Real Simple Lie Algebras Research Articles

An isomorphism between unitals and between related classical groups

Abstract An isomorphism between two hermitian unitals is provided, and used to treat isomorphisms of classical groups that are related to the isomorphism between certain simple real Lie algebras of types A and D (and rank 3).

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.

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.

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.

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.

Octonions, triality, the exceptional Lie algebra F4 and polar actions on the Cayley hyperbolic plane

Using octonions and the triality property of Spin(8), we find explicit formulae for the Lie brackets of the exceptional simple real Lie algebras [Formula: see text] and [Formula: see text], i.e. the Lie algebras of the isometry groups of the Cayley projective plane and the Cayley hyperbolic plane. As an application, we determine all polar actions on the Cayley hyperbolic plane which leave a totally geodesic subspace invariant.

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.

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.

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.

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

A Poisson realization of the simple real Lie algebra so*(4n) on the phase space of each Sp(1)-Kepler problem is exhibited. As a consequence, one obtains the Laplace-Runge-Lenz vector for each classical Sp(1)-Kepler problem. The verification of these Poisson realizations is greatly simplified via an idea of Weinstein. The totality of these Poisson realizations is shown to be equivalent to the canonical Poisson realization of so*(4n) on the Poisson manifold T*H*n/Sp(1). (Here H*n≔Hn\{0} and the Hamiltonian action of Sp(1) on T*H*n is induced from the natural right action of Sp(1) on H*n.)

Leibniz and Hochschild homology

ABSTRACTWe construct and study the map from Leibniz homology HL∗(𝔥) of an abelian extension 𝔥 of a simple real Lie algebra 𝔤 to the Hochschild homology HH∗−1(U(𝔥)) of the universal envelopping algebra U(𝔥). To calculate some homology groups, we use the Hochschild-Serre spectral sequences and Pirashvili spectral sequences. The result shows what part of the non-commutative Leibniz theory is detected by classical Hochschild homology, which is of interest today in string theory.

Nilradicals of parabolic subalgebras admitting symplectic structures

In this paper we describe all the nilradicals of parabolic subalgebras of split real simple Lie algebras admitting symplectic structures.The main tools used to obtain this list are Kostant's description of the highest weight vectors (hwv) of the cohomology of these nilradicals and some necessary conditions obtained for the g-hwv's of H2(n) for a finite dimensional real symplectic nilpotent Lie algebra n with a reductive Lie subalgebra of derivations g acting on it.

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

A PERSPECTIVE ON THE MAGIC SQUARE AND THE "SPECIAL UNITARY" REALIZATION OF REAL SIMPLE LIE ALGEBRAS

This paper contains the last part of the minicourse "Spaces: A Perspective View" delivered at the IFWGP2012. The series of three lectures was intended to bring the listeners from the more naive and elementary idea of space as "our physical Space" (which after all was the dominant one up to the 1820s) through the generalization of the idea of space which took place in the last third of the 19th century. That was a consequence of first the discovery and acceptance of non-Euclidean geometry and second, of the views afforded by the works of Riemann and Klein and continued since then by many others, outstandingly Lie and Cartan. Here we deal with the part of the minicourse which centers on the classification questions associated to the simple real Lie groups. We review the original introduction of the Magic Square "á la Freudenthal", putting the emphasis in the role played in this construction by the four normed division algebras ℝ, ℂ, ℍ, 𝕆. We then explore the possibility of understanding some simple real Lie algebras as "special unitary" over some algebras 𝕂 or tensor products 𝕂1 ⊗ 𝕂2, and we argue that the proper setting for this construction is not to confine only to normed division algebras, but to allow the split versions ℂ′, ℍ′, 𝕆′ of complex, quaternions and octonions as well. This way we get a "Grand Magic Square" and we fill in all details required to cover all real forms of simple real Lie algebras within this scheme. The paper ends with the complete lists of all realizations of simple real Lie algebras as "special unitary" (or only unitary when n = 2) over some tensor product of two *-algebras 𝕂1, 𝕂2, which in all cases are obtained from ℝ, ℂ, ℂ′, ℍ, ℍ′, 𝕆, 𝕆′ as sets, endowing them with a *-conjugation which usually but not always is the natural complex, quaternionic or octonionic conjugation.

Computing with real Lie algebras: Real forms, Cartan decompositions, and Cartan subalgebras

We describe algorithms for performing various tasks related to real simple Lie algebras. These algorithms form the basis of our software package CoReLG, written in the language of the computer algebra system GAP4. First, we describe how to efficiently construct real simple Lie algebras up to isomorphism. Second, we consider a real semisimple Lie algebra g. We provide an algorithm for constructing a maximally (non-)compact Cartan subalgebra of g; this is based on the theory of Cayley transforms. We also describe the construction of a Cartan decomposition g=k⊕p. Using these results, we provide an algorithm to construct all Cartan subalgebras of g up to conjugacy; this is a constructive version of a classification theorem due to Sugiura.

Determinants of sums of two real matrices and their extensions

We study the determinants of the sum of two real matrices under the action of SO(n) ⊗ SO(n). The extremal determinants are determined. The result is a refinement of the results of Li and Mathias [C.K. Li and R. Mathias, The determinant of the sum of two matrices, Bull. Aust. Math. Soc. 52 (1995), pp. 425–429]. We also study the problem in the context of real classical simple Lie algebras. The results of Fiedler [M. Fiedler, Bounds for the determinant of the sum of Hermitian matrices, Proc. Amer. Math. Soc. 30 (1971), pp. 27–31], Li and Mathias 4 and Tam and Thompson [T.Y. Tam and M.C. Thompson, Determinant and Pfaffian of sum of skew symmetric matrices, Linear Algebra Appl. 433 (2010), pp. 412–423] are some special cases in this context. Complete solutions are obtained.

A structure theorem for Leibniz homology

Presented is a structure theorem for the Leibniz homology, HL⁎, of an Abelian extension of a simple real Lie algebra g. As applications, results are stated for affine extensions of the classical Lie algebras sln(R), son(R), and spn(R). Furthermore, HL⁎(h) is calculated when h is the Lie algebra of the Poincaré group as well as the Lie algebra of the affine Lorentz group. The structure theorem identifies all of these in terms of g-invariants.

Finite order automorphisms and real forms of affine Kac-Moody algebras in the smooth and algebraic category

Let $\mathfrak{g}$ be a real or complex (finite dimensional) simple Lie algebra and $\sigma\in\mathrm{Aut}\mathfrak{g}$. The authors study automorphisms of the twisted loop algebra $L(\mathfrak{g},\sigma)$ of smooth $\sigma$-periodic maps from $\mathbb{R}$ to $\mathfrak{g}$ as well as of the ``smooth'' affine Kac-Moody algebra $\hat L(\mathfrak{g},\sigma)$, which is a $2$-dimensional extension of $L(\mathfrak{g},\sigma)$. It turns out that these automorphisms which either preserve or reverse the orientation of loops, and are correspondingly called to be of first and second kind, can be described essentially by curves of automorphisms of $\mathfrak{g}$. If the order of the automorphisms is finite, then the corresponding curves in $\mathrm{Aut}\mathfrak{g}$ allow us to define certain invariants and these turn out to parametrize the conjugacy classes of the automorphisms. If their order is $2$ the authors carry this out in detail and deduce a complete classification of involutions and real forms (which correspond to conjugate linear involutions) of smooth affine Kac-Moody algebras. The resulting classification can be seen as an extension of Cartan's classification of symmetric spaces, i.e. of involutions on $\mathfrak{g}$. If $\mathfrak{g}$ is compact, then conjugate linear extensions of involutions from $\hat L(\mathfrak{g},\sigma)$ to conjugate linear involutions on $\hat L(\mathfrak{g}_{\mathbb{C}},\sigma_{\mathbb{C}})$ yield a bijection between their conjugacy classes and this gives existence and uniqueness of Cartan decompositions of real forms of complex smooth affine Kac-Moody algebras. The authors show that their methods work equally well also in the algebraic case where the loops are assumed to have finite Fourier expansions.

Construction process for simple Lie algebras

We give here a construction process for the complex simple Lie algebras and the non-Hermitian type real forms which intersect the minimal nilpotent complex adjoint orbit, using a finite dimensional irreducible representation of the conformal group, or of some two-fold covering of it, with highest weight vector a semi-invariant of degree four. This process leads to a five-graded simple complex Lie algebra and the underlying semi-invariant is intimately related to the structure of the minimal nilpotent orbit. We also describe a similar construction process for the simple real Lie algebras of Hermitian type.