Real Lie Algebras Research Articles

The Toda flow on Hessenberg elements of real, split simple Lie algebras

In this work, we consider the Toda flow associated with compact/Borel decompositions of real, split simple Lie algebras. Using the primitive invariant polynomials of Chevalley, we show how to construct integrals in involution which are invariants of the maximal compact subgroup, and moreover, we show that the number of such integrals is given by a formula involving only Lie-theoretic data. We then introduce the space of Hessenberg elements, characterize the generic Hessenberg coadjoint orbits, and show that the dimension of such orbits is precisely twice the number of nontrivial invariants which appeared earlier. For the class of classical, real split simple Lie algebras, we construct angle-type variables which in particular shows that the Toda flow is Liouville integrable on generic Hessenberg coadjoint orbits.

Classification of 7-dimensional solvable Lie algebras having 5-dimensional nilradicals

By combining basic techniques in Lie Theory and a computer algebra tool which is the so-called triangular decomposition, the class of 7-dimensional real and complex indecomposable solvable Lie algebras having 5-dimensional nilradicals is classified up to isomorphism. In association with Gong (1998), Parry (2007), Hindeleh and Thompson (2008), we achieve a full classification of 7-dimensional real and complex indecomposable solvable Lie algebras.

Abelian factors in 2-step nilpotent Lie algebras constructed from graphs

We consider real 2-step metric nilpotent Lie algebras associated to graphs with possibly repeated edge labels as constructed by Ray in 2016. We determine how the structure of the edge labeling within the graph contributes to the abelian factor in these Lie algebras. Furthermore, we explicitly compute the abelian factor of the 2-step nilpotent Lie algebras associated with some families of graphs such as star graphs, cycles, Schreier graphs, and properly edge-colored graphs. We also study the singularity properties of these Lie algebras in certain cases.

Decomposable Five-Dimensional Lie Algebras in the Problem on Holomorphic Homogeneity in ℂ3

In connection with the problem of describing holomorphically homogeneous real hypersurfaces in the space ℂ3, we study five-dimensional real Lie algebras realized as algebras of holomorphic vector fields on such manifolds. We prove the following assertion: If on a holomorphically homogeneous real hypersurface M of the space ℂ3, there is a decomposable, solvable, five-dimensional Lie algebra of holomorphic vector fields having a full rank near some point P ∈ M, then this surface is either degenerate near P in the sense of Levy or is a holomorphic image of an affine-homogeneous surface.

Odd‐dimensional counterparts of abelian complex and hypercomplex structures

AbstractWe introduce the notion of abelian almost contact structures on an odd‐dimensional real Lie algebra . We investigate correspondences with even‐dimensional Lie algebras endowed with an abelian complex structure, and with Kähler Lie algebras when carries a compatible inner product. The classification of 5‐dimensional Sasakian Lie algebras with abelian structure is obtained. Later, we introduce abelian almost 3‐contact structures on real Lie algebras of dimension , obtaining the classification of these Lie algebras in dimension 7. Finally, we deal with the geometry of a Lie group G endowed with a left invariant abelian almost 3‐contact metric structure. We determine conditions for G to admit a canonical metric connection with skew torsion, which plays the role of the Bismut connection for hyperKähler with torsion (HKT) structures arising from abelian hypercomplex structures. We provide examples and discuss the parallelism of the torsion of the canonical connection.

Almost Kaehlerian and Hermitian Structures on Four Dimensional Indecomposable Lie Algebras

It is known that from a given almost Hermitian structure on a simply connected Liegroup, one can obtain left-invariant almost Hermitian structure on its Lie algebra.In this work, we consider Mubarakzyanov’s classification of four-dimensional realLie algebras and evaluate the existence of almost Hermitian structures on four dimensional decomposable real Lie algebras. In particular, we focus on almost Kaehlerian and Hermitian structures on these Lie algebras.

Computer-Aided Analysis of Solvable Rigid Lie Algebras with a Given Eigenvalue Spectrum

With the help of symbolic computer packages, the study of the cohomological rigidity of real solvable Lie algebras of rank one with a maximal torus of derivations t and the eigenvalue spectrum spec(t)=1,k,k+1,⋯,n+k−2 initiated in a previous work is continued for arbitrary values k≥2, obtaining new hierarchies of solvable rigid Lie algebras.

An Atlas Adapted to the Toda Flow

Abstract We describe an atlas adapted to the Toda flow on the manifold of full flags of any non-compact real semisimple Lie algebra and on its Hessenberg-type submanifolds. We show that in the local coordinates of the atlas the Toda flow becomes linear. The local coordinates are also used to show that the Toda flow on the manifold of full flags is Morse–Smale, which generalizes the result for traceless matrices in [27] to arbitrary non-compact real semisimple Lie algebras. As a byproduct we describe new features of classical constructions in matrix theory.

Inner Ideals of Real Simple Lie Algebras

A classification up to automorphism of the inner ideals of the real finite-dimensional simple Lie algebras is given, jointly with precise descriptions in the case of the exceptional Lie algebras.

Higher-Order Relative Kinematics of Rigid Body and Multibody Systems. A Novel Approach with Real and Dual Lie Algebras

In this paper, a computing method for the higher-order acceleration vector field properties in general rigid body motion and multibody system is proposed using algebraic properties of dual number algebras. The higher-order accelerations field of a rigid body in a general motion is uniquely determined of higher- order time derivative of a spatial twist. For the relative kinematics in multibody systems, equations that allow the determination of the higher-order order acceleration vector field are given, using an exponential Brockett-like formula in these algebras. This is obtained with no need for further differentiation of the body pose with respect to time variable. The properties for the fields of velocity, acceleration, jerk, and jounce are presented, as well as cases of the general theory. This approach uses the isomorphism between the Lie algebra of the rigid displacements ▪, of the Special Euclidean group, SE3, and the Lie algebra of dual vectors. The results are coordinate-free and in a closed-form.

Integrable couplings of two expanded non-isospectral soliton hierarchies and their bi-Hamiltonian structures

Starting from two matrix spectral problems associated with the real special orthogonal Lie algebra [Formula: see text], two non-isospectral hierarchies of Ablowitz–Kaup–Newell–Segur (AKNS) type and Kaup–Newell (KN) type are constructed. In addition, we enlarge the two spectral problems and then obtain non-isospectral [Formula: see text] integrable couplings of AKNS type and KN type by solving the expanded non-isospectral zero curvature equation. We find that the obtained four hierarchies have the bi-Hamiltonian structure of the combined form. It follows that all equations in the resulting soliton hierarchy are integrable in the sense of Liouville.

Reduction and reconstruction of multisymplectic Lie systems

A Lie system is a non-autonomous system of first-order ordinary differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional real Lie algebra of vector fields, a so-called Vessiot–Guldberg Lie algebra. In this work, multisymplectic forms are applied to the study of the reduction of Lie systems through their Lie symmetries. By using a momentum map, we perform a reduction and reconstruction procedure of multisymplectic Lie systems, which allows us to solve the original problem by analysing several simpler multisymplectic Lie systems. Conversely, we study how reduced multisymplectic Lie systems allow us to retrieve the form of the multisymplectic Lie system that gave rise to them. Our results are illustrated with examples from physics, mathematics, and control theory.

Classification of four-rebit states

We classify states of four rebits, that is, we classify the orbits of the group Gˆ(R)=SL(2,R)4 in the space (R2)⊗4. This is the real analogon of the well-known SLOCC operations in quantum information theory. By constructing the Gˆ(R)-module (R2)⊗4 via a Z/2Z-grading of the simple split real Lie algebra of type D4, the orbits are divided into three groups: semisimple, nilpotent and mixed. The nilpotent orbits have been classified in Dietrich et al. (2017) [26], yielding applications in theoretical physics (extremal black holes in the STU model of N=2,D=4 supergravity, see Ruggeri and Trigiante (2017) [51]). Here we focus on the semisimple and mixed orbits which we classify with recently developed methods based on Galois cohomology, see Borovoi et al. (2021) [8,9]. These orbits are relevant to the classification of non-extremal (or extremal over-rotating) and two-center extremal black hole solutions in the STU model.

The structure of almost Abelian Lie algebras

An almost Abelian Lie algebra is a non-Abelian Lie algebra with a codimension 1 Abelian ideal. Most 3-dimensional real Lie algebras are almost Abelian, and they appear in every branch of physics that deals with anisotropic media — cosmology, crystallography, etc. In differential geometry and theoretical physics, almost Abelian Lie groups have given rise to some of the simplest solvmanifolds on which various geometric structures such as symplectic, Kähler, spin, etc., are currently studied in explicit terms. However, a systematic study of almost Abelian Lie groups and algebras from mathematics perspective has not been carried out yet, and this paper is the first step in addressing this wide and diverse class of groups and algebras. This paper studies the structure and important algebraic properties of almost Abelian Lie algebras of arbitrary dimension over any field of scalars. A classification of almost Abelian Lie algebras is given. All Lie subalgebras and ideals, automorphisms and derivations, Lie orthogonal operators and quadratic Casimir elements are described exactly.

Lie subalgebras of so(3,1) up to conjugacy

PurposeThis study aims to find all subalgebras up to conjugacy in the real simple Lie algebra so(3,1).Design/methodology/approachThe authors use Lie Algebra techniques to find all inequivalent subalgebras of so(3,1) in all dimensions.FindingsThe authors find all subalgebras up to conjugacy in the real simple Lie algebra so(3,1).Originality/valueThis paper is an original research idea. It will be a main reference for many applications such as solving partial differential equations. If so(3,1) is part of the symmetry Lie algebra, then the subalgebras listed in this paper will be used to reduce the order of the partial differential equation (PDE) and produce non-equivalent solutions.

A Left-Symmetric Structure on The Semi-Direct Sum Real Frobenius Lie Algebra of Dimension 8

Let be the Lie algebra of the semi-direct sum of the real vector space and the Lie algebra of the sets of all real matrices. In this paper, a Frobenius functional is constructed in order for the Lie algebra to be the real Frobenius Lie algebra of dimension 8. Moreover, a bilinear form corresponding to this Frobenius functional is symplectic. Then the obtained symplectic bilinear form induces the left-symmetric algebra structures on . In other words, the Lie algebra is the left-symmetric algebra. In particular, we give the formulas of its left-symmetric algebra structure explicitely. The left-symmetric algebra structures for case of higher dimension of this Lie algebra type are still an open problem to be investigated.

3-Derivations and 3-Automorphisms on Lie Algebras

In this paper, first we establish the explicit relation between 3-derivations and 3- automorphisms of a Lie algebra using the differential and exponential map. More precisely, we show that the Lie algebra of 3-derivations is the Lie algebra of the Lie group of 3-automorphisms. Then we study the derivations and automorphisms of the standard embedding Lie algebra of a Lie triple system. We prove that derivations and automorphisms of a Lie triple system give rise to derivations and automorphisms of the corresponding standard embedding Lie algebra. Finally we compute the 3-derivations and 3-automorphisms of 3-dimensional real Lie algebras.

On the problem of classifying solvable Lie algebras having small codimensional derived algebras

In this paper, we study the classification of finite-dimensional real solvable Lie algebras whose derived algebras are of codimension 1 or 2. We present an effective method to classify -dimensional real solvable Lie algebras having 1-codimensional derived algebras provided that a full classification of n-dimensional nilpotent Lie algebras is given. In addition, the problem of classifying -dimensional real solvable Lie algebras having 2-codimensional derived algebras is proved to be wild. In this case, we classify a subclass of the considered Lie algebras which are extended from their derived algebras by a pair of derivations containing at least one inner derivation.

Representation of real solvable Lie algebras having 2-dimensional derived ideal and geometry of coadjoint orbits of corresponding Lie groups

Let Lie ([Formula: see text]) be the class of all [Formula: see text]-dimensional real solvable Lie algebras having [Formula: see text]-dimensional derived ideals. In 2020, Le et al. gave a classification of all non-2-step nilpotent Lie algebras of Lie ([Formula: see text], 2). We propose in this paper to study representations of these Lie algebras as well as their corresponding connected and simply connected Lie groups. That is, for each algebra, we give an upper bound of the minimal degree of a faithful representation. Then, we give a geometrical description of coadjoint orbits of corresponding groups. Moreover, we show that the characteristic property of the family of maximal dimensional coadjoint orbits of an MD-group studied by Shum et al. is still true for the Lie groups considered here. Namely, we prove that, for each considered group, the family of the maximal dimensional coadjoint orbits forms a measurable foliation in the sense of Connes. The topological classification of these foliations is also provided.

Yang-Mills model for centrally extended 2D gravity

A Yang-Mills theory linear in the scalar curvature for two-dimensional gravity with symmetry generated by the semidirect product formed with the Lie derivative of the algebra of diffeomorphisms with the two-dimensional Abelian algebra is formulated. As compared with dilaton models, the role of the dilaton is played by the dual field strength of a $U(1)$ gauge field. All vacuum solutions are found. They are either black holes or have constant scalar curvature. Those with constant scalar curvature have constant dual field strength. In particular, solutions with vanishing cosmological constant but nonzero scalar curvature exist. In the conformal-Lorenz gauge, the model has a conformal field theory interpretation whose residual symmetry combines holomorphic diffeomorphisms with a subclass of $U(1)$ gauge transformations while preserving two-dimensional de Sitter and anti-de Sitter boundary conditions. This is the same symmetry as in Jackiw-Teitelboim-Maxwell gravity considered by Hartman and Strominger. It is argued that this is the only nontrivial Yang-Mills model linear in the scalar curvature that exists for real Lie algebras of dimension four.