Papers by Emmanuel Lanzmann
In [Zh], R. B. Zhang found a way to link certain formal deformations of the Lie algebra o(2l + 1)... more In [Zh], R. B. Zhang found a way to link certain formal deformations of the Lie algebra o(2l + 1) and the Lie superalgebra osp(1, 2l). The aim of this article is to reformulate the Zhang transformation in the context of the quantum enveloping algebrasà la Drinfeld-Jimbo and to show how this construction can explain the main theorem of [GL2]: the annihilator of a Verma module over the Lie superalgebra osp(1, 2l) is generated by its intersection with the centralizer of the even part of the enveloping algbra. A well known theorem of Duflo claims that the annihilator of a Verma module over a complex semi-simple Lie algebra is generated by its intersection with the centre of the enveloping algbra. In [GL2] we show that in order for this theorem to hold in the case of the Lie superalgebra osp(1, 2l) one has to replace the centre by the centralizer of the even part of the enveloping algbra. The purpose of this article is to show how quantum groups can illucidate this phenomemon. Let k, g b...
Pour Felix Lanzmann
Les Temps Modernes, 2017
Le Théorème D'annulation Dans Le Cadre Des Super-algèbres De Lie Complètement Réductibles Et De Leurs Groupes Quantiques
Algebras and Representation Theory, 2002
R. B. Zhang found a way to link certain formal deformations of the Lie algebra o(2l+1) and the Li... more R. B. Zhang found a way to link certain formal deformations of the Lie algebra o(2l+1) and the Lie superalgebra osp(1,2l). The aim of this article is to reformulate the Zhang transformation in the context of the quantum enveloping algebras à la Drinfeld and Jimbo and to show how this construction can explain the main theorem of Gorelik and Lanzmann:
Analyse spatiale amelioree d'un front d'onde et mesure en 3d
Cette invention concerne une methode d'analyse de front d'onde (100) consistant soumettre... more Cette invention concerne une methode d'analyse de front d'onde (100) consistant soumettre le front d'onde a une transformee, a appliquer une pluralite de changements de phase differents (110, 112, 114) au front d'onde transforme (108) et a etablir une pluralite d'images d'intensite (130, 132, 134). La pluralite des differents changements de phase qui sont appliques au front d'onde transforme correspond a une forme de la source lumineuse.
Scanning, 2017
Phase measurements obtained by high-coherence interferometry are restricted by the 2π ambiguity, ... more Phase measurements obtained by high-coherence interferometry are restricted by the 2π ambiguity, to height differences smaller than λ/2. A further restriction in most interferometric systems is for focusing the system on the measured object. We present two methods that overcome these restrictions. In the first method, different segments of a measured wavefront are digitally propagated and focused locally after measurement. The divergent distances, by which the diverse segments of the wavefront are propagated in order to achieve a focused image, provide enough information so as to resolve the 2π ambiguity. The second method employs an interferogram obtained by a spectrum constituting a small number of wavelengths. The magnitude of the interferogram’s modulations is utilized to resolve the 2π ambiguity. Such methods of wavefront propagation enable several applications such as focusing and resolving the 2π ambiguity, as described in the article.
Improved Spatial Wavefront Analysis and 3D Measurement
Methods and Apparatus for Wavefront Manipulations and Improved 3-D Measurements
Spatial wavefront analysis and 3d measurement
Journal of the Optical Society of America A, 2005
We describe a new wavefront analysis method, in which certain wavefront manipulations are applied... more We describe a new wavefront analysis method, in which certain wavefront manipulations are applied to a spatially defined area in a certain plane along the optical axis. These manipulations replace the reference-beam phase shifting of existing methods, making this method a spatial phase-shift interferometry method. We demonstrate the system's dependence on a defined spatial Airy number, which is the ratio of the characteristic dimension of the manipulated area and the Airy disk diameter of the optical system. We analytically obtain the resulting intensity data of the optical setup and develop various methods to accurately reconstruct the inspected wavefront out of the data. These reconstructions largely involve global techniques, in which the entire wavefront's pattern affects the reconstruction of the wavefront in any given position. The method's noise sensitivity is analyzed, and actual reconstruction results are presented.
Inventiones Mathematicae, 1999
A well known theorem of Duflo claims that the annihilator of a Verma module in the enveloping alg... more A well known theorem of Duflo claims that the annihilator of a Verma module in the enveloping algebra of a complex semisimple Lie algebra is generated by its intersection with the centre. For a Lie superalgebra this result fails to be true. For instance, in the case of the orthosymplectic Lie superalgebra osp(1, 2), Pinczon gave in [Pi] an example of a Verma module whose annihilator is not generated by its intersection with the centre of universal enveloping algebra. More generally, Musson produced in [Mu1] a family of such "singular" Verma modules for osp(1, 2l) cases. In this article we give a necessary and sufficient condition on the highest weight of a osp(1, 2l)-Verma module for its annihilator to be generated by its intersection with the centre. This answers a question of Musson. The classical proof of the Duflo theorem is based on a deep result of Kostant which uses some delicate algebraic geometry reasonings. Unfortunately these arguments can not be reproduced in the quantum and super cases. This obstruction forced Joseph and Letzter, in their work on the quantum case (see [JL]), to find an alternative approach to the Duflo theorem. Following their ideas, we compute the factorization of the Parthasarathy-Ranga-Rao-Varadarajan (PRV) determinants. Comparing it with the factorization of Shapovalov determinants we find, unlike to the classical and quantum cases, that the PRV determinant contains some extrafactors. The set of zeroes of these extrafactors is precisely the set of highest weights of Verma modules whose annihilators are not generated by their intersection with the centre. We also find an analogue of Hesselink formula (see [He]) giving the multiplicity of every simple finite dimensional module in the graded component of the harmonic space in the symmetric algebra. Consider the case l = 1, i.e. g = osp(1, 2), which has been treated by Pinczon. In this case, any g-Verma module M , viewed as a g 0-module, is the direct sum of two g 0-Verma modules M 0 and M 1. Let C 0 be a Casimir element for g 0. Then C 0 acts by scalars c i on M i (i=0,1). In general, c 1 = c 0 , and in this case, Pinczon proved that Ann M = U(g) Ann Z(g) M. However when the highest weight of M equals −ρ, one has c 1 = c 0 and so C 0 − c 0 belongs to the annihilator of M. It is easy to check that (C 0 − c 0) ∈ U(g) Ann Z(g) M. Consequently, the annilihilator theorem doesn't hold in full generality.
The annihilation theorem for the Lie superalgebra osp(1,2ℓ)
Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 1998
Abstract We give necessary and sufficient conditions for the annihilator of a Verma module over a... more Abstract We give necessary and sufficient conditions for the annihilator of a Verma module over a Lie superalgebra osp(1,2l) to be generated by its intersection with the centre of the universal enveloping superalgebra. © Academie des Sciences/Elsevier, Paris
Advances in Mathematics, 2000
A well known theorem of Duflo, the "annihilation theorem", claims that the annihilator of a Verma... more A well known theorem of Duflo, the "annihilation theorem", claims that the annihilator of a Verma module in the enveloping algebra of a complex semisimple Lie algebra is centrally generated. For the Lie superalgebra osp(1, 2l), this result does not hold. In this article, we introduce a "correct" analogue of the centre for which the annihilation theorem does hold in the case osp(1, 2l). This substitute of the centre is the centralizer of the even part of the enveloping algebra. This algebra shares some nice properties with the centre. As a consequence of the annihilation theorem we obtain the description of the minimal primitive spectrum of the enveloping algebra of the Lie superalgebra osp(1, 2l). We also deduce a criterium for a osp(1, 2l)-Verma module to be a direct sum of sp(2l)-Verma modules.
Applied Optics, 2006
Common-path imaging interferometers offer some advantages over other interferometers, such as ins... more Common-path imaging interferometers offer some advantages over other interferometers, such as insensitivity to vibrations and the ability to be attached to any optical system to analyze an imaged wavefront. We introduce the spatial-phase-shift imaging interferometry technique for surface measurements and wavefront analysis in which different parts of the wavefront undergo certain manipulations in a certain plane along the optical axis. These manipulations replace the reference-beam phase shifting of existing interferometry methods. We present the mathematical algorithm for reconstructing the wavefront from the interference patterns and detail the optical considerations for implementing the optical system. We implemented the spatial phase shift into a working system and used it to measure a variety of objects. Measurement results and comparison with other measurement methods indicate that this approach improves measurement accuracy with respect to existing quantitative phase-measurement methods.
Journal of the Optical Society of America A, 2005
We describe a new wavefront analysis method, in which certain wavefront manipulations are applied... more We describe a new wavefront analysis method, in which certain wavefront manipulations are applied to a spatially defined area in a certain plane along the optical axis. These manipulations replace the reference-beam phase shifting of existing methods, making this method a spatial phase-shift interferometry method. We demonstrate the system's dependence on a defined spatial Airy number, which is the ratio of the characteristic dimension of the manipulated area and the Airy disk diameter of the optical system. We analytically obtain the resulting intensity data of the optical setup and develop various methods to accurately reconstruct the inspected wavefront out of the data. These reconstructions largely involve global techniques, in which the entire wavefront's pattern affects the reconstruction of the wavefront in any given position. The method's noise sensitivity is analyzed, and actual reconstruction results are presented.
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Papers by Emmanuel Lanzmann