Finite-dimensional Real Lie Algebra Research Articles

Realization of arbitrary Lie algebras by automorphisms of $\mathrm{CR}$ manifolds and symmetries of differential equations

For any finite-dimensional real Lie algebra $\mathfrak{h}$, we construct a germ of a real analytic hypersurface in complex space such that its Lie algebra of infinitesimal holomorphic automorphisms is isomorphic to $\mathfrak{h}$. For any $\mathfrak{h}$, we also construct a system of partial differential equations whose Lie algebra of symmetries is isomorphic to the complexification of the algebra $\mathfrak{h}$.

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.

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.

Group invariance of integrable Pfaffian systems

Let $${\mathcal {S}}$$ be an integrable Pfaffian system. When it is invariant under a transversally free infinitesimal action of a finite-dimensional real Lie algebra g and consequently invariant under the local action of a Lie group G, we show that the vertical variational cohomology of $${\mathcal {S}}$$ is equal to the Lie algebra cohomology of g with values in the space of the horizontal cohomology in maximum dimension. This result, besides giving an effective algorithm for the computation of the variational cohomology of an invariant Pfaffian system, provides a method for detecting obstructions to the existence of finite or infinitesimal actions leaving a given system invariant.

Lie algebras with Abelian centralizers

In the paper, finite-dimensional real Lie algebras for which the centralizers of all nonzero element are Abelian are studied. These Lie algebras are also characterized by the transitivity condition for the commutation relation for two nonzero elements. A complete description of these Lie algebras up to isomorphism is given. Some results concerning the relationship between the aforementioned Lie algebras and the Lie algebras of vector fields whose orbits are one-dimensional are considered.

Representations of Lie algebras by non-skewselfadjoint operators in Hilbert space

We study non-skewselfadjoint representations of a finite dimensional real Lie algebra g. To this end we embed a non-skewselfadjoint representation of g into a more complicated structure, that we call a g-operator vessel and that is associated to an overdetermined linear conservative input/state/output system on the corresponding simply connected Lie group G. We develop the frequency domain theory of the system in terms of representations of G, and introduce the joint characteristic function of a g-operator vessel which is the analogue of the classical notion of the characteristic function of a single non-selfadjoint operator. As the first non-commutative example, we apply the theory to the Lie algebra of the ax+b group, the group of affine transformations of the line.

Lie–Hamilton systems on the plane: applications and superposition rules

A Lie–Hamilton (LH) 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 real Lie algebra of Hamiltonian vector fields with respect to a Poisson structure. We provide new algebraic/geometric techniques to easily determine the properties of such Lie algebras on the plane, e.g., their associated Poisson bivectors. We study new and known LH systems on with physical, biological and mathematical applications. New results cover Cayley–Klein Riccati equations, the here defined planar diffusion Riccati systems, complex Bernoulli differential equations and projective Schrödinger equations. Constants of motion for planar LH systems are explicitly obtained which, in turn, allow us to derive superposition rules through a coalgebra approach.

The action of a compact Lie group on nilpotent Lie algebras of type {{n,2}}

Abstract We classify finite-dimensional real nilpotent Lie algebras with 2-dimensional central commutator ideals admitting a Lie group of automorphisms isomorphic to SO 2 ⁢ ( ℝ ) ${{\mathrm{SO}}_{2}(\mathbb{R})}$ . This is the first step to extend the class of nilpotent Lie algebras 𝔥 ${{\mathfrak{h}}}$ of type { n , 2 } ${\{n,2\}}$ to solvable Lie algebras in which 𝔥 ${{\mathfrak{h}}}$ has codimension one.

Lie–Hamilton systems on the plane: Properties, classification and applications

We study Lie–Hamilton systems on the plane, i.e. systems of first-order differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of planar Hamiltonian vector fields with respect to a Poisson structure. We start with the local classification of finite-dimensional real Lie algebras of vector fields on the plane obtained in González-López, Kamran, and Olver (1992) [23] and we interpret their results as a local classification of Lie systems. By determining which of these real Lie algebras consist of Hamiltonian vector fields relative to a Poisson structure, we provide the complete local classification of Lie–Hamilton systems on the plane. We present and study through our results new Lie–Hamilton systems of interest which are used to investigate relevant non-autonomous differential equations, e.g. we get explicit local diffeomorphisms between such systems. We also analyse biomathematical models, the Milne–Pinney equations, second-order Kummer–Schwarz equations, complex Riccati equations and Buchdahl equations.

K-Symplectic Lie systems: theory and applications

A Lie system is a system of first-order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of vector fields: a so-called Vessiot–Guldberg Lie algebra. We suggest the definition of a particular class of Lie systems, the k-symplectic Lie systems, admitting a Vessiot–Guldberg Lie algebra of Hamiltonian vector fields with respect to the presymplectic forms of a k-symplectic structure. We devise new k-symplectic geometric methods to study their superposition rules, t-independent constants of motion and general properties. Our results are illustrated through examples of physical and mathematical interest. As a byproduct, we find a new interesting setting of application of the k-symplectic geometry: systems of first-order ordinary differential equations.

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.

Geodesic bases for Lie algebras

For finite dimensional real Lie algebras, we investigate the existence of an inner product having a basis comprised of geodesic elements. We give several existence and nonexistence results in certain cases: unimodular solvable Lie algebras having an abelian nilradical, algebras having an abelian derived algebra, algebras having a codimension one ideal of a particular kind, nonunimodular algebras of dimension and unimodular algebras of dimension .

From constants of motion to superposition rules for Lie–Hamilton systems

A Lie system is a non-autonomous system of first-order differential equations possessing a superposition rule, i.e. a map expressing its general solution in terms of a generic finite family of particular solutions and some constants. Lie–Hamilton systems form a subclass of Lie systems whose dynamics is governed by a curve in a finite-dimensional real Lie algebra of functions on a Poisson manifold. It is shown that Lie–Hamilton systems are naturally endowed with a Poisson coalgebra structure. This allows us to devise methods for deriving in an algebraic way their constants of motion and superposition rules. We illustrate our methods by studying Kummer–Schwarz equations, Riccati equations, Ermakov systems and Smorodinsky–Winternitz systems with time-dependent frequency.

Solutions of the Classical Yang–Baxter Equation and Noncommutative Deformations

We show that given a finite-dimensional real Lie algebra $${\mathcal{G}}$$ acting on a smooth manifold P then, for any solution of the classical Yang–Baxter equation on $${\mathcal{G}}$$ , there is a canonical Poisson tensor on P and an associated canonical torsion-free and flat contravariant connection. Moreover, we prove that the metacurvature of this contravariant connection vanishes if the isotropy Lie subalgebras of the action are trivial. Those results permit to get a large class of smooth manifolds satisfying the necessary conditions, introduced by Eli Hawkins, to the existence of noncommutative deformations.

Metric Lie algebras and quadratic extensions

The present paper contains a systematic study of the structure of metric Lie algebras, i.e., finite-dimensional real Lie algebras equipped with a nondegenerate invariant symmetric bilinear form. We show that any metric Lie algebra g without simple ideals has the structure of a so called balanced quadratic extension of an auxiliary Lie algebra l by an orthogonal l-module a in a canonical way. Identifying equivalence classes of quadratic extensions of l by a with a certain cohomology set H2 Q(l,a), we obtain a classification scheme for general metric Lie algebras and a complete classification of metric Lie algebras of index 3.

Invariant generalized functions on [formula omitted] with values in a [formula omitted]-module

Let g be a finite-dimensional real Lie algebra. Let ρ : g→ End(V) be a representation of g in a finite-dimensional real vector space. Let C V=( End(V)⊗S( g)) g be the algebra of End( V)-valued invariant differential operators with constant coefficients on g . Let U be an open subset of g . We consider the problem of determining the space of generalized functions φ on U with values in V which are locally invariant and such that C Vφ is finite dimensional. In this article we consider the case g= sl(2, R) . Let N be the nilpotent cone of sl(2, R) . We prove that when U is SL(2, R) -invariant, then φ is determined by its restriction to U⧹ N where φ is analytic (cf. Theorem 6.1). In general this is false when U is not SL(2, R) -invariant and V is not trivial. Moreover, when V is not trivial, φ is not always locally L 1. Thus, this case is different and more complicated than the situation considered by Harish-Chandra (Amer. J. Math 86 (1964) 534; Publ. Math. 27 (1965) 5) where g is reductive and V is trivial. To solve this problem we find all the locally invariant generalized functions supported in the nilpotent cone N . We do this locally in a neighborhood of a nilpotent element Z of g (cf. Theorem 4.1) and on an SL(2, R) -invariant open subset U⊂ sl(2, R) (cf. Theorem 4.2). Finally, we also give an application of our main theorem to the Superpfaffian (Superpfaffian, prepublication, e-print math.GR/0402067, 2004).

On the variational cohomology of g-invariant foliations

Let S be an integrable Pfaffian system. If it is invariant under a transversally free infinitesimal action of a finite dimensional real Lie algebra g, we show that the “vertical” variational cohomology of S is equal to the Lie algebra cohomology of g with values in the space of the “horizontal” cohomology in a maximum dimension. This result, besides giving an effective algorithm for the computation of the variational cohomology of an invariant Pfaffian system, provides a method for detecting obstructions to the existence of infinitesimal actions leaving a given system invariant.

Kontsevich quantization and invariant distributions on Lie groups

We study Kontsevich's deformation quantization for the dual of a finite-dimensional real Lie algebra (or superalgebra) g . In this case the Kontsevich ★-product defines a new convolution on S( g) , regarded as the space of distributions supported at 0∈ g . For p∈S( g) , we show that the convolution operator f↦ p★ f is a differential operator with analytic germ. We use this fact to prove a conjecture of Kashiwara and Vergne on invariant distributions on a Lie group G. This implies local solvability of bi-invariant differential operators on a Lie supergroup. In the special case of Lie groups, we get a new proof Duflo's theorem.

Existence of solutions of the classical yang- baxter equation for a real lie Algebra

We characterize finite-dimensional Lie algebras over the real numbers for which the classical Yang-Baxter equation has a non-trivial skew-symmetric solution (resp. a non-trivial solution with invariant symmetric part). Equivalently, we obtain a characterization of those finite-dimensional real Lie algebras which admit a non-trivial (quasi-) triangular Lie bialgebra structure.

Invariants of Real Forms of Affine Kac-Moody Lie Algebras

The theory of Vogan diagrams, which are Dynkin diagrams with an overlay of certain additional information, allows one to give a rapid classification of finite-dimensional real semisimple Lie algebras and to make use of this classification in practice. This paper develops a corresponding theory of Vogan diagrams for “almost compact” real forms of indecomposable nontwisted affine Kac–Moody Lie algebras. In this case also, the equivalence classes of Vogan diagrams correspond to the isomorphism classes of almost compact real forms. Although the real forms of such algebras had already been classified, the theory of Vogan diagrams introduces invariants for such algebras and makes it possible to locate a given real form within the classification.