Research and Exposition in Mathematics

Volume 25

I. Bajo, E. Sanmartin (eds.)
Recent Advances in Lie Theory
406 p., soft cover, ISBN 3-88538-225-3, EUR 44.00, 2002

Contents with Abstracts

D. V. Alekseevsky, A. F. Spiro: Flag Manifolds and Homogeneous CR Structures, 3--44
The first part of the paper is an elementary self-contained introduction to some aspects of the geometry of flag manifolds. The second part gives an exposition of the theory of compact, homogeneous, Levi non-degenerate CR manifolds of hypersurface type (shortly called "homogeneous CR manifolds"). The Lie algebraic approach to the study of flag manifolds, developed in the first part, turns out to be an essential tool to study such manifolds and most of the homogeneous CR manifolds (the so called CR manifolds of the standard type) can be easily described as homogeneous circle bundles over flag manifolds. The classification of homogeneous CR manifold is then easily reduced to the classification of CR manifolds of non standard type. We state the main results on the classification of those manifolds together with their realization as real hypersurfaces in some complex manifold.

Yu. A. Brailov, A. T. Fomenko: Lie Groups and Integrable Hamiltonian Systems, 45--76
The paper consists of two parts. The first part contains the general review of methods for integration of special classes of Hamiltonian systems on Lie groups, symmetric spaces and homogeneous Riemannian spaces. We discuss here the so-called non-commutative integrability of Hamiltonian systems and its connection with Liouville integrability (that is commutative integrability). The present collection of recent results allows to prove complete integrability for a large series of Hamiltonian systems. The second part of the paper contains new results in the singularity theory of integrable Hamiltonian systems. The main observation is that integrable Hamiltonian systems on Lie algebras have a rich algebraic structure, so that the information about the singularities of the momentum mapping in these systems usually can be obtained without huge calculations. It makes possible effective investigation of the degenerations of the momentum mapping and the singularities of Liouville's foliation in the multidimensional cases.

M. Scheunert: Introduction to the Cohomology
of Lie Superalgebras and some Applications, 77--108
The cohomology of colour Lie algebras (which include the Lie superalgebras as special cases) is introduced, and some classical results of Lie algebra cohomology are generalized to the present graded setting. These results are then used to study the formal deformations (in the sense of Gerstenhaber) of the enveloping algebra of a colour Lie algebra. In order to pave the way for this application, the Hochschild cohomology and the formal deformation theory of graded associative algebras is briefly discussed.

V. A. Artamonov: Automorphisms and Derivations of Quantum Polynomials, 109--120
We present a survey of recent results on automorphisms, derivations of general quantum polynomial rings and their division rings of fractions (quantum fields). Furthermore we consider completions of quantum fields with respect to maximal valuations and study their automorphism groups and derivations. We also consider (co)actions of (co)commutative Hopf algebras and finite dimensional pointed Hopf algebras on quantum polynomials. In other words we classify (commutative or finite) quantum groups acting on a general quantum affine space.

I. V. Arzhantsev: Invariant Subalgebras and
Affine Embeddings of Homogeneous Spaces, 121--126
Let G be a semisimple complex Lie group and let the subsset A of C[G], the algebra of polynomial functions on G, be a left-G-invariant finitely generated subalgebra and let I be a G-invariant prime ideal in A. We show that
tr.deg (Q(A/I))G <= (1/2) (dim G - rk G) - 1
and this estimate is sharp for any G. The proof is based on a formula for the maximal value of G-modality over all affine embeddings of a fixed affine homogeneous space G/H.

A. Baklouti, H. Fujiwara: Harmonic Analysis on some Exponential Solvable Homogeneous Spaces, 127--134
Let G = exp g be an exponential solvable Lie group and H = exp h an analytic subgroup of G. Let c = cf, f from g*, be a unitary character of H and let t = IndHGc. Suppose that the multiplicities of all the irreducible components of t are finite. We show that the algebra Dt(G/H) of the G-invariant differential operators on G/H is isomorphic to the algebra of H-invariant polynomials on the affine space f + h^ when t is induced from a Levi component. We also prove the Frobenius reciprocity in this case.

S. Benayadi: Inductive Classification of Quadratic Lie Su
peralgebras, 135--148
We generalize the notion of double extension, introduced by Medina and Revoy to study quadratic Lie algebras, to quadratic Lie superalgebras. We give a sufficient condition for a quadratic Lie superalgebra to be a double extension and we get an inductive classification of this class of quadratic Lie superalgebras g = g0 + g1 such that dim g1 = 2. We obtain an inductive classification of quadratic Lie superalgebras g = g0 + g1 such that the action of g0 on g1 is completely reducible in the following cases: g0 is a reductive Lie algebra, g is solvable, and g is semisimple.

M. Bordemann, A. Medina: Le Groupe des Transformations Affines d'un Groupe de Lie a Structure Affine Bi-Invariante, 149--180
We classify the groups of all affine transformations of a certain connected Lie group admitting a bi-invariant torsion-free flat connection. The fact that the Lie structure of the Lie algebra of such a group comes from an underlying associative structure makes it possible to reduce the geometric problem to the algebraic problem of charactrising all real linear bijections of a real associative algebra with unit inducing a bijection on its group of invertible elements. Several examples like the general linear group are discussed.

J. F. Carinena, A. Ramos: Lie-Scheffers Systems in Physics, 181--188
We recall the Theorem by Lie and Scheffers concerning the characterization of systems of differential equations admitting a superposition function, i.e. those whose general solution can be written in terms of some particular solutions and constants. Each of these systems is related with a Lie algebra, specified by the own Theorem. We expose some recently developed Lie theoretic and geometric techniques, useful for treating such systems, as a reduction property and a generalization of the Wei-Norman method. We illustrate the theory with some applications, which are mainly inspired in physical problems.

J. M. Casas, A. M. Viaites: Central Extensions of Perfect Leibniz Algebras, 189--196
Associated to an extension of Leibniz algebras we obtain an eight-term exact sequence in Leibniz homology with trivial coefficients which we apply to achieve several properties related with simply connected Leibniz algebras and central extensions of perfect Leibniz algebras.

G. Gaeta, N. Rodriguez Quintero: Lie Symmetries of Stochastic Differential Equations, 197--204
We discuss the Lie-point symmetries of stochastic (ordinary) differential equations, and how these are related with the analogous symmetries of the associated Fokker-Planck equation for the probability measure.

H. Glöckner, J. Winkelmann: A Property of Locally Compact Groups, 205--210
Let G be a locally compact group. We show that every identity neighbourhood U of G contains an identity neighbourhood V such that, for every finite sequence g1, ... , gn in V, there exists a permutation p in Sn and signs s 1, ... , sn in {-1, 1} such that gp(1)s1 gp(2)s2 ... gp(n)sn in U.

D. Gomez-Ullate, A. Gonzalez-Lopez, M. A. Rodriguez: Partially Solvable Problems in Quantum Mechanics, 211--232
We develop a systematic procedure for constructing quantum Calogero-Sutherland Hamiltonians whose spectrum can be partially or totally computed by purely algebraic means. The exactly solvable models thus obtained include rational and hyperbolic potentials related to root systems, in some cases with an additional external field. The quasi-exactly solvable models are of two types. The first one can be naturally considered as a deformation of the previosuly mentioned exactly solvable potentials, which share with them their algebraic character. The second type is a novel CN Calogero-Sutherland model in an external field, which is shown to be quasi-exactly solvable for a discrete set of values of the strength of the external potential. While the hyperbolic (or trigonometric) and rational Calogero-Sutherland models have long been known to be exactly-solvable, the elliptic model has defied a full treatment. In this article we give some reasons to explain this fact.

V. V. Gorbatsevich: Isometry Groups of Solvable and Nilpotent Lie Groups, 233--246
Riemannian metrics on solvable Lie groups for which the isometry group is maximal are studied. For a class of nilpotent Lie groups, namely prefiliform and quasifiliform Lie groups, endowed with invariant Riemannian metrics we describe their isometry groups.

G. Hector, E. Macias-Virgos: Diffeological Groups, 247--260
We introduce the basic concepts of the theory of J. M. Souriau's diffeological spaces. As a particular example we study the space of leaves of a Lie foliation on a compact manifold and its group of diffeomorphisms.

K. H. Hofmann: Counting the Topological Dimension of Large Homogeneous Spaces of Compact Groups, 261--270
We define a cardinal valued dimension function dim on topological spaces. Every compact group G has a Lie algebra L(G) whose underlying topological vector space is weakly complete. Quotient spaces of weakly complete spaces are weakly complete; the dimension of a weakly complete vector space is the linear dimension of its dual. Assume that a compact group G acts transitively on a given space X and that H is the isotropy group of the action at an arbitrary point; let L(G) and L(H) denote the Lie algebras of G, respectively, H. We show that dim X = dim L(G) / L(H). Moreover, such an X contains a space homeomorphic to [0,1]dim X; conversely, if X contains a homeomorphic copy of a cube [0,1]aleph, then aleph <= dim X.
Finally, these results are generalized to quotient spaces of locally compact groups. A generalization of a Theorem of Iwasawa is instrumental; it is of considerable independent interest.

Yu. Khakimdjanov: Characteristically Nilpotent, Filiform and Affine Lie Algebras, 271--288
We give a survey of some recent directions of research on characteristically nilpotent, filiform and affine Lie algebras. In particular, we give necessary and sufficient conditions for a filiform Lie algebra to be characteristically nilpotent and show that any irreducible component of the set Fn of filiform Lie algebra laws contains a nonempty Zariski open set, whose elements are characteristically nilpotent Lie algebras. Further, for any natural number n, n not equal to 1 mod 5, we prove that the variety Fn contains a nonempty Zariski open set whose elements are affine Lie algebra laws. We finish the survey with some geometrical aspects; explicitly, we see that Auslander's Conjecture is true for the class of filiform Lie groups whenever the dimension n of the group is odd, and it is false for n even and n > 2.

E. Koelink: Lie Theory and Special Functions, 289--304
A short discussion of the relation between special functions and representation theory of groups is given. An explicit example is given using the discrete series of the Lie algebra su(1,1) and the Meixner-Pollaczek polynomials. Some recent developments in the relation between quantum groups and special functions are discussed.

M. de Leon, J. Cortes, D. M. de Diego, S. Martinez: An Introduction to Mechanics with Symmetry, 305--332
Symmetries are known to be an important instrument to reduce and integrate the equations of motion in Classical Mechanics through Noether theorems which provide conserved quantities. In this paper, some different types of infinitesimal symmetries are reviewed, from the almost classical results for unconstrained systems to the more recent research in nonholonomic mechanics. The case of vakonomic dynamics with some applications to optimal control theory is also discussed.

X. M. Masa: Alexander-Spanier Cohomology of a Lie Foliation, 333--340
We present a construction, for spaces with two topologies, to define continuous basic cohomology and a spectral sequence, similar to the de Rham one for a foliation, to relate continuous basic and foliated cohomology with the cohomology of the space. This construction is based upon the Alexander-Spanier continuous cochains. For a G-Lie foliation, we give an isomorphism between the E2 term of the spectral sequence and the reduced cohomology of G (in the sense of S.-T. Hu) with coefficients in the foliated cohomology of F. This permits us to conclude that both spectral sequences, the de Rham one and the Alexander-Spanier one, are isomorphic for any Riemannian foliation and, in particular, the topological invariance of these cohomologies.

C. Medori, M. Nacinovich: The Levi-Malcev Theorem for Graded CR Lie Algebras, 341--346
The purpose of this paper is to give a detailed proof of the theorem on Levi-Malcev decomposition of Levi-Tanaka algebras which was only outlined in some of our previous papers, while the result is fundamental for the structure theorems for standard CR manifolds proved therein. We give a fairly complete description of those manifolds as Mostow fibrations over minimal orbits (for real Lie groups actions) in complex flag manifolds.

C. Moreno, J. Teles: Preferred Quantizations of Nondegenerate Triangular Lie Bialgebras and Drinfeld Associators, 347--366
Any preferred triangular Hopf quantized universal enveloping algebra, A', which is a quantization of a finite-dimensional nondegenerate triangular Lie bialgebra (a, dcr1) over a field K of characteristic zero and which is a twist of a trivial one, is isomorphic to the one obtained from any fixed Drinfeld associator F through the Etingof- Kazhdan quantization theorem of the classical double of some formal nondegenerate triangular Lie bialgebra (a, dcrt(infinity), rt(infinity) = r1 + Suml=2infinity rl(infinity) tl-1), after identification t = h-bar. We show also the isomorphisms appearing when two different associators F and F' are considered.

K.-H. Neeb: Highest Weight Representations and Infinite-Dimensional Kähler Manifolds, 367--392
We discuss some ideas concerning a geometric analysis of unitary highest weight representations of infinite-dimensional Lie groups. We first describe an approach to highest weight representations of finite-dimensional, not necessarily semisimple, Lie groups G which can be generalized to infinite-dimensional groups. Then we explain the framework for coadjoint orbits in the context of Banach-Lie groups. In Section 3 we briefly discuss those unitary representations of the unitary group G = U(A) of a unital C*-algebra A obtained by restricting an irreducible algebra representation to G. Here the results on representations of C*-algebras provide interesting information which deserves to be considered in the framework of the results described in the first section for finite-dimensional groups. The main point in Section 4 is a description of the elliptic coadjoint orbits of L*-groups which are strong Kähler orbits. In some sense these orbits are the nicest ones and geometrically quite close to the coadjoint Kähler orbits of finite-dimensional semisimple groups. For the compact L*-algebras they are generalizations of the flag manifolds of finite-dimensional classical groups, and for the non-compact L*-algebras (which then must be hermitian), they have the structure of a holomorphic fiber bundle, where the fibers are coadjoint Kähler orbits of compact L*-algebras and the base is a symmetric Hilbert domain. After discussing holomorphic highest weight representations of certain complex classical groups, we conclude this note by explaining why and how these coadjoint orbits correspond to unitary highest weight representations.