Seja d:= \'\\partial\'/\'\\partial IND.x\'+ \'beta\\partial\'/\'partial IND.y\'uma derivação simples de K[x,y], onde K é um corpo de característica zero. Doering, Lequain e Ripoll ([1]) provaram que exite um \'gama\'\'PERTENCE A\' K[x,y] tal que o operador S = \'\\partial\'/\'\\partial x\'+\'beta\\partial\'/\'\\partial y\'+\'gama\'\'PERTENCE A\'\'A IND.2\'\':= K[x,y]\' < \'\\partial\'/partial IND.x\', \'\\partial\'/\'partial\'/\'partial IND y\'\'>\'gera um ideal à esquerda maximal principal de \'A IND.2\'. Desta maneira mostraram, para n=2, que a seguinte conjectura é verdadeira: Seja d:=\'\\partial\'/ \'\\partial IND.x\"IND.1\"+\"alfa\'IND.2\'\'\\partial\'/\'\\partial\'IND.x\'\'IND.2\"+...+ alfa IND.n\"\\partial\'/\'\\partial IND.x\'\'IND.n\" uma derivaçào simples de K[\'x IND.1\'...\'x IND n\']. Então, A IND.n\'(d+\'gama\') é um ideal à esquerda maximal principal de Á IND.n\', para algum \'gama\'\'PERTENCE A\'K[\'x IND.1\',...\'x IND.n\']. Nós mostramos que esta conjectura é verdadeira em alguns casos particulares / Let d: =\'\\partial/\'/\'\\partial IND.x\'+ \'beta\\partial IND.y\' be a simple derivation of K[x,y], where K is a field of characteristic zero. Doering, Lequain e Ripol ([1]) proved that there exists a polynomial um \'gama\'\'IT BELONGS\' K[x,y] such that the operador S =\'\\partial\'/\'\\partial x\'+\'beta\\partial\'/\'\\partial y\'\'gama\'\'IT BELONGS\'\' á ind.2\':= K[x,y]\' < \'\\partial\'/\'partial IND.x\',\'partial\'/\'partial\'/\'partial IND y\'> \'generates a principal maximal left ideal of A IND.2\'. In this way, they showed that, for n=2, the following conjectures is tru: Let d:=\'\\partial\'/\'\\partial IND.x\"+\"alfaÍND.2\"\\partial\'/ \"\\partial\' IND.x\'IND.2\"+ álfa IND.n\"\\partial IND.xÍND.n\"be a simple derivation of K[\'x IND.1\',...,\'x IND n\']. Then, \'A IND.n\'(d+\'gama\') is a principal maximal left ideal of \'A IND.n\',for some \'gama\"IT BELONGS\'K[x IND.1\',...,\'x IND.n\']. We show that this conjecture is true in some cases

The classical Nevanlinna-Pick interpolation theorem, proved in 1915 by Pick and in 1919 by Nevanlinna, gives a condition for when there exists an interpolating function in H∞(D) for a specified set of data in the complex plane. In 1967, Sarason proved his commutant lifting theorem for H∞(D), from which an operator theoretic proof of the classical Nevanlinna-Pick theorem followed. Several competing noncommutative generalizations arose as a consequence of Sarason's result, and two strategies emerged for proving generalized Nevanlinna-Pick theorems: via a commutant lifting theorem or via a resolvent, or displacement, equation. We explore the difference between these two approaches. Specifically, we compare two theorems: one by Constantinescu-Johnson from 2003 and one by Muhly-Solel from 2004. Muhly-Solel's theorem is stated in the highly general context of W*-correspondences and is proved via commutant lifting. Constantinescu-Johnson's theorem, while stated in a less general context, has the advantage of an elegant proof via a displacement equation. In order to make the comparison, we first generalize Constantinescu-Johnson's theorem to the setting of W*-correspondences in Theorem 3.0.1. Our proof, modeled after Constantinescu-Johnson's, hinges on a modified version of their displacement equation. Then we show that Theorem 3.0.1 is fundamentally different from Muhly-Solel's. More specifically, interpolation in the sense of Muhly-Solel's theorem implies interpolation in the sense of Theorem 3.0.1, but the converse is not true. Nevertheless, we identify a commutativity assumption under which the two theorems yield the same result. In addition to the two main theorems, we include smaller results that clarify the connections between the notation, space of interpolating maps, and point evaluation employed by Constantinescu-Johnson and those employed by Muhly-Solel. We conclude with an investigation of the relationship between Theorem 3.0.1 and Popescu's generalized Nevanlinna-Pick theorem proved in 2003.

De nombreuses approches à la gravité quantique suggèrent que la description usuelle de l’espace-temps ne serait pas adaptée à la description des phénomènes physiques impliquant à la fois des processus gravitationnels et quantiques. Une meilleure description pourrait consister à munir l’espace-temps d’une structure non-commutative en remplaçant les coordonnées locales sur la variété par des opérateurs ne commutant pas deux-à-deux. Il s’ensuit que le comportement des théories de champs construites sur de tels espaces diffère en général de celui des théories de champs ordinaires. L’étude de ces possibles nouvelles propriétés est l’objet de la théorie non-commutative des champs (TNCC) dont nous étudions certains des aspects.Dans le présent mémoire, nous considérons deux familles d’espaces quantiques dont l’algèbres de coordonnées admet une structure d’algèbre de Lie. La première famille est caractérisée par l’algèbre su(2) et apparait dans le cadre de modèle de gravité quantique en 3 dimensions, ainsi que dans certains modèles de « brane » et de « group field theory ». La seconde famille d’espaces quantiques est connue sous le nom de kappa-Minkowski. L’intérêt de cet espace réside dans le fait qu’il est défini comme l’espace homogène associé à l’algèbre de Hopf de kappa-Poincaré. Cette dernière définit une déformation, à l’échelle de Planck, de l’algèbre de Poincaré et s’avère être étroitement liée à certains modèles de gravité quantique.Afin d’étudier les TNCC, il est commode de représenter l’espace quantique comme une algèbre non-commutative de fonctions munie d’un produit déformé appelé « star-product ». Une façon canonique de construire un tel produit consiste à se servir d’outils d’analyse harmonique et à adapter le schéma de quantification de Weyl (originellement introduit dans le cadre de la mécanique quantique) à l’algèbre considérée. Les expressions de star-product associé aux espaces susmentionnés sont dérivées de manière explicite. Nous montrons en particulier que des familles de star-product inéquivalents peuvent être classifiées par des considérations cohomologiques. Nous étudions enfin les propriétés quantiques de différents modèles de TNCC scalaire quartique construits à l’aide de ces star-product. Dans le cas où l’espace quantique est caractérisé par l’algèbre su(2), nous trouvons que la fonction 2-point est fini à l’ordre une boucle, le paramètre de déformation jouant le rôle d’une coupure ultraviolette et infrarouge. Dans le cas de kappa-Minkowski, nous insistons sur l’invariance sous kappa-Poincaré de l’action fonctionnelle et montrons que certains modèles de TNCC scalaire quartique divergent moins que dans le cas commutatif. Par ailleurs, la fonction 4-point est trouvée finie à l’ordre une boucle. Nos résultats, ainsi que leurs conséquences, sont finalement discutés. / As many theoretical studies point out, the classical description of spacetime, as a continuum, might be no longer adequate to reconcile gravity with quantum mechanics at very high energy (the relevant energy scale being often regarded as the Planck scale). Instead, a more appropriate description could be provided by the data of a noncommutative algebra of coordinate operators replacing the usual commutative local coordinates on smooth manifold. Once the noncommutative nature of spacetime is assumed, it is to expect that the (classical and quantum) properties of field theories on noncommutative background differ from the ones of field theories on classical background. This is the aim of Non-Commutative Field Theory (NCFT) to explore and study these new properties.In the present dissertation, we consider two families of quantum spacetimes of Lie algebra type noncommutativity. The first family is characterised by su(2) noncommutativity and appears in the description of some models of quantum gravity in 3-dimensions. The other family of quantum spacetimes is known in the physics literature as the 4-d kappa-Minkowski space. The importance of this quantum spacetime lies into the fact that its symmetries are provided by the (quantum) kappa-Poincaré algebra (a deformation of the classical Poincaré algebra) together with the fact that the deformation parameter 'kappa', which is of mass dimension, provides a natural energy scale at which the quantum gravity effects may be relevant (and is often regarded as being related to the Planck scale). For these reasons, the kappa-Minkowski space appears as a good candidate for a spacetime to be involved in the description of Doubly Special Relativity and Relative Locality models.To study NCFT it is often convenient to introduce a star product characterising the (noncommutative) C*-algebra of fields modelling the quantum spacetime under consideration. We emphasise that a canonical star product can be obtained by using the group algebraic structures underlying the construction of such Lie algebra type quantum spaces, namely by making use of harmonic analysis on the corresponding Lie group together with the Weyl quantisation scheme. The explicit derivation of such star product for kappa-Minkowski is given. In addition, we show that su(2) Lie algebras of coordinate operators related to quantum spaces with su(2) noncommutativity can be conveniently represented by SO(3)-equivariant poly-differential involutive representations and show that the quantized plane waves obtained from the quantization map action on the usual exponential functions are determined by polar decomposition of operators combined with constraint stemming from the Wigner theorem for SU(2). We finally indicate a convenient way to extend this construction to other semi-simple but non simply connected Lie groups by making use of results from group cohomology with value in an abelian group that would replace the constraints stemming from the simple Wigner theorem.Then, we investigate the quantum properties of various models of interacting scalar field theory on noncommutative background making use of the aforementioned star product formalism to construct physically reasonable expressions for the action functional. Considering quantum spacetime with su(2) noncommutativity, we find that the one-loop 2-point function for complex scalar field theories with quartic interactions is finite, the deformation parameter playing the role of a natural UV cut-off. Special attention is paid to the derivation of the one-loop corrections to both the 2-point and 4-point functions for various models of kappa-Poincaré invariant scalar field theory with quartic interactions. In that case, we show that for some models the 2-point function divergences linearly thus slightly milder than their commutative counterpart, while the one-loop 4-point function is shown to be finite. The results we obtained together with their consequences are finally discussed.

In this thesis, structures defined via modular operads and properads are generalized to their non-commutative analogs. We define the connected sum for modular operads. This way we are able to construct the graded commutative product on the algebra over Feynman transform of the modular operad. This forms a Batalin-Vilkovisky algebra with symmetry given by the modular operad. We transfer this structure to the cohomology via the Homological perturbation lemma. In particular, we consider these constructions for Quantum closed and Quantum open modular operad. As a parallel project we introduce associative analog of Frobenius properad, called Open Frobenius properad. We construct the cobar complex over it and in the spirit of Barannikov interpret algebras over cobar complex as homological differential operators. Furthermore we present the IBA∞-algebras as analog of well-known IBL∞-algebras. 1

yes / The orthocomplemented modular lattice of subspaces L[H(d)] , of a quantum system with d-dimensional Hilbert space H(d), is considered. A generalized additivity relation which holds for Kolmogorov probabilities is violated by quantum probabilities in the full lattice L[H(d)] (it is only valid within the Boolean subalgebras of L[H(d)] ). This suggests the use of more general (than Kolmogorov) probability theories, and here the Dempster-Shafer probability theory is adopted. An operator D(H1,H2) , which quantifies deviations from Kolmogorov probability theory is introduced, and it is shown to be intimately related to the commutator of the projectors P(H1),P(H2) , to the subspaces H 1, H 2. As an application, it is shown that the proof of the inequalities of Clauser, Horne, Shimony, and Holt for a system of two spin 1/2 particles is valid for Kolmogorov probabilities, but it is not valid for Dempster-Shafer probabilities. The violation of these inequalities in experiments supports the interpretation of quantum probabilities as Dempster-Shafer probabilities.

Most combinatorial optimization problems from theoretical computer science have a natural framing as optimization of polynomials in commuting variables. Noncommutativity is one of the defining features of quantum mechanics. So it is not surprising that noncommutative polynomial optimization plays an equally important role in quantum computer science. Our main goal here is to understand the relative hardness of commutative versus noncommutative polynomial optimization. At a first glance it might seem that noncommutative polynomial optimization must be more complex. However this is not always true and this question of relative hardness is substantially more subtle than might appear at the outset. First in this thesis we show that the general noncommutative polynomial optimization is complete for the class $\Pi_2$; this class is in the second level of the arithmetical hierarchy and strictly contains both the set of recursively enumerable languages and its complement. On the other hand, commutative polynomial optimization is decidable and belongs to $\PSPACE$. We then provide evidence that for polynomials arising from a large class of constraint satisfaction problems the situation is reversed: the noncommutative polynomial optimization is an easier computational problem compared to its commutative analogue. A second question we are interested in is about whether we could extract good commutative solutions from noncommutative solutions? This brings us to the second theme of this thesis which is about understanding the algebraic structure of the solutions of noncommutative polynomial optimization. We show that this structural insight then could shed light on the optimal commutative solutions and thereby paves the path in understanding the relationships between the commutative and noncommutative solutions. Here we first use the sum-of-squares framework to understand the algebraic relationships that are present between operators in any optimal noncommutative solution of a class of polynomial optimization problems arising from certain constraint satisfaction problems. We then show how we can design approximation algorithms for these problems so that some algebraic structures of our choosing is present. Finally we propose a rounding scheme for extracting good commutative solutions from noncommutative ones.

No description available.

We develop a public key cryptosystem whose security is based on the intractability of the ideal membership problem for a noncommutative algebra over a finite field. We show that this system, which is the noncommutative analogue of the Polly Cracker cryptosystem, is more secure than the commutative version. This is due to the fact that there are a number of ideals of noncommutative algebras (over finite fields) that have infinite reduced Groebner bases, and can be used to generate a public key. We present classes of such ideals and prove that they do not have a finite Groebner basis under any admissible order. We also examine various techniques to realize finite Groebner bases, in order to determine whether these ideals can be used effectively in the design of a public key cryptosystem. We then show how some of these classes of ideals, which have infinite reduced Groebner bases, can be used to design a public key cryptosystem. We also study various techniques of encryption. Finally, we study techniques of cryptanalysis that may be used to attack the cryptosystems that we present. We show how poorly constructed public keys can in fact, reveal the private key, and discuss techniques to design public keys that adequately conceal the private key. We also show how linear algebra can be used in ciphertext attacks and present a technique to overcome such attacks. This is different from the commutative version of the Polly Cracker cryptosystem, which is believed to be susceptible to "intelligent" linear algebra attacks. / Ph. D.

In this dissertation, we solve multivariable Nevanlinna-Pick type interpolation problems. Particularly, we consider the left tangential interpolation problems on the commutative or noncommutative unit ball. For the commutative setting, we discuss left-tangential operator-argument interpolation problems for Schur-class multipliers on the Drury-Arveson space and for the noncommutative setting, we discuss interpolation problems for Schur-class multipliers on Fock space. We apply the Krein-space geometry approach (also known as the Grassmannian Approach). To implement this approach J-versions of Beurling-Lax representers for shift-invariant subspaces are required. Here we obtain these J-Beurling-Lax theorems by the state-space method for both settings. We see that the Krein-space geometry method is particularly simple in solving the interpolation problems when the Beurling-Lax representer is bounded. The Potapov approach applies equally well whether the representer is bounded or not. / Ph. D.

Eine Quantengruppe ist eine nichtkommutative und nichtkokommutative Hopfalgebra. In dieser Arbeit konstruieren wir eine Deformation der lokalkonvexen Hopfalgebra der lokalanalytischen Funktionen auf GL(2,O), wobei O hier der Bewertungsring einer endlichen Erweiterung der p-adischen Zahlen ist. Wir zeigen, dass diese Deformation eine nichtkommutative, nichtkokommutative lokalkonvexe Hopfalgebra, also eine p-adische Quantengruppe, ist. Unser Hauptresultat ist, dass das starke Dual dieser Deformation eine Fréchet-Stein Algebra ist. Dies bedeutet, dass das starke Dual ein projektiver Limes von noetherschen Banachalgebren unter rechtsflachen Übergangsabbildungen ist. Im kommutativen Fall wurde dies von P. Schneider und J. Teitelbaum gezeigt. Unser Beweis im nichtkommutativen Fall benutzt Ideen von M. Emerton, der einen alternativen Beweis im kommutativen Fall gefunden hat. Für unseren Beweis beschreiben wir gewisse Vervollständigungen der quanten-einhüllenden Algebra und benutzen die Technik der partiell dividierten Potenzen. Eine wichtige Klasse lokalanalytischer Darstellungen von GL(2,K) wird mithilfe globaler Schnitte von Linienbündeln auf der p-adischen oberen Halbebene konstruiert. Wir konstruieren ein nichtkommutatives Analogon der p-adischen oberen Halbebene, von dem wir erwarten, dass es interessante Darstellungen unserer p-adischen Quantengruppe induziert. Die wichtigsten Hilfsmittel der Konstruktion sind die Maninsche Quantenebene, der Bruhat-Tits Baum für PGL(2,K) und die Theorie der algebraischen Mikrolokalisierung. / A quantum group is a noncommutative noncocommutative Hopf algebra. In this thesis we deform the locally convex Hopf algebra of locally analytic functions on GL(2,O), where O is the valuation ring of a finite extension of the p-adic numbers. We show that this deformation is a noncommutative noncocommutative locally convex Hopf algebra, i.e. a p-adic quantum group. Our main result is that the strong dual of our deformation is a Fréchet Stein algebra, i.e. a projective limit of Noetherian Banach algebras with right flat transition maps. This was shown in the commutative case by P. Schneider and J. Teitelbaum. For our proof in the noncommutative case we use ideas of M. Emerton, who gave an alternative proof of the Fréchet Stein property in the commutative case. For the proof we describe completions of the quantum enveloping algebra and use partial divided powers. An important class of locally analytic representations of GL(2,K) is constructed from global sections of line bundles on the p-adic upper half plane. We construct a noncommutative analogue of an affine version of the p-adic upper half plane which we expect to give rise to interesting representations of our p-adic quantum group. We construct this space by using the Manin quantum plane, the Bruhat-Tits tree for PGL(2,K) and the theory of algebraic microlocalization.