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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.