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Dissertations / Theses on the topic 'Poisson algebras'
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Al-Shujary, Ahmed. "Kähler-Poisson Algebras." Licentiate thesis, Linköpings universitet, Matematik och tillämpad matematik, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-150620.
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The focus of this thesis is to introduce the concept of Kähler-Poisson algebras as analogues of algebras of smooth functions on Kähler manifolds. We first give here a review of the geometry of Kähler manifolds and Lie-Rinehart algebras. After that we give the definition and basic properties of Kähler-Poisson algebras. It is then shown that the Kähler type condition has consequences that allow for an identification of geometric objects in the algebra which share several properties with their classical counterparts. Furthermore, we introduce a concept of morphism between Kähler-Poisson algebras and show its consequences. Detailed examples are provided in order to illustrate the novel concepts.The series name is corrected in the electronic version of the cover.
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El, Hadrami Mohamed Lemine Ould 1962. "Poisson algebras and convexity." Diss., The University of Arizona, 1996. http://hdl.handle.net/10150/290675.
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In this dissertation, we identify a subgroup Tˢ of Dˢ(μ), the group of Sobolev symplectomorphisms of CP (n), n = 1,2 that has all the properties of a torus of a compact finite dimensional Lie group. We prove that Tˢ: (1) topologically is a submanifold of Dˢ(μ); (2) algebraically is a maximal abelian subgroup of Dˢ(μ); (3) geometrically is flat and totally geodesic. We also characterize the doubly stochastic operators on measurable spaces and use this result to extend the convexity Theorem of T. Bloch, H. Flaschka and T. Ratiu.
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Miscione, Steven. "Loop algebras and algebraic geometry." Thesis, McGill University, 2008. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=116115.
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This thesis primarily discusses the results of two papers, [Hu] and [HaHu]. The first is an overview of algebraic-geometric techniques for integrable systems in which the AKS theorem is proven. Under certain conditions, this theorem asserts the commutatvity and (potential) non-triviality of the Hamiltonian flow of Ad*-invariant functions once they're restricted to subalgebras. This theorem is applied to the case of coadjoint orbits on loop algebras, identifying the flow with a spectral curve and a line bundle via the Lax equation. These results play an important role in the discussion of [HaHu], wherein we consider three levels of spaces, each possessing a linear family of Poisson spaces. It is shown that there exist Poisson mappings between these levels. We consider the two cases where the underlying Riemann surface is an elliptic curve, as well as its degeneration to a Riemann sphere with two points identified (the trigonometric case). Background in necessary areas is provided.
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Lecoutre, César. "Polynomial Poisson algebras : Gel'fand-Kirillov problem and Poisson spectra." Thesis, University of Kent, 2014. https://kar.kent.ac.uk/47941/.
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We study the fields of fractions and the Poisson spectra of polynomial Poisson algebras. First we investigate a Poisson birational equivalence problem for polynomial Poisson algebras over a field of arbitrary characteristic. Namely, the quadratic Poisson Gel'fand-Kirillov problem asks whether the field of fractions of a Poisson algebra is isomorphic to the field of fractions of a Poisson affine space, i.e. a polynomial algebra such that the Poisson bracket of two generators is equal to their product (up to a scalar). We answer positively the quadratic Poisson Gel'fand-Kirillov problem for a large class of Poisson algebras arising as semiclassical limits of quantised coordinate rings, as well as for their quotients by Poisson prime ideals that are invariant under the action of a torus. In particular, we show that coordinate rings of determinantal Poisson varieties satisfy the quadratic Poisson Gel'fand-Kirillov problem. Our proof relies on the so-called characteristic-free Poisson deleting derivation homomorphism. Essentially this homomorphism allows us to simplify Poisson brackets of a given polynomial Poisson algebra by localising at a generator. Next we develop a method, the characteristic-free Poisson deleting derivations algorithm, to study the Poisson spectrum of a polynomial Poisson algebra. It is a Poisson version of the deleting derivations algorithm introduced by Cauchon [8] in order to study spectra of some noncommutative noetherian algebras. This algorithm allows us to define a partition of the Poisson spectrum of certain polynomial Poisson algebras, and to prove the Poisson Dixmier-Moeglin equivalence for those Poisson algebras when the base field is of characteristic zero. Finally, using both Cauchon's and our algorithm, we compare combinatorially spectra and Poisson spectra in the framework of (algebraic) deformation theory. In particular we compare spectra of quantum matrices with Poisson spectra of matrix Poisson varieties.
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Zwicknagl, Sebastian. "Equivariant Poisson algebras and their deformations /." view abstract or download file of text, 2006. http://proquest.umi.com/pqdweb?did=1280144671&sid=2&Fmt=2&clientId=11238&RQT=309&VName=PQD.
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Thesis (Ph. D.)--University of Oregon, 2006.Typescript. Includes vita and abstract. "In this dissertation I investigate Poisson structures on symmetric and exterior algebras of modules over complex reductive Lie algebras. I use the results to study the braided symmetric and exterior algebras"--P. 1. Includes bibliographical references (leaves 150-152). Also available for download via the World Wide Web; free to University of Oregon users.
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Martino, Maurizio. "Symplectic reflection algebras and Poisson geometry." Thesis, University of Glasgow, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.426614.
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Walker, Lachlan Duncan. "Deformed Poisson W-algebras of type A." Thesis, University of Aberdeen, 2018. http://digitool.abdn.ac.uk:80/webclient/DeliveryManager?pid=239477.
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For the algebraic group SLl+1(C) we describe a system of positive roots associated to conjugacy classes in its Weyl group Sl+1. Using this we explicitly describe the algebra of regular functions on certain transverse slices to conjugacy classes in SLl+1(C) as a polynomial algebra of invariants. These may be viewed as an algebraic group analogue of certain parabolic invariants that generate the W-algebra in type A found by Brundan and Kleshchev.
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Damianou, Pantelis Andrea. "Nonlinear Poisson brackets." Diss., The University of Arizona, 1989. http://hdl.handle.net/10150/184704.
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A hierarchy of vector fields (master symmetries) and homogeneous nonlinear Poisson structures associated with the Toda lattice are constructed and the various connections between them are investigated. Among their properties: new brackets are generated from old ones by using Lie-derivatives in the direction of certain vector fields; the infinite sequences obtained consist of compatible Poisson brackets in which the constants of motion for the Toda lattice are in involution. The vector fields in the construction are unique up to addition of a Hamiltonian vector field. Similarly the Poisson brackets are unique up to addition of a trivial Poisson bracket. These are Poisson tensors generated by wedge products of Hamiltonian vector fields. The non-trivial brackets may also be obtained by the use of r-matrices; we give formulas and prove this for the quadratic and cubic Toda brackets. We also indicate how these results can be generalized to other (semisimple) Toda flows and we give explicit formulas for the rank 2 Lie algebra of type B₂. The main tool in this calculation is Dirac's constraint bracket formula. Finally we study nonlinear Poisson brackets associated with orbits through nilpotent conjugacy classes in gl(n, R) and formulate some conjectures. We determine the degree of the transverse Poisson structure through such nilpotent elements in gl(n, R) for n ≤ 7. This is accomplished also by the use of Dirac's bracket formula.
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Casati, Matteo. "Multidimensional Poisson Vertex Algebras and Poisson cohomology of Hamiltonian structures of hydrodynamic type." Doctoral thesis, SISSA, 2015. http://hdl.handle.net/20.500.11767/4853.
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The Poisson brackets of hydrodynamic type, also called Dubrovin-Novikov brackets, constitute the Hamiltonian structure of a broad class of evolutionary PDEs, that are ubiquitous in the theory of Integrable Systems, ranging from Hopf equation to the principal hierarchy of a Frobenius manifold. They can be regarded as an analogue of the classical
Poisson brackets, defined on an infinite dimensional space of maps Σ → M between two manifolds. Our main problem is the study of Poisson-Lichnerowicz cohomology of such space when dim Σ > 1. We introduce the notion of multidimensional Poisson Vertex Algebras, generalizing and adapting the theory by A. Barakat, A. De Sole, and V. Kac [Poisson Vertex Algebras in the theory of Hamiltonian equations, 2009]; within this framework we explicitly compute the first nontrivial cohomology groups for an arbitrary Poisson bracket of hydrodynamic type, in the case dim Σ = dim M = 2. For the case of the so-called scalar brackets, namely the ones for which dim M = 1, we give a complete description on their Poisson–Lichnerowicz cohomology. From this computations it
follows, already in the particular case dim Σ = 2, that the cohomology is infinite dimensional.
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Cruz, Ines Maria Bravo de Faria. "The local structure of Poisson manifolds." Thesis, University of Warwick, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.309896.
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Gärtner, Andreas [Verfasser]. "Recurrence, Transience, and Poisson Boundaries in Operator Algebras / Andreas Gärtner." München : Verlag Dr. Hut, 2014. http://d-nb.info/1059329999/34.
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Brandl, Mary-Katherine. "Primitive and Poisson spectra of non-semisimple twists of polynomial algebras /." view abstract or download file of text, 2001. http://wwwlib.umi.com/cr/uoregon/fullcit?p3024507.
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Thesis (Ph. D.)--University of Oregon, 2001.Typescript. Includes vita and abstract. Includes bibliographical references (leaf 49). Also available for download via the World Wide Web; free to University of Oregon users.
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Thiffeault, Jean-Luc. "Classification, Casimir invariants, and stability of lie-poisson systems /." Digital version accessible at:, 1998. http://wwwlib.umi.com/cr/utexas/main.
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Goze, Nicolas. "N-ary algebras. Arithmetic of intervals." Phd thesis, Université de Haute Alsace - Mulhouse, 2011. http://tel.archives-ouvertes.fr/tel-00710165.
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This thesis has two distinguish parts. The first part concerns the study of n-ary algebras. A n-ary algebra is a vector space with a multiplication on n arguments. Classically the multiplications are binary, but the use of ternary multiplication in theoretical physic like for Nambu brackets led mathematicians to investigate these type of algebras. Two classes of n-ary algebras are fundamental: the associative n-ary algebras and the Lie n-ary algebras. We are interested by both classes. Concerning the associative n-ary algebras we are mostly interested in 3-ary partially associative 3-ary algebras, that is, algebras whose multiplication satisfies ((xyz)tu)+(x(yzt)u)+(xy(ztu))=0. This type is interesting because the previous woks on this subject was not distinguish the even and odd cases. We show in this thesis that the case n=3 can not be treated as the even cases. We investigate in detail the free partially associative 3-ary algebra on k generators. This algebra is graded and we compute the dimensions of the 7 first components. In the general case, we give a spanning set such as the sub family of non zero vector is a basis. The main consequences are the free partially associative 3-ary algebra is solvable. In the free commutative partially associative 3-ary algebra any product on 9 elements is trivial. The operad for partially associative 3-ary algebra do not satisfy the Koszul property. Then we study n-ary products on the tensors. The simplest example is given by a internal product of non square matrices. We can define a 3-ary product by taking A . ^tB . C. We show that we have to generalize a bit the definition of partial associativity for n-ary algebras. We then introduce the products -partially associative where is a permutation of the symmetric group of degree n. Concerning the n-ary algebras, two classes have been defined: Filipov algebras (also called recently Lie-Nambu algebras) and some more general class, the n-Lie algebras. Filipov algebras are very important in the study of the mechanic of Nambu-Poisson, and is a particular case of the other. So to define an approach of Maurer-Cartan type, that is, define a scalar cohomology, we consider in this work Fillipov as n-Lie algebras and develop such a calculus in the n-Lie algebras frame work. We also give some classifications of n-ary nilpotent algebras. The last chapter of this part concerns my work in Master on the Poisson algebras on polynomials. We present link with the Lie algebras is clear. Thus we extend our study to Poisson algebras which associated Lie algebra is rigid and we apply these results to the enveloping algebras of rigid Lie algebras. The second part concerns intervals arithmetic. The interval arithmetic is used in a lot of problems concerning robotic, localization of parameters, and sensibility of inputs. The classical operations of intervals are based of the rule : the result of an operation of interval is the minimal interval containing all the result of this operation on the real elements of the concerned intervals. But these operations imply many problems because the product is not distributive with respect the addition. In particular it is very difficult to translate in the set of intervals an algebraic functions of a real variable. We propose here an original model based on an embedding of the set of intervals on an associative algebra. Working in this algebra, it is easy to see that the problem of non distributivity disappears, and the problem of transferring real function in the set of intervals becomes natural. As application, we study matrices of intervals and we solve the problem of reduction of intervals matrices (diagonalization, eigenvalues, and eigenvectors).
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Lano, Ralph Peter. "Application of co-adjoint orbits to the loop group and the diffeomorphism group of the circle." Thesis, University of Iowa, 1994. https://ir.uiowa.edu/etd/5393.
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Ekstrand, Joel. "Going Round in Circles : From Sigma Models to Vertex Algebras and Back." Doctoral thesis, Uppsala universitet, Teoretisk fysik, 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-159918.
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In this thesis, we investigate sigma models and algebraic structures emerging from a Hamiltonian description of their dynamics, both in a classical and in a quantum setup. More specifically, we derive the phase space structures together with the Hamiltonians for the bosonic two-dimensional non-linear sigma model, and also for the N=1 and N=2 supersymmetric models. A convenient framework for describing these structures are Lie conformal algebras and Poisson vertex algebras. We review these concepts, and show that a Lie conformal algebra gives a weak Courant–Dorfman algebra. We further show that a Poisson vertex algebra generated by fields of conformal weight one and zero are in a one-to-one relationship with Courant–Dorfman algebras. Vertex algebras are shown to be appropriate for describing the quantum dynamics of supersymmetric sigma models. We give two definitions of a vertex algebra, and we show that these definitions are equivalent. The second definition is given in terms of a λ-bracket and a normal ordered product, which makes computations straightforward. We also review the manifestly supersymmetric N=1 SUSY vertex algebra. We also construct sheaves of N=1 and N=2 vertex algebras. We are specifically interested in the sheaf of N=1 vertex algebras referred to as the chiral de Rham complex. We argue that this sheaf can be interpreted as a formal quantization of the N=1 supersymmetric non-linear sigma model. We review different algebras of the chiral de Rham complex that one can associate to different manifolds. In particular, we investigate the case when the manifold is a six-dimensional Calabi–Yau manifold. The chiral de Rham complex then carries two commuting copies of the N=2 superconformal algebra with central charge c=9, as well as the Odake algebra, associated to the holomorphic volume form.
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Källén, Johan. "Twisting and Gluing : On Topological Field Theories, Sigma Models and Vertex Algebras." Doctoral thesis, Uppsala universitet, Teoretisk fysik, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-173225.
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This thesis consists of two parts, which can be read separately. In the first part we study aspects of topological field theories. We show how to topologically twist three-dimensional N=2 supersymmetric Chern-Simons theory using a contact structure on the underlying manifold. This gives us a formulation of Chern-Simons theory together with a set of auxiliary fields and an odd symmetry. For Seifert manifolds, we show how to use this odd symmetry to localize the path integral of Chern-Simons theory. The formulation of three-dimensional Chern-Simons theory using a contact structure admits natural generalizations to higher dimensions. We introduce and study these theories. The focus is on the five-dimensional theory, which can be understood as a topologically twisted version of N=1 supersymmetric Yang-Mills theory. When formulated on contact manifolds that are circle fibrations over a symplectic manifold, it localizes to contact instantons. For the theory on the five-sphere, we show that the perturbative part of the partition function is given by a matrix model. In the second part of the thesis, we study supersymmetric sigma models in the Hamiltonian formalism, both in a classical and in a quantum mechanical setup. We argue that the so called Chiral de Rham complex, which is a sheaf of vertex algebras, is a natural framework to understand quantum aspects of supersymmetric sigma models in the Hamiltonian formalism. We show how a class of currents which generate symmetry algebras for the classical sigma model can be defined within the Chiral de Rham complex framework, and for a six-dimensional Calabi-Yau manifold we calculate the equal-time commutators between the currents and show that they generate the Odake algebra.
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Van, De Ven Christiaan Jozef Farielda. "Quantum Systems and their Classical Limit A C*- Algebraic Approach." Doctoral thesis, Università degli studi di Trento, 2021. http://hdl.handle.net/11572/324358.
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In this thesis we develop a mathematically rigorous framework of the so-called ''classical limit'' of quantum systems and their semi-classical properties. Our methods are based on the theory of strict, also called C*- algebraic deformation quantization. Since this C*-algebraic approach encapsulates both quantum as classical theory in one single framework, it provides, in particular, an excellent setting for studying natural emergent phenomena like spontaneous symmetry breaking (SSB) and phase transitions typically showing up in the classical limit of quantum theories. To this end, several techniques from functional analysis and operator algebras have been exploited and specialised to the context of Schrödinger operators and quantum spin systems. Their semi-classical properties including the possible occurrence of SSB have been investigated and illustrated with various physical models. Furthermore, it has been shown that the application of perturbation theory sheds new light on symmetry breaking in Nature, i.e. in real, hence finite materials. A large number of physically relevant results have been obtained and presented by means of diverse research papers.
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Castañeda, Terrones Jose Luis. "Review of geometric quantization and WKB method." Universidade Estadual Paulista (UNESP), 2018. http://hdl.handle.net/11449/157267.
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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)
A quantização geométrica é um procedimento para construir uma teoria quântica a partir de elementos geométricos de um sistema clássico considerado como uma variedade simplética. Ele fornece uma abordagem matemática para uma teoria quântica com uma ampla gama de aplicações que vão desde sistemas com partículas até teorias de campo quântico, para as quais a variedade simplética é o espaço cotangente do espaço de campos (elementos do espaço cotangente são variações infinitesimais). Por outro lado, o método WKB fornece uma maneira de construir uma solução aproximada para a equação de Schrödinger na mecânica quântica a partir de elementos geométricos no espaço de fase de soluções de um sistema clássico. Estas notas são uma revisão de alguns artigos sobre essas duas abordagens da mecânica quântica.
Geometric quantization is a procedure to construct a quantum theory from geometric elements of a classical system regarded as a symplectic manifold. It provides a mathematical approach to a quantum theory with a wide range of applications that go from systems with particles to quantum field theories, for which the symplectic manifold is the cotangent space of the space of fields (elements of the cotangent space are infinitesimal variations). On the other side, WKB method provides a way to construct an approximate solution to the Schrödinger equation in quantum mechanics from geometric elements on the phase space of solutions of a classical system. These notes are a review of some papers on those two approaches to quantum mechanics.
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Aminou, Adérodjou A. Rachidi. "Groupes de Lie-Poisson et bigèbres de Lie." Lille 1, 1988. http://www.theses.fr/1988LIL10139.
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Un groupe de lie-poisson est un groupe de lie g muni d'une structure de poisson telle que la multiplication soit un morphisme de poisson de g x g dans g. L'algebre de lie d'un groupe de lie-poisson porte une structure supplementaire qui en fait une bigebre de lie. Nous etudions les bigebres de lie (autodualite, triplets de manin) et les algebres de lie bicroisees qui generalisent des bigebres de lie. Nous considerons le cas des bigebres de lie exactes, en particulier des bigebres de lie quasitriangulaires et nous etudions plusieurs exemples. Nous montrons que la categorie des bigebres de lie quasi-triangulaires est isomorphe a la categorie des algebres de lie-semenov. Nous comparons la notion de carre d'une algebre de lie-semenov due a semenov-trian-shansky et la notion de double d'une bigebre de lie due a drinfeld. Enfin, nous demontrons le "troisieme theoreme de lie" pour les groupes de lie-poisson et nous etudions les structures de poisson sur un groupe de lie definies par des solutions des equations de yang-baxter classique, generalisee ou modifiee.
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Saint-Germain, Michel. "Algebres de poisson et structures transverses." Paris 7, 1997. http://www.theses.fr/1997PA077150.
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Pour une algebre de lie nilpotente reelle, on considere une orbite coadjointe dans le dual de cette algebre. C'est une feuille symplectique de la structure de poisson de ce dual. L'algebre des fonctions regulieres sur l'orbite est une algebre de poisson-weyl. On a ainsi l'existence de fonctions regulieres sur l'orbite verifiant les relations de darboux. Le premier probleme etudie est celui du prolongement de ces fonctions par des fonctions sur le dual, verifiant encore les relations de darboux. On montre l'existence de relevements (homomorphismes d'algebres de poisson) de l'algebre de poisson-weyl precitee dans le complete de l'algebre symetrique de l'algebre de lie (complete pour la topologie associee a la filtration par les puissances de l'ideal attache a l'orbite). Il est muni naturellement d'une structure d'algebre de poisson. On rapproche ces resultats de ceux obtenus par alan weinstein. Le second probleme est l'etude du commutant de l'image de l'algebre de poisson-weyl attachee a l'orbite par un relevement. C'est une algebre de series formelles dont la classe d'isomorphie ne depend pas du relevement choisi. On compare cette algebre a celle donnee par la structure de poisson transverse a l'orbite, calculee par le theoreme de decomposition de alan weinstein ou la formule de p. A. M. Dirac. La methode des orbites associe a l'orbite coadjointe un ideal primitif de l'algebre enveloppante. Cette these est inspiree de l'etude faite par fokko du cloux du voisinage d'un tel ideal primitif. Celui est decrit par une algebre. Notre philosophie est que cette algebre est une quantification de la structure de poisson transverse a l'orbite.
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Zhang, Pumei. "Algebraic aspects of compatible poisson structures." Thesis, Loughborough University, 2012. https://dspace.lboro.ac.uk/2134/10110.
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This thesis consists of three chapters. In Chapter one, we introduce some notions and definitions for basic concepts of the theory of integrable bi-Hamiltonian systems. Brief statements of several open problems related to our main results are also mentioned in this part. In Chapter two, we applied the so-called Jordan-Kronecker decomposition theorem to study algebraic properties of the pencil P generated by two constant compatible Poisson structures on a vector space. In particular, we study the linear automorphism group GP that preserves P. In classical symplectic geometry, many fundamental results are based on the symplectic group, which preserves the symplectic structure. Therefore in the theory of bi-Hamiltonian structures, we hope GP also plays a fundamental role. In Chapter three, we study one of the famous Poisson pencils which is sometimes called 'argument shift pencil'. This pencil is defined on the dual space g * of an arbitrary Lie algebra g. This pencil is generated by the Lie-Poisson bracket { , } and constant bracket { , }a for a ε g * . Thus we may apply the Jordan-Kronecker decomposition theorem to introduce the so-called Jordan-Kronecker invariants of a finite-dimensional Lie algebra g. These invariants can be understood as the algebraic type of the canonical Jordan-Kronecker form for the 'argument shift pencil' at a generic point. Jordan-Kronecker invariants are found for all low-dimensional Lie algebras (dim g ≤ 5) and can be used to construct the families of polynomials in bi-involution. The results are found to be useful in the discussion of the existence of a complete family of polynomials in bi-involution w.r.t. these two brackets { , } and { , }a.
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Bäck, Viktor. "Localization of Multiscale Screened Poisson Equation." Thesis, Uppsala universitet, Algebra och geometri, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-180928.
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Dresse, Alain M. G. "Polynomial Poisson structures and dummy variables in computer algebra." Doctoral thesis, Universite Libre de Bruxelles, 1993. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/212775.
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Kunz, Daniel. "Lieovy grupy a jejich fyzikální aplikace." Master's thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2020. http://www.nusl.cz/ntk/nusl-417088.
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In this thesis I describe construction of Lie group and Lie algebra and its following usage for physical problems. To be able to construct Lie groups and Lie algebras we need define basic terms such as topological manifold, tensor algebra and differential geometry. First part of my thesis is aimed on this topic. In second part I am dealing with construction of Lie groups and algebras. Furthermore, I am showing different properties of given structures. Next I am trying to show, that there exists some connection among Lie groups and Lie algebras. In last part of this thesis is used just for showing how this apparat can be used on physical problems. Best known usage is to find physical symmetries to establish conservation laws, all thanks to famous Noether theorem.
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Melani, Valerio. "Poisson and coisotropic structures in derived algebraic geometry." Thesis, Sorbonne Paris Cité, 2016. http://www.theses.fr/2016USPCC299/document.
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Dans cette thèse, on définit et on étudie les notions de structure de Poisson et coïsotrope sur un champ dérivé, dans le contexte de la géométrie algébrique dérivée. On considère deux présentations différentes de structure de Poisson : la première est purement algébrique, alors que la deuxième est plus géométrique. On montre que les deux approches sont en fait équivalentes. On introduit aussi la notion de structure coïsotrope sur un morphisme de champs dérivés, encore une fois en présentant deux définitions équivalentes : la première est basée sur une généralisation appropriée de l'opérade Swiss-Cheese de Voronov, tandis que la deuxième est formulée en termes de champs de multivecteurs rélatifs. En particulier, on montre que le morphisme identité admet une unique structure coïsotrope ; cela produit une application d'oubli des structures de Poisson n-décalées aux structures de Poisson (n-1)-décalées. On montre aussi que l'intersection de deux morphismes coïsotropes dans un champ de Poisson n-décalée est naturellement equipée d'une structure de Poisson (n-1)-décalée canonique. En outre, on fournit une équivalence entre l'espace de structures coïsotropes non-dégénérées et l'espace des structures Lagrangiennes en géométrie dérivée, introduites dans les travaux de Pantev-Toën-Vaquié-VezzosiIn this thesis, we define and study Poisson and coisotropic structures on derived stacks in the framework of derived algebraic geometry. We consider two possible presentations of Poisson structures of different flavour: the first one is purely algebraic, while the second is more geometric. We show that the two approaches are in fact equivalent. We also introduce the notion of coisotropic structure on a morphism between derived stacks, once again presenting two equivalent definitions: one of them involves an appropriate generalization of the Swiss Cheese operad of Voronov, while the other is expressed in terms of relative polyvector fields. In particular, we show that the identity morphism carries a unique coisotropic structure; in turn, this gives rise to a non-trivial forgetful map from n-shifted Poisson structures to (n-1)-shifted Poisson structures. We also prove that the intersection of two coisotropic morphisms inside a n-shifted Poisson stack is naturally equipped with a canonical (n-1)-shifted Poisson structure. Moreover, we provide an equivalence between the space of non-degenerate coisotropic structures and the space of Lagrangian structures in derived geometry, as introduced in the work of Pantev-Toën-Vaquié-Vezzosi
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GAVARINI, FABIO. "Quantizzazione di gruppi di Poisson." Doctoral thesis, Università degli Studi di Roma "Tor Vergata", 1996. http://hdl.handle.net/2108/40610.
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Sia G ^\tau un gruppo algebrico semisemplice connesso e semplicemente connesso, dotato della struttura di gruppo di Poisson di Sklyanin-Drinfel’d generalizzata; sia H^\tau il suo gruppo di Poisson duale.
Mediante la costruzione del doppio quantico e la dualizzazione tramite algebre di Hopf formali, costruiamo nuovi gruppi quantici U^M_{q,\varphi}(h) — duali dei gruppi quantici multiparametrici U^{M'}_{q,\varphi}(g) costruiti su g^\tau , con g = Lie(G) — che danno quantizzazioni infinitesimali di H ^\tau e G^\tau ; studiamo le loro specializzazioni alle radici dell'unità (in particolare, i loro limiti classici), scoprendo così nuovi morfismi di Frobenius quantici. L'intera descrizione dualizza per H ^\tau quel che era noto per G^\tau , completando la quantizzazione della coppia (G^\tau,H^\tau).Let G ^\tau be a connected simply connected semisimple algebraic group, endowed with generalized Sklyanin-Drinfel’d structure of Poisson group; let H^\tau be its dual Poisson group. By means of quantum double construction and dualization via formal Hopf algebras, we construct new quantum groups U^M_{q,\varphi}(h) — dual to the multiparameter quantum group U^{M'}_{q,\varphi}(g) built upon g^\tau , with g = Lie(G) — which yield infinitesimal quantization of H ^\tau and G^\tau ; we study their specializations at roots of 1 (in particular, their classical limits), thus discovering new quantum Frobenius morphisms. The whole description dualize for H ^\tau what was known for G^\tau , completing the quantization of the pair (G^\tau , H^\tau).
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Elek, Balázes. "Computing the standard Poisson structure on Bott-Samelson varieties incoordinates." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2012. http://hub.hku.hk/bib/B4833005X.
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Bott-Samelson varieties associated to reductive algebraic groups are much studied in representation theory and algebraic geometry. They not only provide resolutions of singularities for Schubert varieties but also have interesting geometric properties of their own. A distinguished feature of Bott-Samelson varieties is that they admit natural affine coordinate charts, which allow explicit computations of geometric quantities in coordinates.
Poisson geometry dates back to 19th century mechanics, and the more recent theory of quantum groups provides a large class of Poisson structures associated to reductive algebraic groups. A holomorphic Poisson structure Π on Bott-Samelson varieties associated to complex semisimple Lie groups, referred to as the standard Poisson structure on Bott-Samelson varieties in this thesis, was introduced and studied by J. H. Lu. In particular, it was shown by Lu that the Poisson structure Π was algebraic and gave rise to an iterated Poisson polynomial algebra associated to each affine chart of the Bott-Samelson variety. The formula by Lu, however, was in terms of certain holomorphic vector fields on the Bott-Samelson variety, and it is much desirable to have explicit formulas for these vector fields in coordinates.
In this thesis, the holomorphic vector fields in Lu’s formula for the Poisson structure Π were computed explicitly in coordinates in every affine chart of the Bott-Samelson variety, resulting in an explicit formula for the Poisson structure Π in coordinates. The formula revealed the explicit relations between the Poisson structure and the root system and the structure constants of the underlying Lie algebra in any basis. Using a Chevalley basis, it was shown that the Poisson structure restricted to every affine chart of the Bott-Samelson variety was defined over the integers. Consequently, one obtained a large class of iterated Poisson polynomial algebras over any field, and in particular, over fields of positive characteristic. Concrete examples were given at the end of the thesis.published_or_final_version
Mathematics
Master
Master of Philosophy
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Leray, Johan. "Approche fonctorielle et combinatoire de la propérade des algèbres double Poisson." Thesis, Angers, 2017. http://www.theses.fr/2017ANGE0027/document.
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On construit et étudie la généralisation des algèbres double Poisson décalées à toute catégorie monoïdale symétrique additive. On s’intéresse notamment aux algèbres double Poisson linéaires et quadratiques. Dans un second temps, on étudie la koszulité des propérades DLie et DPois = As ⮽c DLie qui encodent respectivement les algèbres double Lie et les algèbres doubles Poisson. On associe à chacune de ces propérades, un S-module muni d’une structure de monoïde pour un nouveau produit monoïdal dit de composition connexe : on appelle de tels monoïdes protopérades. On montre notamment l’existence, pour toutS-module, d’une protopérade libre associée et l’on explicite la combinatoire sous-jacente en terme de briques et de murs. On définit une adjonction bar-cobar, une dualité de Koszul et une notion de base PBW pour les protopérades. On présente également une tentative de théorème PBW à la Hoffbeck pour les protopérades, de laquelle on déduit la koszulité de la diopérade associée à la propérade DLieWe construct and study the generalization of shifted double Poisson algebras to all additive symmetric monoidal categories. We are especially interested in linear and quadratic double Poisson algebras. We then study the koszulity of the properads DLie and DPois = As ⮽c DLie which encode double Lie algebras and double Poisson algebras respectively. We associate to each, a S-module with a monoidal structure for a new monoïdal product call the connected composition product : we call such monoids protoperads. We show, for any S-module, the existence of the associated free protoperad and we make explicit the underlying combinatorics. We define a bar-cobar adjunction, the notion of Koszul duality and PBW bases for protoperads. We present an attempt of prove a PBW theorem à la Hoffbeck for protoperads, and prove the koszulity of the dioperad associated to the properad DLie
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Dahamna, Khaled. "Classification des algèbres de Lie sous-riemanniennes et intégrabilité des équations géodésiques associées." Phd thesis, INSA de Rouen, 2011. http://tel.archives-ouvertes.fr/tel-00769931.
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Dans cette thèse, on s'intéresse en premier aux problèmes sous-riemanniens sur un groupe de Lie nilpotent d'ordre 2. Dans un premier temps, on réalise la classification complète des algèbres de Lie sous-riemanniennes (SR-algèbres de Lie) nilpotentes d'ordre 2 de dimension n compris entre 3 et 7, et celles de dimension arbitraire n telle que l'algèbre dérivée est de dimension une.De plus, nous avons distingué les SR-algèbres de Lie de contact et de quasi-contact et nous avons calculé, en dimension 5, le groupe des SR-symétries infinitésimales. Une fois cette classification réalisée, on étudie les géodésiques sous-riemanniennes associées aux SR-algèbres de Lie nilpotentes d'ordre 2 obtenues dans notre classification. Nous avons étudié l'intégrabilité des équations géodésiques adjointes et donné les contrôles optimaux ainsi que les trajectoires optimales dans chacun des cas. Dans une seconde partie de la thèse, on étudie les géodésiques sous-riemanniennes pour un groupe de Lie sous-riemannien (G;D;B) où G = SO(4) ou G = SO(2; 2) et D est de codimension2 (donnant des espaces SR-homogènes de contact). Nous avons donné un modèle canonique de ces espaces et ensuite montré que les systèmes adjoints de Lie-Poisson associés au modèle étaient toujours intégrables au sens de Liouville. De plus, nous montrons que le système de Lie-Poisson est soit un système linéaire qui est super-intégrable en fonctions trigonométriques du temps ou constantes ; soit un système non linéaire intégrable au sens de Liouville et dont les solutions sont exprimables à l'aide de la fonction elliptique de Weierstrass.
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Valvo, Lorenzo. "Hamiltonian perturbation theory on a poisson algebra : application to a throbbing top and to magnetically confined particles." Thesis, Aix-Marseille, 2019. http://www.theses.fr/2019AIXM0498.
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La théorie de perturbation Hamiltonienne de la mécanique classique est basé sur la structure d'algébre de Lie. Mais on trouve des structures de Lie dans tout les systèmes dit ``de Poisson''. Dans la première partie de cette thèse, on propose une approche purement algébrique à la théorie classique des perturbations, qui s'applique donc à tout les système de Poisson. Dans cette méthode, introduit en [Vittot, 2004] une transformation (de Lie) permet de diviser la perturbation en un terme préservant le flot non perturbé, et une correction quadratique.Dans l'exemple d'une Toupie Pulsante (un corps rigide non autonome) symétrique et périodique on montre que notre théorème s'applique et reproduit le théorème de KAM de la mécanique classique. Puis on considere une Toupie non symétrique avec moments d'inertie qui presentent des fluctuations quelconques: dans ce cas, on etudie sous quelles conditions les trajectoires du systeme sont proches de celle du système statique.Dans la troisième partie de ce mémoire, on étudie la dynamique d'une particule chargée dans un champ électromagnétique donné arbitraire. Par par la théorie des perturbations on peut réduire la dimensionnalité de la dynamique, ou étudier la rétroaction de la particule sur le champ. Cependant, fournir une description du flot non perturbé est une tâche redoutable, liée à la question de longue date de la théorie du centre-guide en physique des plasmas. Nous dérivons les équations du mouvement et leur structure de Poisson dans une nuovelle description relativiste et non perturbative de cette théorieThe Hamiltonian perturbation theory of classical mechanics is based on the underlying Lie algebraic structure. But Lie structures are met in a wider class of dynamical systems, called Poisson systems. In the first part of this thesis, we propose a purely algebraic approach to classical perturbation theory to extend its scope to any Poisson system. In this method, introduced in [Vittot, 2004], a (Lie) transform allows to split the perturbation into a term reserving the unperturbed flow, and a smaller correction, quadratic in the original perturbation strength.The second part of the dissertation is about the dynamics of a non-autonomous Top. We consider first a symmetric Top with periodically dependent moments of inertia; in this case, our theorem applies and reproduces the KAM theorem of classical mechanics. Then we switch to a non symmetric Top with non-periodically fluctuating moments of inertia: in this case we study for which conditions the static trajectories give a good approximation to those of the non-autonomous system.In the third part of this work we study the dynamics of a magnetically confined particle. By perturbation theory one may reduce the dimensionality of the dynamics, or study the retroaction of the particle on the field. However, providing an efficient description of the unperturbed flow is a formidable task, related to the long-standing issue of Guiding Centre Theory in plasma physics. Recently a novel relativistic and non-perturbative approach to Guiding Centre theory has been proposed [Di Troia, 2018]. We derive the equations of motion and their Poisson structure in this description
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Elchinger, Olivier. "Formalité liée aux algèbres enveloppantes et étude des algèbres Hom-(co)Poisson." Phd thesis, Université de Haute Alsace - Mulhouse, 2012. http://tel.archives-ouvertes.fr/tel-00857460.
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Le but de cette thèse est d'étudier quelques aspects algébriques du problème de quantification par déformation. On considère d'une part la formalité dans le cas des algèbres libres et de l'algèbre de Lie so(3), et on s'intéresse d'autre part à la quantification par déformation pour des structures Hom-algébriques. Suivant le résultat de formalité de Kontsevich en 1997 pour les algèbres symétriques, on étudie dans la première partie de cette thèse les algèbres libres, qui sont un cas particulier d'algèbres enveloppantes, et on montre qu'il n'y a pas formalité en général, sauf dans les cas triviaux. On montre aussi qu'il n'y a pas formalité pour l'algèbre de Lie so(3). Les techniques utilisées sont de type homologiques. On calcule la cohomologie de ces algèbres et on procède à la construction du L-infini-quasi-isomorphisme entre l'algèbre de Lie différentielle graduée des cochaînes de Hochschild munie du crochet de Gerstenhaber et l'algèbre de la cohomologie munie du crochet de Schouten. Dans la seconde partie de ce travail, on utilise un principe de déformation par twist pour les structures Hom-algébriques, pour construire de nouvelles structures de même type, ou encore pour déformer une structure classique en une Hom-structure correspondante à l'aide d'un morphisme d'algèbres. En particulier, on applique ce procédé aux structures de Poisson et aux star-produits de Moyal-Weyl. Par ailleurs, on établit une correspondance entre les algèbres enveloppantes d'algèbres Hom-Lie possédant une structure Hom-coPoisson et les bigèbres Hom-Lie.
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Fauquant-Millet, Florence. "Sur la polynomialité de certaines algèbres d'invariants d'algèbres de Lie." Habilitation à diriger des recherches, Université Jean Monnet - Saint-Etienne, 2014. http://tel.archives-ouvertes.fr/tel-00994655.
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Ce mémoire étudie la polynomialité de l'algèbre des invariants de l'algèbre des fonctions polynomiales sur le dual d'une certaine algèbre de Lie, lorsque cette dernière est la troncation canonique d'une sous-algèbre biparabolique d'une algèbre de Lie semi-simple complexe.
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Garcia, Hilares Nilton Alan. "A Parallel Aggregation Algorithm for Inter-Grid Transfer Operators in Algebraic Multigrid." Thesis, Virginia Tech, 2019. http://hdl.handle.net/10919/94618.
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As finite element discretizations ever grow in size to address real-world problems, there is an increasing need for fast algorithms. Nowadays there are many GPU/CPU parallel approaches to solve such problems.
Multigrid methods can be used to solve large-scale problems, or even better they can be used to precondition the conjugate gradient method, yielding better results in general. Capabilities of multigrid algorithms rely on the effectiveness of the inter-grid transfer operators. In this thesis we focus on the aggregation approach, discussing how different aggregation strategies affect the convergence rate. Based on these discussions, we propose an alternative parallel aggregation algorithm to improve convergence. We also provide numerous experimental results that compare different aggregation approaches, multigrid methods, and conjugate gradient iteration counts, showing that our proposed algorithm performs better in serial and parallel.Modeling real-world problems incurs a high computational cost because these mathematical models involve large-scale data manipulation. Thus we need fast and efficient algorithms. Nowadays there are many high-performance approaches for these problems. One such method is called the Multigrid algorithm. This approach models a physical domain using a hierarchy of grids, and so the effectiveness of these approaches relies on how well data can be transferred from grid to grid. In this thesis, we focus on the aggregation approach, which clusters a grid’s vertices according to its connections. We also provide an alternative parallel aggregation algorithm to give a faster solution. We show numerous experimental results that compare different aggregation approaches and multigrid methods, showing that our proposed algorithm performs better in serial and parallel than other popular implementations.
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Santos, Caio Fernando Rodrigues dos 1986. "Funções de interpolação e técnicas de solução para problemas de poisson usando método de elementos finitos de alta ordem." [s.n.], 2011. http://repositorio.unicamp.br/jspui/handle/REPOSIP/263524.
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Orientador: Marco Lúcio BittencourtDissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica
Made available in DSpace on 2018-08-17T22:43:41Z (GMT). No. of bitstreams: 1 Santos_CaioFernandoRodriguesdos_M.pdf: 3714047 bytes, checksum: 27c280eb98d3fe8f79e3d49756adf322 (MD5) Previous issue date: 2011
Resumo: Esse trabalho apresenta uma nova técnica de solução para o problema de Poisson, via problemas de projeção local, baseada na equivalência dos coeficientes para os problemas de Poisson e projeção. Um método de construção de matrizes de massa e rigidez, para triângulos, através do produto de matrizes unidimensionais de massa, mista e rigidez, usando-se coordenadas baricêntricas, é também apresentado. Dois novos conjuntos de funções de interpolação para triângulos, baseado em coordenadas de área, são considerados. Discute-se a propriedade de ortogonalidade dos polinômios de Jacobi, no domínio de integração de um triângulo na direção L2 = (0, 1- L1) e ponderações ótimas dos polinômios de Jacobi para as matrizes de massa são determinadas
Abstract: This work presents a new solution technique to Poisson problems, using local projection solution, based on the equivalence of the coefficients for the Poisson and projection problems. A calculation method for the mass and stiffness matrices of triangles, based on the product of one-dimensional mass, mixed and stiffness matrices, using barycentric coordinates is also proposed. Two new sets of interpolation functions for triangles, based on area coordinates, are considered. The orthogonality property of Jacobi polynomials in the triangle integration domain is discussed for the direction L2 = (0, 1 - L1) and optimal weights of Jacobi polynomials for the mass matrices are determined
Mestrado
Mecanica dos Sólidos e Projeto Mecanico
Mestre em Engenharia Mecânica
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Butin, Frédéric. "Structures de Poisson sur les Algèbres de Polynômes, Cohomologie et Déformations." Thesis, Lyon 1, 2009. http://www.theses.fr/2009LYO10192/document.
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La quantification par déformation et la correspondance de McKay forment les grands thèmes de l'étude qui porte sur des variétés algébriques singulières, des quotients d'algèbres de polynômes et des algèbres de polynômes invariants sous l'action d'un groupe fini. Nos principaux outils sont les cohomologies de Poisson et de Hochschild et la théorie des représentations. Certains calculs formels sont effectués avec Maple et GAP. Nous calculons les espaces d'homologie et de cohomologie de Hochschild des surfaces de Klein, en développant une généralisation du Théorème de HKR au cas de variétés non lisses et utilisons la division multivariée et les bases de Gröbner. La clôture de l'orbite nilpotente minimale d'une algèbre de Lie simple est une variété algébrique singulière sur laquelle nous construisons des star-produits invariants, grâce à la décomposition BGS de l'homologie et de la cohomologie de Hochschild, et à des résultats sur les invariants des groupes classiques. Nous explicitons les générateurs de l'idéal de Joseph associé à cette orbite et calculons les caractères infinitésimaux. Pour les algèbres de Lie simples B, C, D, nous établissons des résultats généraux sur l'espace d'homologie de Poisson en degré 0 de l'algèbre des invariants, qui vont dans le sens de la conjecture d'Alev et traitons les rangs 2 et 3. Nous calculons des séries de Poincaré à 2 variables pour des sous-groupes finis du groupe spécial linéaire en dimension 3, montrons que ce sont des fractions rationnelles, et associons aux sous-groupes une matrice de Cartan généralisée pour obtenir une correspondance de McKay algébrique en dimension 3. Toute l'étude a donné lieu à 4 articlesDeformation quantization and McKay correspondence form the main themes of the study which deals with singular algebraic varieties, quotients of polynomial algebras, and polynomial algebras invariant under the action of a finite group. Our main tools are Poisson and Hochschild cohomologies and representation theory. Certain calculations are made with Maple and GAP. We calculate Hochschild homology and cohomology spaces of Klein surfaces by developing a generalization of HKR theorem in the case of non-smooth varieties and use the multivariate division and the Groebner bases. The closure of the minimal nilpotent orbit of a simple Lie algebra is a singular algebraic variety : on this one we construct invariant star-products, with the help of the BGS decomposition of Hochschild homology and cohomology, and of results on the invariants of the classical groups. We give the generators of the Joseph ideal associated to this orbit and calculate the infinitesimal characters. For simple Lie algebras of type B, C, D, we establish general results on the Poisson homology space in degree 0 of the invariant algebra, which support Alev's conjecture, then we are interested in the ranks 2 and 3. We compute Poincaré series of 2 variables for the finite subgroups of the special linear group in dimension 3, show that they are rational fractions, and associate to the subgroups a generalized Cartan matrix in order to obtain a McKay correspondence in dimension 3. All the study comes from 4 papers
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Tagne, Pelap Serge Roméo. "Les propriétés homologiques des algèbres elliptiques de petite dimension." Phd thesis, Université d'Angers, 2008. http://tel.archives-ouvertes.fr/tel-00599328.
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Cette thèse est consacrée à l'étude des propriétés homologiques d'une famille d'algèbres associatives attachée aux courbes elliptiques. Chaque algèbre de cette famille admet un nombre ni de générateurs subordonnés aux relations quadratiques. Elles sont aujourd'hui connues sous le nom d'algèbres elliptiques de Sklyanin-Odesskii- Feigin. Il convient toutefois de souligner que le cas le plus simple, la famille d'algèbres elliptiques avec trois générateurs, était déjà connue de Artin et Shelter.
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Lemarié, Caroline. "Quelques structures de Poisson et équations de Lax associées au réseau de Toeplitz et au réseau de Schur." Thesis, Poitiers, 2012. http://www.theses.fr/2012POIT2286/document.
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Le réseau de Toeplitz est un système hamiltonien dont la structure de Poisson est connue. Dans cette thèse, nous donnons l'origine de cette structure de Poisson et nous en déduisons des équations de Lax associées au réseau de Toeplitz. Nous construisons tout d'abord une sous-variété de Poisson Hn de GLn(C), ce dernier étant vu comme un groupe de Lie-Poisson réel ou complexe dont la structure de Poisson provient d'un R-crochet quadratique sur gln(C) pour une R-matrice fixée. L'existence d'hamiltoniens associés au réseau de Toeplitz pour la structure de Poisson sur Hn ainsi que les propriétés du R-crochet quadratique permettent alors d'expliciter des équations de Lax du système. On en déduit alors l'intégrabilité au sens de Liouville du réseau de Toeplitz. Dans le point de vue réel, nous pouvons ensuite construire une sous-variété de Poisson Han du groupe Un qui est lui-même une sous-variété de Poisson-Dirac de GLR n(C). Nous construisons alors un hamiltonien, pour la structure de Poisson induite sur Han, correspondant à un autre système déduit du réseau de Toeplitz : le réseau de Schur modifié. Grâce aux propriétés des sous-variétés de Poisson-Dirac, nous explicitons une équation de Lax pour ce nouveau système et nous en déduisons une équation de Lax pour le réseau de Schur. On en déduit également l'intégrabilité au sens de Liouville du réseau de Schur modifiéThe Toeplitz lattice is a Hamiltonian system whose Poisson structure is known. In this thesis, we reveil the origins of this Poisson structure and we derive from it the associated Lax equations for this lattice. We first construct a Poisson subvariety Hn of GLn(C), which we view as a real or complex Poisson-Lie group whose Poisson structure comes from a quadratic R-bracket on gln(C) for a fixed R-matrix. The existence of Hamiltonians, associated to the Toeplitz lattice for the Poisson structure on Hn, combined with the properties of the quadratic R-bracket allow us to give explicit formulas for the Lax equation. Then, we derive from it the integrability in the sense of Liouville of the Toeplitz lattice. When we view the lattice as being defined over R, we can construct a Poisson subvariety Han of Un which is itself a Poisson-Dirac subvariety of GLR n(C). We then construct a Hamiltonian for the Poisson structure induced on Han, corresponding to another system which derives from the Toeplitz lattice : the modified Schur lattice. Thanks to the properties of Poisson-Dirac subvarieties, we give an explicit Lax equation for the new system and derive from it a Lax equation for the Schur lattice. We also deduce the integrability in the sense of Liouville of the modified Schur lattice
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pl, tomasz@uci agh edu. "A Lie Group Structure on Strict Groups." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1076.ps.
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Brenčys, Liutauras. "Puasono lygties sprendimas naudojantis šaltinio apibendrintomis hiperbolinės funkcijomis." Master's thesis, Lithuanian Academic Libraries Network (LABT), 2011. http://vddb.laba.lt/obj/LT-eLABa-0001:E.02~2011~D_20110804_100133-71588.
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Sudarytas Puasono lygties sprendimo per „rutuliukų“ potencialus algoritmas. Šiuo metodu Puasono lygties sprendimo uždavinys suvedamas į tiesinių algebrinių lygčių sistemos sprendimą. Sudaryta ir išbandyta matematiniu paketu MATHCAD to sprendimo programa. Palyginti gauti sprendiniai su tais, kurie gaunami analiziškai, įvertintas gautų sprendinių tikslumas. Šį sprendimo būdą galima panaudoti realiems fizikiniams potencialams paskaičiuoti, turint galvoje realų potencialą su kuriuo realūs krūviai.It consists of Poisson equation solution in the "ball" potential algorithm. In this method the Poisson equation, the decision problem are reduced to linear algebraic equations system solution. Created and tested a mathematical package MATHCAD program for that decision. Compared to solutions with those obtained analytically, estimated to obtain accurate solutions. This solution can be used to calculate the real physical potentials, given the real potential of the real workloads.
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Paolantoni, Thibault. "Application de Riemann-Hilbert-Birkhoff." Thesis, Université Paris-Saclay (ComUE), 2017. http://www.theses.fr/2017SACLS410/document.
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L'application exponentielle duale est une façon d'encoder les matrices de Stokes d'une connexion sur un fibré trivial sur la sphère de Riemann avec deux pôles : un pôle double en 0 et un pôle simple en l'infini.On donne ici une formule pour l'application exponentielle duale comme une série formelle non commutative. D'autres généralisations de cette formule sont donnéesThe exponential dual map is a way to encode Stokes data of a connection on a trivial vector bundle on the Riemann sphere with two poles: one double pole at 0 and one simple pole at infinity.We give here a formula for the exponential dual map expressed as a non commutative serie. Others generalizations of this formula are given
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Villoutreix, Paul. "Aléatoire et variabilité dans l’embryogenèse animale, une approche multi-échelle." Thesis, Sorbonne Paris Cité, 2015. http://www.theses.fr/2015PA05T016/document.
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Nous proposons dans cette thèse de caractériser quantitativement la variabilité à différentes échelles au cours de l'embryogenèse. Pour ce faire, nous utilisons une combinaison de modèles mathématiques et de résultats expérimentaux. Dans la première partie, nous utilisons une petite cohorte d'oursins digitaux pour construire une représentation prototypique du lignage cellulaire, reliant les caractéristiques des cellules individuelles avec les dynamiques à l'échelle de l'embryon tout entier. Ce modèle probabiliste multi-niveau et empirique repose sur les symétries des embryons et sur les identités cellulaires; cela permet d'identifier un niveau de granularité générique pour observer les distributions de caractéristiques cellulaires individuelles. Le prototype est défini comme le barycentre de la cohorte dans la variété statistique correspondante. Parmi plusieurs résultats, nous montrons que la variabilité intra-individuelle est impliquée dans la reproductibilité du développement embryonnaire. Dans la seconde partie, nous considérons les mécanismes sources de variabilité au cours du développement et leurs relations à l'évolution. En nous appuyant sur des résultats expérimentaux montrant une pénétrance incomplète et une expressivité variable de phénotype dans une lignée mutante du poisson zèbre, nous proposons une clarification des différents niveaux de variabilité biologique reposant sur une analogie formelle avec le cadre mathématique de la mécanique quantique. Nous trouvons notamment une analogie formelle entre l'intrication quantique et le schéma Mendélien de transmission héréditaire. Dans la troisième partie, nous étudions l'organisation biologique et ses relations aux trajectoires développementales. En adaptant les outils de la topologie algébrique, nous caractérisons des invariants du réseaux de contacts cellulaires extrait d'images de microscopie confocale d'épithéliums de différentes espèces et de différents fonds génétiques. En particulier, nous montrons l'influence des histoires individuelles sur la distribution spatiales des cellules dans un tissu épithélialWe propose in this thesis to characterize variability quantitatively at various scales during embryogenesis. We use a combination of mathematical models and experimental results. In the first part, we use a small cohort of digital sea urchin embryos to construct a prototypical representation of the cell lineage, which relates individual cell features with embryo-level dynamics. This multi-level data-driven probabilistic model relies on symmetries of the embryo and known cell types, which provide a generic coarse-grained level of observation for distributions of individual cell features. The prototype is defined as the centroid of the cohort in the corresponding statistical manifold. Among several results, we show that intra-individual variability is involved in the reproducibility of the developmental process. In the second part, we consider the mechanisms sources of variability during development and their relations to evolution. Building on experimental results showing variable phenotypic expression and incomplete penetrance in a zebrafish mutant line, we propose a clarification of the various levels of biological variability using a formal analogy with quantum mechanics mathematical framework. Surprisingly, we find a formal analogy between quantum entanglement and Mendel’s idealized scheme of inheritance. In the third part, we study biological organization and its relations to developmental paths. By adapting the tools of algebraic topology, we compute invariants of the network of cellular contacts extracted from confocal microscopy images of epithelia from different species and genetic backgrounds. In particular, we show the influence of individual histories on the spatial distribution of cells in epithelial tissues
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Sen, Suparna. "Segal-Bargmann Transform And Paley Wiener Theorems On Motion Groups." Thesis, 2010. https://etd.iisc.ac.in/handle/2005/2267.
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Sen, Suparna. "Segal-Bargmann Transform And Paley Wiener Theorems On Motion Groups." Thesis, 2010. http://etd.iisc.ernet.in/handle/2005/2267.
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Vignoli, Veronica. "On Poisson vertex algebra cohomology." Doctoral thesis, 2019. http://hdl.handle.net/11573/1322598.
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We construct a canonical map from the Poisson vertex algebra cohomology complex to the differential Harrison cohomology complex, which restricts to an isomorphism on the top degree.
This is an important step in the computation of Poisson vertex algebra and vertex algebra cohomologies.
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"Algebraic Multigrid Poisson Equation Solver." Master's thesis, 2015. http://hdl.handle.net/2286/R.I.29693.
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abstract: From 2D planar MOSFET to 3D FinFET, the geometry of semiconductor devices is getting more and more complex. Correspondingly, the number of mesh grid points increases largely to maintain the accuracy of carrier transport and heat transfer simulations. By substituting the conventional uniform mesh with non-uniform mesh, one can reduce the number of grid points. However, the problem of how to solve governing equations on non-uniform mesh is then imposed to the numerical solver. Moreover, if a device simulator is integrated into a multi-scale simulator, the problem size will be further increased. Consequently, there exist two challenges for the current numerical solver. One is to increase the functionality to accommodate non-uniform mesh. The other is to solve governing physical equations fast and accurately on a large number of mesh grid points.
This research rst discusses a 2D planar MOSFET simulator and its numerical solver, pointing out its performance limit. By analyzing the algorithm complexity, Multigrid method is proposed to replace conventional Successive-Over-Relaxation method in a numerical solver. A variety of Multigrid methods (standard Multigrid, Algebraic Multigrid, Full Approximation Scheme, and Full Multigrid) are discussed and implemented. Their properties are examined through a set of numerical experiments. Finally, Algebraic Multigrid, Full Approximation Scheme and Full Multigrid are integrated into one advanced numerical solver based on the exact requirements of a semiconductor device simulator. A 2D MOSFET device is used to benchmark the performance, showing that the advanced Multigrid method has higher speed, accuracy and robustness.Dissertation/Thesis
Masters Thesis Materials Science and Engineering 2015
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TORTORELLA, ALFONSO GIUSEPPE. "Deformations of coisotropic submanifolds in Jacobi manifolds." Doctoral thesis, 2017. http://hdl.handle.net/2158/1077777.
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In this thesis, we investigate deformation theory and moduli theory of coisotropic submanifolds in Jacobi manifolds. Originally introduced by Kirillov as local Lie algebras with one dimensional fibers, Jacobi manifolds encompass, unifying and generalizing, locally conformal symplectic manifolds, locally conformal Poisson manifolds, and non-necessarily coorientable contact manifolds.
We attach an L-infinity-algebra to any coisotropic submanifold in a Jacobi manifold. Our construction generalizes and unifies analogous constructions by Oh-Park (symplectic case), Cattaneo-Felder (Poisson case), and Le-Oh (locally conformal symplectic case). As a completely new case we also associate an L-infinity-algebra with any coisotropic submanifold in a contact manifold. The L-infinity-algebra of a coisotropic submanifold S controls the formal coisotropic deformation problem of S, even under Hamiltonian equivalence, and provides criteria both for the obstructedness and for the unobstructedness at the formal level. Additionally we prove that if a certain condition ("fiberwise entireness") is satisfied then the L-infinity-algebra controls the non-formal coisotropic deformation problem, even under Hamiltonian equivalence.
We associate a BFV-complex with any coisotropic submanifold in a Jacobi manifold. Our construction extends an analogous construction by Schatz in the Poisson setting, and in particular it also applies in the locally conformal symplectic/Poisson setting and the contact setting. Unlike the L-infinity-algebra, we prove that, with no need of any restrictive hypothesis, the BFV-complex of a coisotropic submanifold S controls the non-formal coisotropic deformation problem of S, even under both Hamiltonian equivalence and Jacobi equivalence.
Notwithstanding the differences there is a close relation between the approaches to the coisotropic deformation problem via L-infinity-algebra and via BFV-complex. Indeed both the L-infinity-algebra and the BFV-complex of a coisotropic submanifold S provide a cohomological reduction of S. Moreover they are L-infinity quasi-isomorphic and so they encode equally well the moduli space of formal coisotropic deformations of S under Hamiltonian equivalence.
In addition we exhibit two examples of coisotropic submanifolds in the contact setting whose coisotropic deformation problem is obstructed at the formal level. Further we provide a conceptual explanation of this phenomenon both in terms of the L-infinity-algebra and in terms of the BFV-complex.
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Zung, Jea-Ming, and 張志明. "On the Poisson algebra related B background field." Thesis, 2002. http://ndltd.ncl.edu.tw/handle/15938716921571365564.
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碩士國立臺灣大學
物理學研究所
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The noncommutative theory which says that the commutator of coordinates has a non-zero value, say $\theta$, in some sense and the Poisson algebra generated by differential forms imply that one may have a Yang-Mills theory described in noncommutative space. Some proposals marks the onset of these jobs: Low energy effective theory lives on a noncommutative space. For a D-brane in a constant $B$ field background, one has $\theta=B^{-1}$ in the zero slope limit. One will have an associative algebra if the open string is quantized. Concerned with the constant background field, for noncommutative space, Seiberg and Witten have constructed the picture for Seiberg-Witten map connecting noncommutative space with commutative space. Using a suitable differential calculus defined in the noncommutative space with the result of the open string quantization for the non-constant background field, we are devoted to generalized results, similar to the work of Seiberg and Witten, about Seiberg-Witten map, noncommutative gauge theory and match the Yang-Mills action with the Dirac-Born-Infeld action. We make the conclusions that by the quantization for the open string and the definitions, the commutator of function and a differential form and the commutator of functions with a $\theta$, alone one can construct a general operation for differential calculus without any other definition for differential form, and that, with the modified field strength, we require that the NCYM action matches with DBI action in the leading order in $\alpha^{'}$ at the Poisson level.
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Ochoa, Arango Jesús Alonso. "Grupoides y algebroides dobles de Lie /." Doctoral thesis, 2010. http://hdl.handle.net/11086/144.
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Tesis (Doctor en Matemática)--Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física, 2010.En este trabajo demostramos que todo grupoide doble de Lie con acción medular propia esta completamente determinado por una factorización de un cierto grupoide de Lie diagonal canónicamente definido. Tambien, estudiamos la versión infinitesimal de este concepto, la de algebroide doble de Lie y como resultado introducimos una nueva clase de ejemplos construidos a partir de ciertos diagramas de álgebras de Lie. En la parte final, proponemos los conceptos de biálgebra infinitesimál de multiplicadores y de bialgebra de Lie de derivadores. Presentamos algunos ejemplos y como resultado principal demostramos, bajo ciertas condiciones, como obtener a partir de una biálgebra infinitesimál de multiplicadores una biálgebra de Lie de derivadores.
Jesús Alonso Ochoa Arango.
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Pym, Brent. "Poisson Structures and Lie Algebroids in Complex Geometry." Thesis, 2013. http://hdl.handle.net/1807/43695.
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This thesis is devoted to the study of holomorphic Poisson structures and Lie algebroids, and their relationship with differential equations, singularity theory and noncommutative algebra.
After reviewing and developing the basic theory of Lie algebroids in the framework of complex analytic and algebraic geometry, we focus on Lie algebroids over complex curves and their application to the study of meromorphic connections. We give concrete constructions of the corresponding Lie groupoids, using blowups and the uniformization theorem. These groupoids are complex surfaces that serve as the natural domains of definition for the fundamental solutions of ordinary differential equations with singularities. We explore the relationship between the convergent Taylor expansions of these fundamental solutions and the divergent asymptotic series that arise when one attempts to solve an ordinary differential equation at an irregular singular point.
We then turn our attention to Poisson geometry. After discussing the basic structure of Poisson brackets and Poisson modules on analytic spaces, we study the geometry of the degeneracy loci---where the dimension of the symplectic leaves drops. We explain that Poisson structures have natural residues along their degeneracy loci, analogous to the Poincar\'e residue of a meromorphic volume form. We discuss the local structure of degeneracy loci that have small codimensions, and place strong constraints on the singularities of the degeneracy hypersurfaces of log symplectic manifolds. We use these results to give new evidence for a conjecture of Bondal.
Finally, we discuss the problem of quantization in noncommutative projective geometry. Using Cerveau and Lins Neto's classification of degree-two foliations of projective space, we give normal forms for unimodular quadratic Poisson structures in four dimensions, and describe the quantizations of these Poisson structures to noncommutative graded algebras. As a result, we obtain a (conjecturally complete) list of families of quantum deformations of projective three-space. Among these algebras is an ``exceptional'' one, associated with a twisted cubic curve. This algebra has a number of remarkable properties: for example, it supports a family of bimodules that serve as quantum analogues of the classical Schwarzenberger bundles.