Real Lie Research Articles

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.

Foliations Formed by Generic Coadjoint Orbits of a Class of Real Seven-Dimensional Solvable Lie Groups

In this paper, we consider exponential, connected and simply connected Lie groups which are corresponding to seven-dimensional Lie algebras such that their nilradical is a five-dimensional nilpotent Lie algebra $\mathfrak{g}_{5,2}$ given in Table~\ref{tab1}. In particular, we give a description of the geometry of the generic orbits in the coadjoint representation of some considered Lie groups. We prove that, for each considered group, the family of the generic coadjoint orbits forms a measurable foliation in the sense of Connes. The topological classification of these foliations is also provided.

KILLING FORM ALJABAR LIE FROBENIUS BERDIMENSI ≤4 UNTUK MENENTUKAN KESEMISEDERHANAANNYA

We study the notion of the Killing form for Frobenius Lie algebras of dimension . The Killing form is a symmetric bilinear form on a finite dimensional Lie algebra over a field defined by where is denoted the trace and is an adjoint representation of . A Lie algebras is said to be semisimple if it has the nondegenerate Killing form. The research aims to consider the criterion for semisimplicity of Frobenius Lie algebras of dimension by using the Killing form. The results show that each Frobenius Lie algebra of dimension and is not semisimple since the the Killing form is degenerate or in other words, a determinant of a representation matrix of the Killing form is equal to zero. For the future research, it is still an open problem to consider the general formulas of the Killing form for higher dimensional Frobenius Lie algebra whether degenerate or nondegenerate such that the semisimplicity of a Lie algebra can be considered. We conjecture that each finite dimensional real Frobenius Lie algebra is not semisimple.

Darboux Families and the Classification of Real Four-Dimensional Indecomposable Coboundary Lie Bialgebras

This work introduces a new concept, the so-called Darboux family, which is employed to determine coboundary Lie bialgebras on real four-dimensional, indecomposable Lie algebras, as well as geometrically analysying, and classifying them up to Lie algebra automorphisms, in a relatively easy manner. The Darboux family notion can be considered as a generalisation of the Darboux polynomial for a vector field. The classification of r-matrices and solutions to classical Yang–Baxter equations for real four-dimensional indecomposable Lie algebras is also given in detail. Our methods can further be applied to general, even higher-dimensional, Lie algebras. As a byproduct, a method to obtain matrix representations of certain Lie algebras with a non-trivial center is developed.

Lie superalgebras based on Heisenberg Lie algebras

We give a complete classification of real and complex Lie superalgebras based on the Heisenberg Lie algebra . This classification generalizes previous classifications.

Classification of left invariant Hermitian structures on 4-dimensional non-compact rank one symmetric spaces

The only 4-dimensional non-compact rank one symmetric spaces are CH2 and RH4. By the classical results of Heintze, one can model these spaces by real solvable Lie groups with left invariant metrics. In this paper we classify all possible left invariant Hermitian structures on these Lie groups, i.e., left invariant Riemannian metrics and the corresponding Hermitian complex structures. We show that each metric from the classification on CH2 admits at least four Hermitian complex structures. One class of metrics on CH2 and all the metrics on RH4 admit 2-spheres of Hermitian complex structures. The standard metric of CH2 is the only Einstein metric from the classification, and also the only metric that admits K¨ahler structure, while on RH4 all the metrics are Einstein. Finally, we examine the geometry of these Lie groups: curvature properties, self-duality, and holonomy.

Isometry Lie algebras of indefinite homogeneous spaces of finite volume

Let $\mathfrak{g}$ be a real finite-dimensional Lie algebra equipped with a symmetric bilinear form $\langle\cdot,\cdot\rangle$. We assume that $\langle\cdot,\cdot\rangle $ is nil-invariant. This means that every nilpotent operator in the smallest algebraic Lie subalgebra of endomomorphims containing the adjoint representation of $\mathfrak{g}$ is an infinitesimal isometry for $\langle\cdot,\cdot\rangle $. Among these Lie algebras are the isometry Lie algebras of pseudo-Riemannian manifolds of finite volume. We prove a strong invariance property for nil-invariant symmetric bilinear forms, which states that the adjoint representations of the solvable radical and all simple subalgebras of non-compact type of $\mathfrak{g} $ act by infinitesimal isometries for $\langle\cdot,\cdot\rangle $. Moreover, we study properties of the kernel of $\langle\cdot,\cdot\rangle $ and the totally isotropic ideals in $\mathfrak{g} $ in relation to the index of $\langle\cdot,\cdot\rangle $. Based on this, we derive a structure theorem and a classification for the isometry algebras of indefinite homogeneous spaces of finite volume with metric index at most two. Examples show that the theory becomes significantly more complicated for index greater than two.

On the irreducibility of some induced representations of real reductive Lie groups

We prove the irreducibility of some standard modules of the metaplectic group Mp2n(ℝ) and some nonstandard modules of the split odd special orthogonal group SO2n+1(ℝ).

Foundations of the Theory of Dual Lie Algebras

In this paper we introduce and study dual Lie algebras, i.e., Lie algebras over the algebra of dual numbers. We establish some fundamental properties of such Lie algebras and compare them with the corresponding properties of real and complex Lie algebras. We discuss the question of classification of dual Lie algebras of small dimension and consider the connection of dual Lie algebras with approximate Lie algebras.

Bi-integrable and tri-integrable couplings of a soliton hierarchy associated with SO(4)

Abstract In our paper, the theory of bi-integrable and tri-integrable couplings is generalized to the discrete case. First, based on the six-dimensional real special orthogonal Lie algebra SO(4), we construct bi-integrable and tri-integrable couplings associated with SO(4) for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Moreover, Hamiltonian structures of the obtained bi-integrable and tri-integrable couplings are constructed by the variational identities.

Some relations between Hodge numbers and invariant complex structures on compact nilmanifolds

Abstract Let N be a simply connected real nilpotent Lie group, n its Lie algebra, and € a lattice in N. If a left-invariant complex structure on N is Γ-rational, then HƏ̄ s,t(Γ/N) ≃ HƏ̄ s,t(nC) for each s; t. We can construct different left-invariant complex structures on one nilpotent Lie group by using the complexification and the scalar restriction. We investigate relationships to Hodge numbers of associated compact complex nilmanifolds.

Bi-Integrable and Tri-Integrable Couplings of a Soliton Hierarchy Associated with SO(3)

Based on the three-dimensional real special orthogonal Lie algebra SO(3), by zero curvature equation, we present bi-integrable and tri-integrable couplings associated with SO(3) for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Moreover, Hamiltonian structures of the obtained bi-integrable and tri-integrable couplings are constructed by applying the variational identities.

Remarks on Hodge numbers and invariant complex structures of compact nilmanifolds

Abstract If N is a simply connected real nilpotent Lie group with a Γ-rational complex structure, where Γ is a lattice in N, then for each s, t.We study relations between invariant complex structures and Hodge numbers of compact nilmanifolds from a viewpoint of Lie algberas.

The portion of matrices with real spectrum in the real orthogonal Lie algebra

The portion of matrices with real spectrum in a matrix Lie algebra is the ratio of the volume of the set of matrices with real spectrum in a ball centered at the zero of the algebra to the volume of the whole ball. We calculate the portion for the real orthogonal Lie algebra.

A New Soliton Hierarchy Associated withso⁡3,Rand Its Conservation Laws

Based on the three-dimensional real special orthogonal Lie algebraso(3,R), we construct a new hierarchy of soliton equations by zero curvature equations and show that each equation in the resulting hierarchy has a bi-Hamiltonian structure and thus integrable in the Liouville sense. Furthermore, we present the infinitely many conservation laws for the new soliton hierarchy.

Adjoint Orbits of sl(2, ℝ) on Real Simple Lie Algebras and Controllability

Adjoint Orbits of sl(2, ℝ) on Real Simple Lie Algebras and Controllability

The $$C^{*}$$ C ∗ -algebras of connected real two-step nilpotent Lie groups

Using the operator valued Fourier transform, the \(C^{*}\)-algebras of connected real two-step nilpotent Lie groups are characterized as algebras of operator fields defined over their spectra. In particular, it is shown by explicit computations, that the Fourier transform of such \(C^{*}\)-algebras fulfills the norm controlled dual limit property.

Fantappiè’s “final relativity” and deformations of Lie algebras

The rigidity of the real semisimple Lie algebra \({\mathfrak {so}}(4,1)\) was first proved in a brief paper published by Fantappie in 1954 (Rend. Accad. Lincei Ser. VIII 17:158–165, 1954). The purpose of this note is to provide some historical context for this work and discuss why no further developments of this result were pursued by Italian mathematicians at the time.

Pseudo-Riemannian Metric with Neutral Signature Induced by Solutions to Euler–Lagrange Equations for a Field of Complex Linear Frames

We investigate a model of the field of complex linear frames E on the product manifold M = R × G, where G is a real semisimple Lie group. The model is invariant under the natural action of the group GL( n , C ) ( n = dim M ). It results in a modified Born–Infeld-type nonlinearity of Euler–Lagrange equations. We analyse a family of solutions to Euler–Lagrange equations. Each solution E belonging to this family induces a pseudo-Riemannian metric γ[ E ] on M = R × G. In the physical case where n = dim M = 4, among these solutions there exist ones for which the signature of γ[ E ] is neutral (++− −). The existence of solutions leading to the neutral signature of γ[ E ] is interesting in itself. Additionally, it can shed new light onto the theory of generally-relativistic spinors and the conformal U(2, 2)-symmetry.

A note on the isometry group of the hyperbolic plane

This note concerns \(G\), the connected component of the identity of the isometries of the hyperbolic plane. \(G\) operates on the closed unit disk. Here we characterize \(G\) both as a Lie group among the centerless, connected, real semisimple Lie groups of noncompact type as having no two dimensional abelian Lie subgroups, then as a transformation group: any two nontrivial isometries of \(G\) commute if and only if they have the same fixed point set, this property being particular to \(G\) among all isometry groups of irreducible symmetric spaces of noncompact type. From this it follows that any subgroup of \(G\) with a nontrivial center is abelian. In particular, each nilpotent subgroup of \(G\) is abelian. Finally, we uniquely characterize SO\((2,1)\) among the centerless, connected, real simple Lie groups of noncompact type with this property.