Geometric and Harmonic Analysis on Homogeneous Spaces
Geometric and Harmonic Analysis on Homogeneous Spaces
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齐次空间的几何和调和分析
DOI：
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发表时间：
2013
期刊：
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通讯作者：
L. Accardi
中科院分区：
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作者：
L. Abdelmoula;I. Kédim;L. Accardi
指数李群的Selberg-Weil-Kobayashi局部刚性定理由A.Selberg和A.Weil证明的关于黎曼对称空间的局部刚性定理，并由T.Kobayashi推广到非黎曼齐次空间G/H，断言在线性非紧半单李群G的背景下，G/H的上紧不连续子群Γ不存在连续形变，除非少数情况：对于H紧的G不局部同构于SL2(R)，或者G不局部同构于SO(n，1)或SU(n，1)。1)对于G×G和H=∆G，当G是指数可解的且H⊂G是极大子群时，我们证明了一个类似的定理，即局部刚性在参数空间R(Γ，G，H)上成立当且仅当G同构于直线ax+b的变换的二维群。值得注意的是，我们也放弃了关于G/H的Γ一致的假设。这是Ali Baklti和Imed Kedim的共同工作。Luigi Accardi：在nite维李代数和重整化高次方的白噪声中。过去50年来，自由量子场的重整化幂(数学语言中的白噪声)的德宁问题一直是数学物理中的中心问题(有关综述，请参阅论文[3])。1999年，Accardi，Lu和Volovich提出了一种新的李代数方法来解决这个问题，并通过许多作者的贡献，构造了(量子)白噪声(RSWN)的重整化平方，并用sl(2，R)的R上当前代数的么正表示理论证明了它的恒等式。这一理论在二次情形中的成功和非平凡自然引发了将这些结果推广到2以上的问题。在过去的14年里，Accardi和Boukas一直在系统地追求这一计划，并得出结论：对于更高的幂，sl(2，R)的作用是在nite维李代数中与完全不同目的的弦理论中引入的Virasoro Zamolodzhikov李代数族严格相关的(关系
Lobna Abdelmoula : The Selberg-Weil-Kobayashi local rigidity Theorem for exponential Lie groups. A local rigidity Theorem proved by A. Selberg and A. Weil for Riemannian symmetric spaces and generalized by T. Kobayashi for a non-Riemannian homogeneous space G/H, asserts that there are no continuous deformations of a cocompact discontinuous subgroup Γ for G/H in the setting of a linear non-compact semi-simple Lie group G except some few cases : G is not locally isomorphic to SL2(R) for H compact or G is not locally isomorphic to SO(n, 1) or SU(n, 1) for G × G and H = ∆G. When in large contrast G is assumed to be exponential solvable and H ⊂ G a maximal subgroup, we prove an analogue of such a Theorem stating that the local rigidity holds on the parameter space R(Γ, G,H) if and only if G is isomorphic to the two-dimensional group of a ne transformations of the line ax+ b. Remarkably, we do also drop the assumption on Γ to be uniform for G/H. This is a joint work with Ali Baklouti and Imed Kedim. Luigi Accardi : In nite dimensional Lie algebras and renormalized higher powers of white noise. The problem of de ning, in an operational way, renormalized powers of free quantum elds (white noises in mathematical language) has been a central one in mathematical physics for the past 50 years (see the paper [3] for a survey). In 1999 a new, Lie algebraic, approach to this problem was proposed by Accardi, Lu and Volovich and has led, through the contributions of a multiplicity of authors, to the construction of the renormalized square of (quantum) white noise (RSWN) and to its identi cation with the theory of unitary representations of the current algebra over R of sl(2,R). The success and the non-triviality of the theory in the quadratic case naturally rose the problem of extending these results to powers higher than 2. This program has been systematically pursued by Accardi and Boukas in the past 14 years and has led to the conclusion that the role of sl(2,R), for higher powers is an in nite dimensional Lie algebra 'strictly related' the Virasoro Zamolodzhikov hierarchy of Lie algebras, introduced in the theory of strings for totally di erent purposes (the relation