in the conjecture of Yau.

Computational Construction of Weierstrass Sections for Semi-Invariant Polynomial Functions on Lie Algebras

In this paper, we present a computational approach to construct Weierstrass sections for semi-invariant polynomial functions on Lie algebras, extending the foundational work of Bourbaki and Popov. We focus on simple Lie algebras of type B, C, or D, and their associated parabolic subalgebras, particularly those with Levi factors composed of successive blocks of size two. Our method extends the notion of Weierstrass sections introduced by Popov, enabling us to explicitly construct these sections and establish their polynomiality. Furthermore, we demonstrate how these sections facilitate the linearization of semi-invariant generators. Central to our approach is the construction of an adapted pair, akin to a principal sl_2-triple in the non-reductive case. We provide computational algorithms and implementations for constructing these Weierstrass sections, offering a novel avenue for research in Lie algebra theory and algebraic geometry.

Double extensions of left-symmetric structures

The notion of double extension for symplectic Lie algebras was introduced by Medina and Revoy in 1985. In 2021, Valencia gave a condition for symplectic Lie algebras which are obtained through double extension to admit compatible left-symmetric structures. In this paper, we formulate the notion of double extension for left-symmetric structures and give a condition for left-symmetric structures to be obtained through double extension. Moreover, we prove that right nilpotent left-symmetric structures on some solvable Lie algebras are obtained through double extension. We also give a condition for left-symmetric structures which are obtained through double extension to be complete.

A Class of Multi-Component Non-Isospectral TD Hierarchies and Their Bi-Hamiltonian Structures

By using the classical Lie algebra, the stationary zero curvature equation, and the Lenard recursion equations, we obtain the non-isospectral TD hierarchy. Two kinds of expanding higher-dimensional Lie algebras are presented by extending the classical Lie algebra. By solving the expanded non-isospectral zero curvature equations, the multi-component non-isospectral TD hierarchies are derived. The Hamiltonian structure for one of them is obtained by using the trace identity.

On decompositions and transitive actions of nilpotent Lie groups

The article considers decompositions of nilpotent Lie algebras and nilpotent Lie groups, and connections between them. Also, descriptions of irreducible transitive actions of nilpotent Lie groups on the plane and on three-dimensional space are given.

Classification of flat Lorentzian nilpotent Lie algebras

AbstractWe give a complete classification of flat Lorentzian nilpotent Lie algebras, this is to say of pseudo‐Euclidean Lie algebras associated to nilpotent Lie groups endowed with a left‐invariant Lorentzian metric of vanishing curvature. We prove that every such a Lie algebra is a direct sum of an indecomposable flat Lorentzian Lie algebra and an abelian Euclidean summand and show that, if denotes the ‐dimensional Heisenberg Lie algebra, then the only non‐abelian Lie algebras admitting flat Lorentzian metrics which are indecomposable are and the semidirect products and , defined by some particular derivations . In all those cases we also find the equivalence classes of flat Lorentzian products.

Faithful Representations of Finite Type for Conformal Lie Algebras

Faithful Representations of Finite Type for Conformal Lie Algebras

On quadratic Hom-Lie algebras with twist maps in their centroids and their relationship with quadratic Lie algebras

Hom-Lie algebras having non-invertible twist maps in their centroids are studied. Central extensions of Hom-Lie algebras having these properties are obtained and shown how the same properties are preserved. Conditions are given so that the produced central extension has an invariant metric with respect to its Hom-Lie product making its twist map self-adjoint when the original Hom-Lie algebra has such a metric. This work is focused on algebras with these properties and following Benayadi and Makhlouf we call them quadratic Hom-Lie algebras. It is shown how a quadratic Hom-Lie algebra gives rise to a quadratic Lie algebra and that the Lie algebra associated to the given Hom-Lie central extension is a Lie algebra central extension of it. It is also shown that if the Hom-Lie product is not a Lie product, there exists a non-abelian algebra, which is in general non-associative too, the commutator of whose product is precisely the Hom-Lie product of the Hom-Lie central extension. Moreover, the algebra whose commutator realizes this Hom-Lie product is shown to be simple if the associated Lie algebra is nilpotent. Non-trivial examples are provided.

Relativistic Formulation in Dual Minkowski Spacetime

The objective of this work is to derive the structure of Minkowski spacetime using a Hermitian spin basis. This Hermitian spin basis is analogous to the Pauli spin basis. The derived Minkowski metric is then employed to obtain the corresponding Lorentz factors, potential Lie algebra, effects on gamma matrices and complex representations of relativistic time dilation and length contraction. The main results, a discussion of the potential applications and future research directions are provided.

Leibniz cohomology with adjoint coefficients

With the Poincaré group R 3 , 1 ⋊ O ( 3 , 1 ) as the model of departure, we focus, for n = p + q , p + q ≥ 4 , on the affine indefinite orthogonal group R p , q ⋊ SO ( p , q ) . Denote by h p , q the Lie algebra of the affine indefinite orthogonal group. We compute the Leibniz cohomology of h p , q with adjoint coefficients, written H L * ( h p , q ; h p , q ) . We calculate several indefinite orthogonal invariants, and h p , q -invariants.

The u(2|2)1 WZW model

WZW models based on super Lie algebras play an important role for the description of string theory on AdS spaces. In particular, for the case of AdS3×S3 with pure NS–NS flux the super Lie algebra of psu(1,1|2)k appears in the hybrid formalism, and higher dimensional AdS spaces can be described in terms of related supergroup cosets. In this paper we study the WZW models based on u(2|2)1 and psu(2|2)1 that may play a role for the worldsheet theory that is dual to free super Yang–Mills in 4D.

On the N-waves hierarchy with constant boundary conditions. Spectral properties

This paper is devoted to [Formula: see text]-wave equations with constant boundary conditions related to symplectic Lie algebras. We study the spectral properties of a class of Lax operators [Formula: see text], whose potentials [Formula: see text] tend to constants [Formula: see text] for [Formula: see text]. For special choices of [Formula: see text], we outline the spectral properties of [Formula: see text], the direct scattering transform and construct its fundamental analytic solutions. We generalize Wronskian relations for the case of CBC — this allows us to analyze the mapping between the scattering data and the [Formula: see text]-derivative of the potential [Formula: see text]. Next, using the Wronskian relations, we derive the dispersion laws for the [Formula: see text]-wave hierarchy and describe the NLEE related to the given Lax operator.