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1.
Roger W. Richardson proved that any parabolic subgroup of a complex semisimple Lie group admits an open dense orbit in the nilradical of its corresponding parabolic subalgebra. In the case of complex symmetric spaces we show that there exist some large classes of parabolic subgroups for which the analogous statement which fails in general, is true. Our main contribution is the extension of a theorem of Peter E. Trapa (in 2005) to real semisimple exceptional Lie groups.
2.
Towards a Lie theory of locally convex groups 总被引:3,自引:0,他引:3
Karl-Hermann Neeb 《Japanese Journal of Mathematics》2006,1(2):291-468
In this survey, we report on the state of the art of some of the fundamental problems in the Lie theory of Lie groups modeled
on locally convex spaces, such as integrability of Lie algebras, integrability of Lie subalgebras to Lie subgroups, and integrability
of Lie algebra extensions to Lie group extensions. We further describe how regularity or local exponentiality of a Lie group
can be used to obtain quite satisfactory answers to some of the fundamental problems. These results are illustrated by specialization
to some specific classes of Lie groups, such as direct limit groups, linear Lie groups, groups of smooth maps and groups of
diffeomorphisms. 相似文献
3.
A. G. Reiman 《Journal of Mathematical Sciences》1982,19(5):1507-1545
In this paper there is given a geometric scheme for constructing integrable Hamiltonian systems based on Lie groups, generalizing the construction of M. Adler. The operation of this scheme is considered for parabolic decompositions of semisimple Lie groups. Fundamental examples of integrable systems are connected with graded Lie algebras. Among them are the generalized periodic chains of Toda, multidimensional tops, and the motion of a point on various homogeneous spaces in a quadratic potential.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 95, pp. 3–54, 1980. 相似文献
4.
Mark Craddock 《Journal of Differential Equations》2009,246(6):2538-90
In this paper we present some new applications of Lie symmetry analysis to problems in stochastic calculus. The major focus is on using Lie symmetries of parabolic PDEs to obtain fundamental solutions and transition densities. The method we use relies upon the fact that Lie symmetries can be integrated with respect to the group parameter. We obtain new results which show that for PDEs with nontrivial Lie symmetry algebras, the Lie symmetries naturally yield Fourier and Laplace transforms of fundamental solutions, and we derive explicit formulas for such transforms in terms of the coefficients of the PDE. 相似文献
5.
A. V. Bukhvalov 《Journal of Mathematical Sciences》1989,44(6):749-762
Banach spaces with unconditional martingale differences are investigated. In Sec. 1 a survey of their fundamental properties and connections with vector-valued harmonic analysis is given. In Sec. 2 new results are obtained regarding bases in the spaces E(X), where E is a symmetric space.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 5–22, 1987. 相似文献
6.
Riemannian supergeometry 总被引:1,自引:0,他引:1
O. Goertsches 《Mathematische Zeitschrift》2008,260(3):557-593
Motivated by Zirnbauer in J Math Phys 37(10):4986–5018 (1996), we develop a theory of Riemannian supermanifolds up to a definition
of Riemannian symmetric superspaces. Various fundamental concepts needed for the study of these spaces both from the Riemannian
and the Lie theoretical viewpoint are introduced, e.g., geodesics, isometry groups and invariant metrics on Lie supergroups
and homogeneous superspaces.
Research supported by the DFG, SFB TR/12 “Symmetries and Universality in Mesoscopic Systems”. 相似文献
7.
Minimal, strongly proximal actions of locally compact groups on compact spaces, also known asboundary actions, were introduced by Furstenberg in the study of Lie groups. In particular, the action of a semi-simple real Lie groupG on homogeneous spacesG/Q, whereQ ⊂G is a parabolic subgroup, are boundary actions. Countable discrete groups admit a wide variety of boundary actions. In this
note we show that ifX is a compact manifold with a faithful boundary action of some locally compact groupH, then (under some mild regularity assumption) the groupH, the spaceX, and the action split into a direct product of a semi-simple Lie groupG acting onG/Q and a boundary action of a discrete countable group.
The author was partially supported by NSF grants DMS-0049069, 0094245 and GIF grant G-454-213.06/95. 相似文献
8.
Brett Milburn 《Differential Geometry and its Applications》2011,29(5):615-641
The aim of this paper is to study generalized complex geometry (Hitchin, 2002) [6] and Dirac geometry (Courant, 1990) [3], (Courant and Weinstein, 1988) [4] on homogeneous spaces. We offer a characterization of equivariant Dirac structures on homogeneous spaces, which is then used to construct new examples of generalized complex structures. We consider Riemannian symmetric spaces, quotients of compact groups by closed connected subgroups of maximal rank, and nilpotent orbits in sln(R). For each of these cases, we completely classify equivariant Dirac structures. Additionally, we consider equivariant Dirac structures on semisimple orbits in a semisimple Lie algebra. Here equivariant Dirac structures can be described in terms of root systems or by certain data involving parabolic subalgebras. 相似文献
9.
A. P. Norden 《Journal of Mathematical Sciences》1980,14(3):1241-1259
A survey of papers on the theory of compositions and their application to the investigation of manifolds in spaces with fundamental groups is presented.Translated from Itogi Nauki i Tekhniki. Problemy Geometrii, Vol. 10, pp. 117–145, 1978.I thank Érnst Gergardovich Neifel'd, whose paper [12] stimulated much of this survey, and who, moreover, showed me inestimable help in the formulation of my paper. In particular, he wrote Sec. 17. 相似文献
10.
N. R. Wallach 《Commentarii Mathematici Helvetici》2003,78(2):266-307
The conjugacy class of parabolic subgroups with Heisenberg unipotent radical in a simple Lie groups over ³ not of type CnC_{n} contains an element defined over  for each quaternionic real form. In this paper we study the Whittaker models for quaternionic discrete series of these real forms and prove results analogous and by analogous methods to the case of simple Lie groups over  that are the automorphism groups of tube type Hermitian symmetric domain and (so-called Bessel models) for holomorphic representations. In particular we calculate the decomposition of the space of Whittaker vectors under the action of the stabilizer of the corresponding character in a Levi factor of the Heisenberg parabolic subgroup. 相似文献
11.
We consider an algebraic parametrization for the set of (Mal'cev completed) fundamental groups of the spaces with fixed first
two Betti numbers, having in mind applications in low-dimensional topology and especially in link theory. The factor set of
(restricted) isomorphism types of these groups acquires the structure of a ‘moduli space’, giving rise to invariants which,
in the case of links, detect the isotopy type. We indicate two methods of computation for these invariants. We also prove
a rigidity result for the associated graded Lie algebra of the fundamental group. A lot of examples are given. 相似文献
12.
Karl-Hermann Neeb 《Monatshefte für Mathematik》1991,112(4):303-321
13.
We consider parabolic operators of fractional order and their iterates on the upper half space of the euclidean space. We deal with Hilbert spaces of solutions of those parabolic equations. We shall show, in this note, the existence of reproducing kernels and give a formula by using their fundamental solutions. As an application, we also discuss the polyharmonic Bergman spaces and give their reproducing kernels by using the Poisson kernel on the upper half space. 相似文献
14.
In this paper, we study naturally reductive Finsler metrics. We first give a sufficient and necessary condition for a Finsler metric to be naturally reductive with respect to certain transitive group of isometries. Then we study in detail the left invariant naturally reductive metrics on compact Lie groups and give a method to construct the non-Riemannian ones. Further, we give a classification of left invariant naturally reductive metrics on nilpotent Lie groups. Finally, we give a classification of all the naturally reductive Finsler spaces of dimension less or qual to 4. As applications, we obtain some rigidity theorems about naturally reductive Finsler metrics. Namely, any left invariant non-symmetric naturally reductive Finsler metric on a compact simple Lie group or an indecomposable nilpotent Lie group must be Riemannian. On the other hand, we provide a very convenient method to construct non-symmetric Berwald spaces which are neither Riemannian nor locally Minkowskian, a kind of spaces which are sought after in the book by Bao et al. (An introduction to Riemann–Finsler geometry, GTM 200, 2000). 相似文献
15.
Hiroshi Tamaru 《Mathematische Annalen》2011,351(1):51-66
In this paper, we study the solvmanifolds constructed from any parabolic subalgebras of any semisimple Lie algebras. These
solvmanifolds are naturally homogeneous submanifolds of symmetric spaces of noncompact type. We show that the Ricci curvatures
of our solvmanifolds coincide with the restrictions of the Ricci curvatures of the ambient symmetric spaces. Consequently,
all of our solvmanifolds are Einstein, which provide a large number of new examples of noncompact homogeneous Einstein manifolds.
We also show that our solvmanifolds are minimal, but not totally geodesic submanifolds of symmetric spaces. 相似文献
16.
李群表示论和Schubert条件 总被引:2,自引:0,他引:2
本文将Grassmann流形上的Schubert子簇所满足的经典的Schubert条件推广到一般的复半单李群G的广义旗流形.利用复半单李群的表示理论,我们首先在李群的权空间上引入自然的Ehresman偏序.这一偏序可以导出李群的最高权表示的权系、Weyl群及其陪集空间上的Ehresman偏序.然后我们对一般的复表示定义了相应的射影空间,Grassmann流形和旗流形.这使得能够像经典的情形一样来分析广义旗流形的Schubert子簇满足的Schubert条件.在讨论中,我们还给出了李群G的Weyl群及其陪集空间中的Bruhat-Chevalley偏序的简单判别条件.我们的结果应用到例外群,给出了Fulton提出的关于例外群的Schubert分析的问题的部分回答. 相似文献
17.
Poincaré and Sobolev Inequalities for Vector Fields Satisfying Hrmander's Condition in Variable Exponent Sobolev Spaces
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In this paper,we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces.The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebesgue spaces on spaces of homogeneous type.We obtain the first order Poincare inequalities for vector fields satisfying Hrmander's condition in variable non-isotropic Sobolev spaces.We also set up the higher order Poincare inequalities with variable exponents on stratified Lie groups.Moreover,we get the Sobolev inequalities in variable exponent Sobolev spaces on whole stratified Lie groups.These inequalities are important and basic tools in studying nonlinear subelliptic PDEs with variable exponents such as the p(x)-subLaplacian.Our results are only stated and proved for vector fields satisfying Hrmander's condition,but they also hold for Grushin vector fields as well with obvious modifications. 相似文献
18.
We study the spaces of left-invariant Riemannian metrics on a Lie group up to isometry, and up to isometry and scaling. In this paper, we see that such spaces can be identified with the orbit spaces of certain isometric actions on noncompact symmetric spaces. We also study some Lie groups whose spaces of left-invariant metrics up to isometry and scaling are small. 相似文献
19.
Jan Gregorovič 《Differential Geometry and its Applications》2012,30(5):450-476
First we introduce a generalization of symmetric spaces to parabolic geometries. We provide construction of such parabolic geometries starting with classical symmetric spaces and we show that all regular parabolic geometries with smooth systems of involutive symmetries can be obtained in this way. Further, we investigate the case of parabolic contact geometries in great detail and we provide the full classification of those with semisimple groups of symmetries without complex factors. Finally, we explicitly construct all non-trivial contact geometries with non-complex simple groups of symmetries. We also indicate geometric interpretations of some of them. 相似文献
20.
We compute the fundamental group of the spaces of ordered commutingn-tuples of elements in the Lie groups SU(2), U(2) and SO(3).For SO(3) the mod-2 cohomology of the components of these spacesis also obtained. 相似文献