Real Lie Algebras Research Articles

Generalized derivations and some structure theorems for Lie algebras

In this work, we show that the existence of invertible generalized derivations impose strong restrictions on the structure of a complex finite-dimensional Lie algebra. In particular, we recover the fact that a real Lie algebra admitting an abelian complex structure is necessarily solvable. On the other hand, we state a structure theorem for a Lie algebra [Formula: see text] admitting a periodic generalized derivation [Formula: see text].

Cartan subalgebras for quantum symmetric pair coideals

There is renewed interest in the coideal subalgebras used to form quantum symmetric pairs because of recent discoveries showing that they play a fundamental role in the representation theory of quantized enveloping algebras. However, there is still no general theory of finite-dimensional modules for these coideals. In this paper, we establish an important step in this direction: we show that every quantum symmetric pair coideal subalgebra admits a quantum Cartan subalgebra which is a polynomial ring that specializes to its classical counterpart. The construction builds on Kostant and Sugiura’s classification of Cartan subalgebras for real semisimple Lie algebras via strongly orthogonal systems of positive roots. We show that these quantum Cartan subalgebras act semisimply on finite-dimensional unitary modules and identify particularly nice generators of the quantum Cartan subalgebra for a family of examples.

Deformation of the Poisson Structure Related to Algebroid Bracket of Differential Forms and Application to Real Low Dimentional Lie Algebras

The main goal of this paper is to present the possibility of application of some well known tools of Poisson geometry to classification of real low dimensional Lie algebras.

EXACT SOLUTIONS FOR A DIRAC-TYPE EQUATION WITH N-FOLD DARBOUX TRANSFORMATION

Based on matrix spectral problems associated with the real special orthogonal Lie algebra so(3, $\mathbb{R}$), a Dirac-type equation is derived by virtue of the zero-curvature equation. Further, an N-fold Darboux transformation for the Dirac-type equation is constructed by means of the gauge transformation. Finally, as its application, some exact solutions and their figures are obtained via symbolic computation software (Maple).

Bruhat order in the Toda system on [formula omitted]: An example of non-split real form

In our previous papers (Chernyakov et al., 2014; 2016; 2017) we described the structure of trajectories of the symmetric Toda system on normal real forms of various Lie algebras and showed that it was totally determined by the Hasse diagram of the Bruhat order on the corresponding Weyl group. This note deals with the simplest non-split real Lie algebra, so(2,4). It turns out, that the phase diagram in this case is also closely related with the Bruhat order of the relative Weyl group, but a bit more information is necessary to describe the dimensions of the trajectory spaces.

New examples of rank one solvable real rigid Lie algebras possessing a nonvanishing Chevalley cohomology

The class of rank one solvable Lie algebras possessing a maximal torus t with eigenvalue spectrum spec(t)=(1,4,5,…,n+2) is studied in the context of rigidity. It is shown that from the value n ≥ 18, three isomorphism classes of rigid Lie algebras exist, two of them being algebraically rigid, and the third being geometrically rigid with a two-dimensional cohomology space H2(g,g).

Gradings on the simple real Lie algebras of types G2 and D4

We classify group gradings on the simple Lie algebras of types G2 and D4 over the field of real numbers (or any real closed field): fine gradings up to equivalence and G-gradings, with a fixed group G, up to isomorphism.

Complex structures on the complexification of a real Lie algebra

Abstract Let g = a+b be a Lie algebra with a direct sum decomposition such that a and b are Lie subalgebras. Then, we can construct an integrable complex structure J̃ on h = ℝ(gℂ) from the decomposition, where ℝ(gℂ) is a real Lie algebra obtained from gℂby the scalar restriction. Conversely, let J̃ be an integrable complex structure on h = ℝ(gℂ). Then, we have a direct sum decomposition g = a + b such that a and b are Lie subalgebras. We also investigate relations between the decomposition g = a + b and dim Hs.t ∂̄ J̃ (hℂ).

Classification of three-dimensional zeropotent algebras over the real number field

ABSTRACTA nonassociative algebra is defined to be zeropotent if the square of any element is zero. In this paper, we give a complete classification of three-dimensional zeropotent algebras over the real number field up to isomorphism. By restricting the result to the subclass of Lie algebras, we can obtain a classification of three-dimensional real Lie algebras, which is in accordance with the Bianchi classification. Moreover, three-dimensional zeropotent algebras over a real closed field are classified in the same manner as those over the real number field.

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.

Invariant Poisson–Nijenhuis structures on Lie groups and classification

We study right-invariant (respectively, left-invariant) Poisson–Nijenhuis structures ([Formula: see text]-[Formula: see text]) on a Lie group [Formula: see text] and introduce their infinitesimal counterpart, the so-called r-n structures on the corresponding Lie algebra [Formula: see text]. We show that [Formula: see text]-[Formula: see text] structures can be used to find compatible solutions of the classical Yang–Baxter equation (CYBE). Conversely, two compatible [Formula: see text]-matrices from which one is invertible determine an [Formula: see text]-[Formula: see text] structure. We classify, up to a natural equivalence, all [Formula: see text]-matrices and all [Formula: see text]-[Formula: see text] structures with invertible [Formula: see text] on four-dimensional symplectic real Lie algebras. The result is applied to show that a number of dynamical systems which can be constructed by [Formula: see text]-matrices on a phase space whose symmetry group is Lie group a [Formula: see text], can be specifically determined.

Anti-Kählerian Geometry on Lie Groups

Let $G$ be a Lie group of even dimension and let $(g,J)$ be a left invariant anti-K\"ahler structure on $G$. In this article we study anti-K\"{a}hler structures considering the distinguished cases where the complex structure $J$ is abelian or bi-invariant. We find that if $G$ admits a left invariant anti-K\"ahler structure $(g,J)$ where $J$ is abelian then the Lie algebra of $G$ is unimodular and $(G,g)$ is a flat pseudo-Riemannian manifold. For the second case, we see that for any left invariant metric $g$ for which $J$ is an anti-isometry we obtain that the triple $(G, g, J)$ is an anti-K\"ahler manifold. Besides, given a left invariant anti-Hermitian structure on $G$ we associate a covariant $3$-tensor $\theta$ on its Lie algebra and prove that such structure is anti-K\"ahler if and only if $\theta$ is a skew-symmetric and pure tensor. From this tensor we classify the real 4-dimensional Lie algebras for which the corresponding Lie group has a left invariant anti-K\"ahler structure and study the moduli spaces of such structures (up to group isomorphisms that preserve the anti-K\"ahler structures).

Classical r-matrices for the generalised Chern–Simons formulation of 3d gravity

We study the conditions for classical r-matrices to be compatible with the generalised Chern–Simons action for 3d gravity. Compatibility means solving the classical Yang–Baxter equations with a prescribed symmetric part for each of the real Lie algebras and bilinear pairings arising in the generalised Chern–Simons action. We give a new construction of r-matrices via a generalised complexification and derive a non-linear set of matrix equations determining the most general compatible r-matrix. We exhibit new families of solutions and show that they contain some known r-matrices for special parameter values.

A reducible Weil representation of sp(4) realized by differential operators in the space of smooth functions on H2 × S1

This work presents a novel way to obtain the associated Romanovski functions Rn,m(x) with n ≥ m in the three separate regions in terms of n and m. We obtain the raising and lowering relations with respect to the both indices, simultaneously, in the three regions. Then, a reducible Weil representation of the real Lie algebra sp(4) is realized in the space of complex-valued smooth functions on H2 × S1 by differential forms for the Cartan-Weyl basis. Its invariant subspace is the second rare instance of the highest weight irreducible representation of sp(4) all whose weight spaces are one-dimensional.

FUNCTIONAL REALIZATIONS OF LIE ALGEBRAS AS NOETHER POINT SYMMETRIES OF SYSTEMS

<p>Functional realizations of Lie algebras are applied to the problem of determining Lie and Noether point symmetries of Lagrangian systems in <em>N</em> dimensions, particularly in the plane. This encompasses both the case of symmetry-preserving perturbations of a given system, as well as the generic analysis on the structure of (regular) Lagrangians in order to admit a symmetry algebra belonging to a specific isomorphy class.</p>

CLASSIFICATION OF REDUCTIVE REAL SPHERICAL PAIRS I. THE SIMPLE CASE

This paper gives a classification of all pairs $(\mathfrak g, \mathfrak h)$ with $\mathfrak g$ a simple real Lie algebra and $\mathfrak h < \mathfrak g$ a reductive subalgebra for which there exists a minimal parabolic subalgebra $\mathfrak p < \mathfrak g$ such that $\mathfrak g = \mathfrak h + \mathfrak p$ as vector sum.

Notes on G2: The Lie algebra and the Lie group

These notes have been prepared for the Workshop on “(Non)-existence of complex structures on S6”, celebrated in Marburg in March, 2017. The material is not intended to be original. It contains a survey about the smallest of the exceptional Lie groups: G2, its definition and different characterizations as well as its relationship to the spheres S6 and S7. With the exception of the summary of the Killing–Cartan classification, this survey is self-contained, and all the proofs are provided. Although these proofs are well-known, they are scattered, some of them are difficult to find, and others require stronger background, while we will usually stick to linear algebra arguments. The approach is algebraical, working at the Lie algebra level most often. We analyze the complex Lie algebra (and group) of type G2 as well as the two real Lie algebras of type G2, the split and the compact one. The octonion algebra will play its role, but it is not the starting point. Also, both the 3-forms approach and the spinorial approach are viewed and connected. Special emphasis is put on relating all the viewpoints by providing precise models.

Basic quantizations of D = 4 Euclidean, Lorentz, Kleinian and quaternionic {mathfrak{o}}^{star }(4) symmetries

We construct firstly the complete list of five quantum deformations of D = 4 complex homogeneous orthogonal Lie algebra mathfrak{o}left(4;mathbb{C}right)cong mathfrak{o}left(3;mathbb{C}right)oplus mathfrak{o}left(3;mathbb{C}right) , describing quantum rotational symmetries of four-dimensional complex space-time, in particular we provide the corresponding universal quantum R-matrices. Further applying four possible reality conditions we obtain all sixteen Hopf-algebraic quantum deformations for the real forms of mathfrak{o}left(4;mathbb{C}right) : Euclidean mathfrak{o}(4) , Lorentz mathfrak{o}left(3, 1right) , Kleinian mathfrak{o}left(2, 2right) and quaternionic {mathfrak{o}}^{star }(4) . For mathfrak{o}left(3, 1right) we only recall well-known results obtained previously by the authors, but for other real Lie algebras (Euclidean, Kleinian, quaternionic) as well as for the complex Lie algebra mathfrak{o}left(4;mathbb{C}right) we present new results.

The classical dynamic symmetry for the U(1)-Kepler problems

For the Jordan algebra of hermitian matrices of order $n\ge 2$, we let $X$ be its submanifold consisting of rank-one semi-positive definite elements. The composition of the cotangent bundle map $\pi_X$: $T^*X\to X$ with the canonical map $X\to \mathbb{C}P^{n-1}$ (i.e., the map that sends a hermitian matrix to its column space), pulls back the K\"{a}hler form of the Fubini-Study metric on $\mathbb{C}P^{n-1}$ to a real closed differential two-form $\omega_K$ on $T^*X$. Let $\omega_X$ be the canonical symplectic form on $T^*X$ and $\mu$ be a real number. A standard fact says that $\omega_\mu:=\omega_X+2\mu\,\omega_K$ turns $T^*X$ into a symplectic manifold, hence a Poisson manifold with Poisson bracket $\{\, ,\,\}_\mu$. In this article we exhibit a Poisson realization of the simple real Lie algebra $\mathfrak {su}(n, n)$ on the Poisson manifold $(T^*X, \{\, ,\,\}_\mu)$, i.e., a Lie algebra homomorphism from $\mathfrak {su}(n, n)$ to $\left(C^\infty(T^*X, \mathbb R), \{\, ,\,\}_\mu\right)$. Consequently one obtains the Laplace-Runge-Lenz vector for the classical $\mathrm{U}(1)$-Kepler problem with level $n$ and magnetic charge $\mu$. Since the McIntosh-Cisneros-Zwanziger-Kepler problems (MICZ-Kepler Problems) are the $\mathrm{U}(1)$-Kepler problems with level $2$, the work presented here is a direct generalization of the work by A. Barut and G. Bornzin [ J. Math. Phys. $\bf 12$ (1971), 841-843] on the classical dynamic symmetry for the MICZ-Kepler problems.

Parabolic conformally symplectic structures I; definition and distinguished connections

Abstract We introduce a class of first order G-structures, each of which has an underlying almost conformally symplectic structure. There is one such structure for each real simple Lie algebra which is not of type C n {C_{n}} and admits a contact grading. We show that a structure of each of these types on a smooth manifold M determines a canonical compatible linear connection on the tangent bundle TM {\mathrm{TM}} . This connection is characterized by a normalization condition on its torsion. The algebraic background for this result is proved using Kostant’s theorem on Lie algebra cohomology. For each type, we give an explicit description of both the geometric structure and the normalization condition. In particular, the torsion of the canonical connection naturally splits into two components, one of which is exactly the obstruction to the underlying structure being conformally symplectic. This article is the first in a series aiming at a construction of differential complexes naturally associated to these geometric structures.