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.

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.

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.

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.

Thesis (M.Sc.) - Carleton University, 2008. / Includes bibliographical references (p. 116-119). Also available in electronic format on the Internet.

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).

Ce mémoire comporte deux parties distinctes. La première partie concerne une étude d'algèbres n-aires. Une algèbre n-aire est un espace vectoriel sur lequel est définie une multiplication sur n arguments. Classiquement les multiplications sont binaires, mais depuis l'utilisation en physique théorique de multiplications ternaires, comme les produits de Nambu, de nombreux travaux mathématiques se sont focalisés sur ce type d'algèbres. Deux classes d'algèbres n-aires sont essentielles: les algèbres n-aires associatives et les algèbres n-aires de Lie. Nous nous intéressons aux deux classes. Concernant les algèbres n-aires associatives, on s'intéresse surtout aux algèbres 3-aires partiellement associatives, c'est-à-dire dont la multiplication vérifie l'identité ((xyz)tu)+(x(yzt)u)+(xy(ztu))=0 Ce cas est intéressant car les travaux connus concernant ce type d'algèbres ne distinguent pas les cas n pair et n-impair. On montre dans cette thèse que le cas n=3 ne peut pas être traité comme si n était pair. On étudie en détail l'algèbre libre 3-aire partiellement associative sur un espace vectoriel de dimension finie. Cette algèbre est graduée et on calcule précisément les dimensions des 7 premières composantes. On donne dans le cas général un système de générateurs ayant la propriété qu'une base est donnée par la sous famille des éléments non nuls. Les principales conséquences sont L'algèbre libre 3-aire partiellement associative est résoluble. L'algèbre libre commutative 3-aire partiellement associative est telle que tout produit concernant 9 éléments est nul. L'opérade quadratique correspondant aux algèbres 3-aires partiellement associatives ne vérifient pas la propriété de Koszul. On s'intéresse ensuite à l'étude des produits n-aires sur les tenseurs. L'exemple le plus simple est celui d'un produit interne sur des matrices non carrées. Nous pouvons définir le produit 3aire donné par A . ^tB . C. On montre qu'il est nécessaire de généraliser un peu la définition de partielle associativité. Nous introduisons donc les produits -partiellement associatifs où  est une permutation de degré p. Concernant les algèbres de Lie n-aires, deux classes d'algèbres ont été définies: les algèbres de Fillipov (aussi appelées depuis peu les algèbres de Lie-Nambu) et les algèbres n-Lie. Cette dernière notion est très générale. Cette dernière notion, très important dans l'étude de la mécanique de Nambu-Poisson, est un cas particulier de la première. Mais pour définir une approche du type Maurer-Cartan, c'est-à-dire définir une cohomologie scalaire, nous considérons dans ce travail les algèbres de Fillipov comme des algèbres n-Lie et développons un tel calcul dans le cadre des algèbres n-Lie. On s'intéresse également à la classification des algèbres n-aires nilpotentes. Le dernier chapitre de cette partie est un peu à part et reflète un travail poursuivant mon mémoire de Master. Il concerne les algèbres de Poisson sur l'algèbre des polynômes. On commence à présenter le crochet de Poisson sous forme duale en utilisant des équations de Pfaff. On utilise cette approche pour classer les structures de Poisson non homogènes sur l’algèbre des polynômes à trois variables . Le lien avec les algèbres de Lie est clair. Du coup on étend notre étude aux algèbres de Poisson dont l'algèbre de Lie sous jacent est rigide et on applique les résultats aux algèbres enveloppantes des algèbres de Lie rigides. La partie 2 concerne l'arithmétique des intervalles. Cette étude a été faite suite à une rencontre avec une société d'ingénierie travaillant sur des problèmes de contrôle de paramètres, de problème inverse (dans quels domaines doivent évoluer les paramètres d'un robot pour que le robot ait un comportement défini). [...] / 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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