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1.
Results on characterization of manifolds in terms of certain Lie algebras growing on them, especially Lie algebras of differential
operators, are reviewed and extended. In particular, we prove that a smooth (real-analytic, Stein) manifold is characterized
by the corresponding Lie algebra of linear differential operators, i.e. isomorphisms of such Lie algebras are induced by the
appropriate class of diffeomorphisms of the underlying manifolds.
The research of Janusz Grabowski supported by the Polish Ministry of Scientific Research and Information Technology under
the grant No. 2 P03A 020 24, that of Norbert Poncin by grant C.U.L./02/010. 相似文献
2.
A Lie algebra g is called two step nilpotent if g is not abelian and [g, g] lies in the center of g. Two step nilpotent Lie algebras are useful in the study of some geometric problems, such as commutative Riemannian manifolds, weakly symmetric Riemannian manifolds, homogeneous Einstein manifolds, etc. Moreover, the classification of two-step nilpotent Lie algebras has been an important problem in Lie theory. In this paper, we study two step nilpotent indecomposable Lie algebras of dimension 8 over the field of complex numbers. Based on the study of minimal systems of generators, we choose an appropriate basis and give a complete classification of two step nilpotent Lie algebras of dimension 8. 相似文献
3.
M. Boucetta 《Differential Geometry and its Applications》2004,20(3):279-291
In a previous paper (C. R. Acad. Sci. Paris Sér. I 333 (2001) 763–768), the author introduced a notion of compatibility between a Poisson structure and a pseudo-Riemannian metric. In this paper, we introduce a new class of Lie algebras called pseudo-Riemannian Lie algebras. The two notions are closely related: we prove that the dual of a Lie algebra endowed with its canonical linear Poisson structure carries a compatible pseudo-Riemannian metric if and only if the Lie algebra is a pseudo-Riemannian Lie algebra. Moreover, the Lie algebra obtained by linearizing at a point a Poisson manifold with a compatible pseudo-Riemannian metric is a pseudo-Riemannian Lie algebra. We also give some properties of the symplectic leaves of such manifolds, and we prove that every Poisson manifold with a compatible Riemannian metric is unimodular. Finally, we study Poisson Lie groups endowed with a compatible pseudo-Riemannian metric, and we give the classification of all pseudo-Riemannian Lie algebras of dimension 2 and 3. 相似文献
4.
On the Construction of Geometric Integrators in the RKMK Class 总被引:2,自引:0,他引:2
Kenth Engø 《BIT Numerical Mathematics》2000,40(1):41-61
We consider the construction of geometric integrators in the class of RKMK methods. Any differential equation in the form of an infinitesimal generator on a homogeneous space is shown to be locally equivalent to a differential equation on the Lie algebra corresponding to the Lie group acting on the homogeneous space. This way we obtain a distinction between the coordinate-free phrasing of the differential equation and the local coordinates used. In this paper we study methods based on arbitrary local coordinates on the Lie group manifold. By choosing the coordinates to be canonical coordinates of the first kind we obtain the original method of Munthe-Kaas [16]. Methods similar to the RKMK method are developed based on the different coordinatizations of the Lie group manifold, given by the Cayley transform, diagonal Padé approximants of the exponential map, canonical coordinates of the second kind, etc. Some numerical experiments are also given. 相似文献
5.
I. V. Shirokov 《Theoretical and Mathematical Physics》2000,123(3):754-767
We propose an algorithm for obtaining the spectra of Casimir (Laplace) operators on Lie groups. We prove that the existence
of the normal polarization associated with a linear functional on the Lie algebra is necessary and sufficient for the transition
to local canonical Darboux coordinates (p, q) on the coadjoint representation orbit that is linear in the “momenta.” We show
that the λ-representations of Lie algebras (which are used, in particular, in integrating differential equations) result from
the quantization of the Poisson bracket on the coalgebra in canonical coordinates.
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 3, pp. 407–423, June, 2000. 相似文献
6.
A-扩张Lie Rinehart代数 总被引:1,自引:0,他引:1
The purpose of this paper is to give a brief introduction to the category of Lie Rinehart algebras and introduces the concept of smooth manifolds associated with a unitary, commutative,associative algebra A.It especially shows that the A-extended algebra as well as the action algebra can be realized as the space of A-left invariant vector fields on a Lie group,analogous to the well known relationship of Lie algebras and Lie groups. 相似文献
7.
Stefan Waldmann 《Bulletin of the Brazilian Mathematical Society》2011,42(4):831-852
Various aspects of Morita theory of deformed algebras and in particular of star product algebras on general Poisson manifolds
are discussed. We relate the three flavours ring-theoretic Morita equivalence, *-Morita equivalence, and strong Morita equivalence
and exemplify their properties for star product algebras. The complete classification of Morita equivalent star products on
general Poisson manifolds is discussed as well as the complete classification of covariantly Morita equivalent star products
on a symplectic manifold with respect to some Lie algebra action preserving a connection. 相似文献
8.
Karl-Hermann Neeb 《manuscripta mathematica》2001,104(3):359-381
In this paper we essentially classify all locally finite Lie algebras with an involution and a compatible root decomposition
which permit a faithful unitary highest weight representation. It turns out that these Lie algebras have many interesting
relations to geometric structures such as infinite-dimensional bounded symmetric domains and coadjoint orbits of Banach–Lie
groups which are strong K?hler manifolds. In the present paper we concentrate on the algebraic structure of these Lie algebras,
such as the Levi decomposition, the structure of the almost reductive and locally nilpotent part, and the structure of the
representation of the almost reductive algebra on the locally nilpotent ideal.
Received: 2 August 2000 / Revised version: 10 January 2001 相似文献
9.
Hans Z. Munthe-Kaas Alexander Lundervold 《Foundations of Computational Mathematics》2013,13(4):583-613
Pre-Lie (or Vinberg) algebras arise from flat and torsion-free connections on differential manifolds. These algebras have been extensively studied in recent years, both from algebraic operadic points of view and through numerous applications in numerical analysis, control theory, stochastic differential equations and renormalization. Butcher series are formal power series founded on pre-Lie algebras, used in numerical analysis to study geometric properties of flows on Euclidean spaces. Motivated by the analysis of flows on manifolds and homogeneous spaces, we investigate algebras arising from flat connections with constant torsion, leading to the definition of post-Lie algebras, a generalization of pre-Lie algebras. Whereas pre-Lie algebras are intimately associated with Euclidean geometry, post-Lie algebras occur naturally in the differential geometry of homogeneous spaces, and are also closely related to Cartan’s method of moving frames. Lie–Butcher series combine Butcher series with Lie series and are used to analyze flows on manifolds. In this paper we show that Lie–Butcher series are founded on post-Lie algebras. The functorial relations between post-Lie algebras and their enveloping algebras, called D-algebras, are explored. Furthermore, we develop new formulas for computations in free post-Lie algebras and D-algebras, based on recursions in a magma, and we show that Lie–Butcher series are related to invariants of curves described by moving frames. 相似文献
10.
S. Faltinsen 《BIT Numerical Mathematics》2000,40(4):652-670
Backward error analysis has proven to be very useful in stability analysis of numerical methods for ordinary differential equations. However the analysis has so far been undertaken in the Euclidean space or closed subsets thereof. In this paper we study differential equations on manifolds. We prove a backward error analysis result for intrinsic numerical methods. Especially we are interested in Lie-group methods. If the Lie algebra is nilpotent a global stability analysis can be done in the Lie algebra. In the general case we must work on the nonlinear Lie group. In order to show that there is a perturbed differential equation on the Lie group with a solution that is exponentially close to the numerical integrator after several steps, we prove a generalised version of Alekseev-Gr: obner's theorem. A major motivation for this result is that it implies many stability properties of Lie-group methods. 相似文献
11.
Triangulated categories and Kac-Moody algebras 总被引:7,自引:0,他引:7
By using the Ringel-Hall algebra approach, we find a Lie algebra arising in each triangulated category with T
2=1, where T is the translation functor. In particular, the generic form of the Lie algebras determined by the root categories, the 2-period
orbit categories of the derived categories of finite dimensional hereditary associative algebras, gives a realization of all
symmetrizable Kac-Moody Lie algebras.
Oblatum 4-XII-1998 & 11-XI-1999?Published online: 21 February 2000 相似文献
12.
13.
We construct the class of integrable classical and quantum systems on the Hopf algebras describing n interacting particles.
We obtain the general structure of an integrable Hamiltonian system for the Hopf algebra A(g) of a simple Lie algebra g and
prove that the integrals of motion depend only on linear combinations of k coordinates of the phase space, 2·ind g≤k≤g·ind g, whereind g andg are the respective index and Coxeter number of the Lie algebra g. The standard procedure of q-deformation results in the
quantum integrable system. We apply this general scheme to the algebras sl(2), sl(3), and o(3, 1). An exact solution for the
quantum analogue of the N-dimensional Hamiltonian system on the Hopf algebra A(sl(2)) is constructed using the method of noncommutative
integration of linear differential equations.
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 3, pp. 373–390, September, 2000 相似文献
14.
Daniel Beltiţa 《Mathematische Annalen》2002,324(2):405-429
The aim of the paper is to investigate spectral properties of the Lie algebras corresponding to the symmetry groups of certain
flags of vector bundles over a compact space. Under natural hypotheses, such Lie algebras are solvable, being in general infinite
dimensional. The spectral theory of finite-dimensional solvable Lie algebras of operators is extended to this natural class
of infinite-dimensional solvable Lie algebras. The discussion uses the language of continuous fields of -algebras. The flag manifolds in -algebraic framework are naturally involved here, they providing the basic method for obtaining flags of vector bundles.
Received: 8 October 2001 / Revised version: 4 February 2002 / Published online: 6 August 2002
Research supported from the contract ICA1–CT–2000–70022 with the European Commission. 相似文献
15.
We propose a notion of algebra of twisted chiral differential operators over algebraic manifolds with vanishing 1st Pontrjagin class. We show that such algebras possess
families of modules depending on infinitely many complex parameters, which we classify in terms of the corresponding algebra
of twisted differential operators. If the underlying manifold is a flag manifold, our construction recovers modules over an
affine Lie algebra parameterized by opers over the Langlands dual Lie algebra. The spaces of global sections of “smallest”
such modules are irreducible
[^(\mathfrakg)]{{\hat{{\mathfrak{g}}}}} -modules, and all irreducible
\mathfrakg{{\mathfrak{g}}} -integrable
[^(\mathfrakg)]{{\hat{{\mathfrak{g}}}}} -modules at the critical level arise in this way. 相似文献
16.
We define a class of infinite-dimensional Lie algebras that generalize the universal enveloping algebra of the algebra sl(2,
ℂ) regarded as a Lie algebra. These algebras are a special case of ℤ-graded Lie algebras with a continuous root system, namely,
their Cartan subalgebra is the algebra of polynomials in one variable. The continuous limit of these algebras defines new
Poisson brackets on algebraic surfaces.
In memory of M. V. Saveliev
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 2, pp. 345–352, May, 2000. 相似文献
17.
We define HNN-extensions of Lie algebras and study their properties. In particular, a sufficient condition for freeness of subalgebras is obtained. We also study differential HNN-extensions of associative rings. These constructions are used to give short proofs of Malcev's and Shirshov's theorems that an associative or Lie algebra of finite or countable dimension is embeddable into a two-generator algebra.
18.
《Journal of Pure and Applied Algebra》2023,227(6):107311
The present paper, though inspired by the use of tensor hierarchies in theoretical physics, establishes their mathematical credentials, especially as genetically related to Lie algebra crossed modules. Gauging procedures in supergravity rely on a pairing – the embedding tensor – between a Leibniz algebra and a Lie algebra. Two such algebras, together with their embedding tensor, form a triple called a Lie-Leibniz triple, of which Lie algebra crossed modules are particular cases. This paper is devoted to showing that any Lie-Leibniz triple induces a differential graded Lie algebra – its associated tensor hierarchy – whose restriction to the category of Lie algebra crossed modules is the canonical assignment associating to any Lie algebra crossed module its corresponding unique 2-term differential graded Lie algebra. This shows that Lie-Leibniz triples form natural generalizations of Lie algebra crossed modules and that their associated tensor hierarchies can be considered as some kind of ‘lie-ization’ of the former. We deem the present construction of such tensor hierarchies clearer and more straightforward than previous derivations. We stress that such a construction suggests the existence of further well-defined Leibniz gauge theories. 相似文献
19.
20.
People studied the properties and structures of restricted Lie algebras all whose elements are semisimple. It is the main
objective of this paper to continue the investigation in order to obtain deeper structure theorems. We obtain some sufficient
conditions for the commutativity of restricted Lie algebras, generalize some results of R. Farnsteiner and characterize some
properties of a finite-dimensional semisimple restricted Lie algebra all whose elements are semisimple. Moreover, we show
that a centralsimple restricted Lie algebra all whose elements are semisimple over a field of characteristic p > 7 is a form of a classical Lie algebra. 相似文献