Contractions Of Lie Algebras Research Articles

Automorphisms of finite order, periodic contractions, and Poisson-commutative subalgebras of {mathcal {S}}({mathfrak g})

Let {mathfrak g} be a semisimple Lie algebra, vartheta in textsf{Aut}({mathfrak g}) a finite order automorphism, and {mathfrak g}_0 the subalgebra of fixed points of vartheta . Recently, we noticed that using vartheta one can construct a pencil of compatible Poisson brackets on {mathcal {S}}({mathfrak g}), and thereby a ‘large’ Poisson-commutative subalgebra {mathcal {Z}}({mathfrak g},vartheta ) of {mathcal {S}}({mathfrak g})^{{mathfrak g}_0}. In this article, we study invariant-theoretic properties of ({mathfrak g},vartheta ) that ensure good properties of {mathcal {Z}}({mathfrak g},vartheta ). Associated with vartheta one has a natural Lie algebra contraction {mathfrak g}_{(0)} of {mathfrak g} and the notion of a good generating system (=g.g.s.) in {mathcal {S}}({mathfrak g})^{mathfrak g}. We prove that in many cases the equality mathrm{ind,}{mathfrak g}_{(0)}=mathrm{ind,}{mathfrak g} holds and {mathcal {S}}({mathfrak g})^{mathfrak g} has a g.g.s. According to V. G. Kac’s classification of finite order automorphisms (1969), vartheta can be represented by a Kac diagram, mathcal {K}(vartheta ), and our results often use this presentation. The most surprising observation is that {mathfrak g}_{(0)} depends only on the set of nodes in mathcal {K}(vartheta ) with nonzero labels, and that if vartheta is inner and a certain label is nonzero, then {mathfrak g}_{(0)} is isomorphic to a parabolic contraction of {mathfrak g}.

Applying chaos indicators to Bianchi cosmological models

Using standard chaos indicators, we perform a numerical investigation of the Bianchi IX Universe in the limit where one of the structure constants tends to zero. In this limit, we recover Bianchi VII0, which can be obtained from a Lie algebra contraction. We find that in this limit, the chaoticity of the system is preserved, but the basin structures of the grid classification are changed.

PERIODIC AUTOMORPHISMS, COMPATIBLE POISSON BRACKETS, AND GAUDIN SUBALGEBRAS

Let \U0001d52e be a finite-dimensional Lie algebra. The symmetric algebra (\U0001d52e) is equipped with the standard Lie–Poisson bracket. In this paper, we elaborate on a surprising observation that one naturally associates the second compatible Poisson bracket on (\U0001d52e) to any finite order automorphism ϑ of \U0001d52e. We study related Poisson-commutative subalgebras (\U0001d52e; ϑ) of \U0001d4ae(\U0001d52e) and associated Lie algebra contractions of \U0001d52e. To obtain substantial results, we have to assume that \U0001d52e = \U0001d524 is semisimple. Then we can use Vinberg’s theory of ϑ-groups and the machinery of Invariant Theory.If \U0001d524 = \U0001d525⊕⋯⊕\U0001d525 (sum of k copies), where \U0001d525 is simple, and ϑ is the cyclic permutation, then we prove that the corresponding Poisson-commutative subalgebra (\U0001d52e; ϑ) is polynomial and maximal. Furthermore, we quantise this (\U0001d524; ϑ) using a Gaudin subalgebra in the enveloping algebra \U0001d4b0(\U0001d524).

Контракции алгебр Ли и уравнение Линдблада

The dynamics of an open quantum system leads to decoherence, which is accompanied by limiting transitions in the Lie algebra of observables and appearance of abelian subalgebras. It is possible to set an inverse problem as well – by a given Lie algebra contraction to find the dynamics of an open quantum system given by the Lindblad equation. The paper proposes examples of finding the Lindblad equation by the known contractions of algebra su(3).

The method of virtual copies and contractions of simple Lie algebras

Contractions of simple Lie algebras onto semidirect sums are analyzed in the context of the method of virtual copies, and various existence conditions are obtained in terms of the branching rules of representations and the properties of the characteristic representation associated to an embedding of semisimple algebras. Contractions of the Lie algebra with respect to nonmaximal symplectic subalgebras are studied in some detail. In particular, a realization of the subalgebra in terms of cubic polynomial operators is constructed.

Separations of Variables and Analytic Contractions on Two-Dimensional Hyperboloids

In this review we present recent results in the field of analytical contraction of Lie algebra in two-dimensional hyperbolic space. A complete geometric description for all possible orthogonal and nonorthogonal (related to the first order symmetries) systems of coordinates, which allow separation of variables of two-dimensional Laplace–Beltrami or Helmholtz equation on the two-sheeted (upper sheet) $${{H}_{2}}$$ and the one-sheeted $${{\tilde {H}}_{2}}$$ hyperboloids is given. The limiting transition between non subgroup (mostly parametric) and subgroup systems is conducted. The analytic contractions between various systems of coordinates in two hyperbolic spaces and Euclidean $${{E}_{2}}$$ and Minkowski $${{E}_{{1,1}}}$$ spaces are presented.

Conformal Lie algebras via deformation theory

We discuss possible notions of conformal Lie algebras, paying particular attention to graded conformal Lie algebras with d-dimensional space isotropy: namely, those with a co(d) subalgebra acting in a prescribed way on the additional generators. We classify those Lie algebras up to isomorphism for all d ⩾ 2 following the same methodology used recently to classify kinematical Lie algebras, as deformations of the “most abelian” graded conformal algebra. We find 17 isomorphism classes of Lie algebras for d ≠ 3 and 23 classes for d = 3. Lie algebra contractions define a partial order in the set of isomorphism classes, and this is illustrated via the corresponding Hasse diagram. The only metric graded conformal Lie algebras are the simple Lie algebras, isomorphic to either so(d+1,2) or so(d+2,1). We also work out the central extensions of the graded conformal algebras and study their invariant inner products. We find that central extensions of a Lie algebra in d = 3 and two Lie algebras in d = 2 are metric. We then discuss several other notions of conformal Lie algebras (generalised conformal, Schrödinger, and Lifshitz Lie algebras), and we present some partial results on their classification. We end with some open problems suggested by our results.

Диссипация кубита и контракции алгебр Ли

Диссипация кубита и контракции алгебр Ли

Reduction by invariants and projection of linear representations of Lie algebras applied to the construction of nonlinear realizations

A procedure for the construction of nonlinear realizations of Lie algebras in the context of Vessiot–Guldberg–Lie algebras of first-order systems of ordinary differential equations (ODEs) is proposed. The method is based on the reduction of invariants and projection of lowest-dimensional (irreducible) representations of Lie algebras. Applications to the description of parameterized first-order systems of ODEs related by contraction of Lie algebras are given. In particular, the kinematical Lie algebras in (2 + 1)- and (3 + 1)-dimensions are realized simultaneously as Vessiot–Guldberg–Lie algebras of parameterized nonlinear systems in R3 and R4, respectively.

Graded contractions of filiform Lie algebras

The study of contractions of Lie algebras is profusely extended in the last decades. In this paper we study the graded contractions of some lower‐dimensional filiform Lie algebras which have not been studied earlier. Particularly, we deal with graded contractions of model filiform Lie algebras of dimension less than or equal to 6 and with the ones of a nonmodel 6‐dimensional filiform Lie algebra.

Invariant functions and contractions of certain types of Lie algebras of lower dimensions

In this paper, we deal with contractions of Lie algebras. We use two invariant functions of Lie algebras as a tool, named ψ and φ function, respectively, which have a great application in Physics due to their remarkable properties. We focus the study of these functions in the frame of the filiform Lie algebras, trying to extend to these algebras some of the properties of such functions over semi-simple Lie algebras.

Lie–Hamilton systems on curved spaces: a geometrical approach

A Lie–Hamilton system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional Lie algebra, a Vessiot–Guldberg Lie algebra, of Hamiltonian vector fields relative to a Poisson structure. Its general solution can be written as an autonomous function, the superposition rule, of a generic finite family of particular solutions and a set of constants. We pioneer the study of Lie–Hamilton systems on Riemannian spaces (sphere, Euclidean and hyperbolic plane), pseudo-Riemannian spaces (anti-de Sitter, de Sitter, and Minkowski spacetimes) as well as on semi-Riemannian spaces (Newtonian spacetimes). Their corresponding constants of motion and superposition rules are obtained explicitly in a geometric way. This work extends the (graded) contraction of Lie algebras to a contraction procedure for Lie algebras of vector fields, Hamiltonian functions, and related symplectic structures, invariants, and superposition rules.

Formula omitted]-contractions of classical Lie algebras and symmetric polynomials

We consider Z2-contractions of classical Lie algebras and the behaviour of the symmetric invariants under these contractions. It is demonstrated on three different examples how the theory of symmetric polynomials works in this invariant-theoretic problem.

LAPLACE EQUATIONS, CONFORMAL SUPERINTEGRABILITY AND BÔCHER CONTRACTIONS

Quantum superintegrable systems are solvable eigenvalue problems. Their solvability is due to symmetry, but the symmetry is often ``hidden''.<p style="-qt-block-indent: 0; text-indent: 0px; margin: 0px;">The symmetry generators of 2nd order superintegrable systems in 2 dimensions close under commutation to define quadratic algebras, a generalization of Lie algebras. Distinct systems and their algebras are related by geometric limits, induced by generalized Inönü-Wigner Lie algebra contractions of the symmetry algebras of the underlying spaces. These have physical/geometric implications, such as the Askey scheme for hypergeometric orthogonal polynomials. The systems can be best understood by transforming them to Laplace conformally superintegrable systems and using ideas introduced in the 1894 thesis of Bôcher to study separable solutions of the wave equation. The contractions can be subsumed into contractions of the conformal algebra <em>so</em>(4,C) to itself. Here we announce main findings, with detailed classifications in papers under preparation.</p>

Bôcher Contractions of Conformally Superintegrable Laplace Equations

The explicit solvability of quantum superintegrable systems is due to symmetry, but the symmetry is often "hidden". The symmetry generators of 2nd order superintegrable systems in 2 dimensions close under commutation to define quadratic algebras, a generalization of Lie algebras. Distinct systems on constant curvature spaces are related by geometric limits, induced by generalized In\"on\"u-Wigner Lie algebra contractions of the symmetry algebras of the underlying spaces. These have physical/geometric implications, such as the Askey scheme for hypergeometric orthogonal polynomials. However, the limits have no satisfactory Lie algebra contraction interpretations for underlying spaces with 1- or 0-dimensional Lie algebras. We show that these systems can be best understood by transforming them to Laplace conformally superintegrable systems, with flat space conformal symmetry group ${\rm SO}(4,{\mathbb C})$, and using ideas introduced in the 1894 thesis of B\^ocher to study separable solutions of the wave equation in terms of roots of quadratic forms. We show that B\^ocher's prescription for coalescing roots of these forms induces contractions of the conformal algebra $\mathfrak{so}(4,{\mathbb C})$ to itself and yields a mechanism for classifying all Helmholtz superintegrable systems and their limits. In the paper [Acta Polytechnica, to appear, arXiv:1510.09067], we announced our main findings. This paper provides the proofs and more details.

On Contractions of Lie Algebras

It is well-known that application of invariant functions in various fields of pure or applied mathematics, physics and computer calculations is certainly of interest. In this paper, we study application of the invariant functions of Lie algebras, named \(\psi \) and \(\varphi \) function, respectively, to contractions of some lower-dimensional Lie algebras, with the goal of contributing to progress in any of these fields, particularly in physics and engineering.

Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems

Two-dimensional quadratic algebras are generalizations of Lie algebras that include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. Distinct superintegrable systems and their quadratic algebras can be related by geometric contractions, induced by In\"on\"u-Wigner type Lie algebra contractions. These geometric contractions have important physical and geometric meanings, such as obtaining classical phenomena as limits of quantum phenomena as ${\hbar}\to 0$ and nonrelativistic phenomena from special relativistic as $c\to \infty$, and the derivation of the Askey scheme for obtaining all hypergeometric orthogonal polynomials as limits of Racah/Wilson polynomials. In this paper we show how to simplify the structure relations for abstract nondegenerate and degenerate quadratic algebras and their contractions. In earlier papers we have classified contractions of 2nd order superintegrable systems on constant curvature spaces and have shown that all results are derivable from free quadratic algebras contained in the enveloping algebras of the Lie algebras $e(2,{\mathbb C})$ in flat space and $o(3,{\mathbb C})$ on nonzero constant curvature spaces. The quadratic algebra contractions are induced by generalizations of In\"on\"u-Wigner contractions of these Lie algebras. As a special case we obtained the Askey scheme for hypergeometric orthogonal polynomials. Here we complete this theoretical development for 2D superintegrable systems by showing that the Darboux superintegrable systems are also characterized by free quadratic algebras contained in the symmetry algebras of these spaces and that their contractions are also induced by In\"on\"u-Wigner contractions. We present tables of the contraction results.

Wilson polynomials/functions and intertwining operators for the generic quantum superintegrable system on the 2-sphere

The Wilson and Racah polynomials can be characterized as basis functions for irreducible representations of the quadratic symmetry algebra of the quantum superintegrable system on the 2-sphere, HΨ = EΨ, with generic 3-parameter potential. Clearly, the polynomials are expansion coefficients for one eigenbasis of a symmetry operator L2 of H in terms of an eigenbasis of another symmetry operator L1, but the exact relationship appears not to have been made explicit. We work out the details of the expansion to show, explicitly, how the polynomials arise and how the principal properties of these functions: the measure, 3-term recurrence relation, 2nd order difference equation, duality of these relations, permutation symmetry, intertwining operators and an alternate derivation of Wilson functions - follow from the symmetry of this quantum system. This paper is an exercise to show that quantum mechancal concepts and recurrence relations for Gausian hypergeometrc functions alone suffice to explain these properties; we make no assumptions about the structure of Wilson polynomial/functions, but derive them from quantum principles. There is active interest in the relation between multivariable Wilson polynomials and the quantum superintegrable system on the n-sphere with generic potential, and these results should aid in the generalization. Contracting function space realizations of irreducible representations of this quadratic algebra to the other superintegrable systems one can obtain the full Askey scheme of orthogonal hypergeometric polynomials. All of these contractions of superintegrable systems with potential are uniquely induced by Wigner Lie algebra contractions of so(3, C) and e(2,C). All of the polynomials produced are interpretable as quantum expansion coefficients. It is important to extend this process to higher dimensions.

(Quasi)-exact-solvability on the sphere Sn

An Exactly Solvable (ES) potential on the sphere Sn is reviewed and a related Quasi-Exactly Solvable (QES) potential is found and studied. After mapping the sphere to a simplex, it is found that the metric (of constant curvature) is in polynomial form, and both the ES and the QES potentials are rational functions. Their hidden algebra is gln, realized in a finite-dimensional representation by first order differential operators acting on RPn. It is shown that variables in the Schrödinger eigenvalue equation can be separated in polyspherical coordinates and there is always complete integrability. The QES system is completely integrable for n = 2 and non-maximally superintegrable for n ≥ 3. There is no separable coordinate system in which it is exactly solvable. We point out that by taking contractions of superintegrable systems, such as induced by Inönü-Wigner Lie algebra contractions, we can find other QES superintegrable systems, and we illustrate this by contracting our Sn system to a QES non-maximal superintegrable system on Euclidean space En, an extension of the Smorodinsky-Winternitz potential.

Contractions of Lie algebras with 2-dimensional generic coadjoint orbits

We determine all the contractions within the class of finite-dimensional real Lie algebras whose coadjoint orbits have dimensions ≤2.