On simple Lie algebras over a field of characteristic 2

We prove that the simple Lie algebras constructed by G. Jurman (2004) in 121 are isomorphic to Hamiltonian algebras. As a corollary we answer all questions formulated in G. Jurman (2004) [2] about isomorphisms of these algebras. (C) 2012 Elsevier Inc. All rights reserved.

ON SPECIALITY OF BINARY-LIE ALGEBRAS

In the present work, binary-Lie, assocyclic, and binary (-1,1) algebras are studied. We prove that, for every assocyclic algebra A, the algebra A(-) is binary-Lie. We find a simple non-Malcev binary-Lie superalgebra T that cannot be embedded in A(-s) for an assocyclic superalgebra A. We use the Grassmann envelope of T to prove the similar result for algebras. This solve negatively a problem by Filippov (see [1, Problem 2.108]). Finally, we prove that the superalgebra T is isomorphic to the commutator superalgebra A(-s) for a simple binary (-1,1) superalgebra A.

COMMUTATIVE ALGEBRAS IN DRINFELD CATEGORIES OF ABELIAN LIE ALGEBRAS

We describe (braided-) commutative algebras with non-degenerate multiplicative form in certain braided monoidal categories, corresponding to abelian metric Lie algebras (so-called Drinfeld categories). We also describe local modules over these algebras and classify commutative algebras with a finite number of simple local modules.

Representations of semisimple Lie algebras

La tesi è dedicata allo studio delle rappresentazioni delle algebre di Lie semisemplici su un campo algebricamente chiuso di caratteristica zero. Mediante il teorema di Weyl sulla completa riducibilità, ogni rappresentazione di dimensione finita di una algebra di Lie semisemplice è scrivibile come somma diretta di sottorappresentazioni irriducibili. Questo permette di poter concentrare l'attenzione sullo studio delle rappresentazioni irriducibili. Inoltre, mediante il ricorso all'algebra inviluppante universale si ottiene che ogni rappresentazione irriducibile è una rappresentazione di peso più alto. Perciò è naturale chiedersi quando una rappresentazione di peso più alto sia di dimensione finita ottenendo che condizione necessaria e sufficiente perché una rappresentazione di peso più alto sia di dimensione finita è che il peso più alto sia dominante. Immediata è quindi l'applicazione della teoria delle rappresentazioni delle algebre di Lie semisemplici nello studio delle superalgebre di Lie, in quanto costituite da un'algebra di Lie e da una sua rappresentazione, dove viene utilizzata la tecnica della Z-graduazione che viene utilizzata per la prima volta da Victor Kac nello studio delle algebre di Lie di dimensione infinita nell'articolo ''Simple irreducible graded Lie algebras of finite growth'' del 1968.

Embedding Of Non-Simple Lie-Groups, Coupling-Constant Relations And Non-Uniqueness Of Models Of Unification

A general derivation of the coupling constant relations which result on embedding a non-simple group like SU L (2) @ U(1) in a larger simple group (or graded Lie group) is given. It is shown that such relations depend only on the requirement (i) that the multiplet of vector fields form an irreducible representation of the unifying algebra and (ii) the transformation properties of the fermions under SU L (2). This point is illustrated in two ways, one by constructing two different unification groups containing the same fermions and therefore have same Weinberg angle; the other by putting different SU L (2) structures on the same fermions and consequently have different Weinberg angles. In particular the value sin~0=3/8 is characteristic of the sequential doublet models or models which invoke a large number of additional leptons like E 6, while addition of extra charged fermion singlets can reduce the value of sin ~ 0 to 1/4. We point out that at the present time the models of grand unification are far from unique.

Real time feature-based facial tracking using Lie algebras

We have developed a novel human facial tracking system that operates in real time at a video frame rate without needing any special hardware. The approach is based on the use of Lie algebra, and uses three-dimensional feature points on the targeted human face. It is assumed that the roughly estimated facial model (relative coordinates of the three-dimensional feature points) is known. First, the initial feature positions of the face are determined using a model fitting technique. Then, the tracking is operated by the following sequence: (1) capture the new video frame and render feature points to the image plane; (2) search for new positions of the feature points on the image plane; (3) get the Euclidean matrix from the moving vector and the three-dimensional information for the points; and (4) rotate and translate the feature points by using the Euclidean matrix, and render the new points on the image plane. The key algorithm of this tracker is to estimate the Euclidean matrix by using a least square technique based on Lie algebra. The resulting tracker performed very well on the task of tracking a human face.

Spectral isometries on non-simple C*-algebras

We prove that unital surjective spectral isometries on certain non-simple unital C*-algebras are Jordan isomorphisms. Along the way, we establish several general facts in the setting of semisimple Banach algebras.

Brisure de symétrie par la réduction des groupes de Lie simples à leurs sous-groupes de Lie réductifs maximaux

Dans ce travail, nous exploitons des propriétés déjà connues pour les systèmes de poids des représentations afin de les définir pour les orbites des groupes de Weyl des algèbres de Lie simples, traitées individuellement, et nous étendons certaines de ces propriétés aux orbites des groupes de Coxeter non cristallographiques. D'abord, nous considérons les points d'une orbite d'un groupe de Coxeter fini G comme les sommets d'un polytope (G-polytope) centré à l'origine d'un espace euclidien réel à n dimensions. Nous introduisons les produits et les puissances symétrisées de G-polytopes et nous en décrivons la décomposition en des sommes de G-polytopes. Plusieurs invariants des G-polytopes sont présentés. Ensuite, les orbites des groupes de Weyl des algèbres de Lie simples de tous types sont réduites en l'union d'orbites des groupes de Weyl des sous-algèbres réductives maximales de l'algèbre. Nous listons les matrices qui transforment les points des orbites de l'algèbre en des points des orbites des sous-algèbres pour tous les cas n<=8 ainsi que pour plusieurs séries infinies des paires d'algèbre-sous-algèbre. De nombreux exemples de règles de branchement sont présentés. Finalement, nous fournissons une nouvelle description, uniforme et complète, des centralisateurs des sous-groupes réguliers maximaux des groupes de Lie simples de tous types et de tous rangs. Nous présentons des formules explicites pour l'action de tels centralisateurs sur les représentations irréductibles des algèbres de Lie simples et montrons qu'elles peuvent être utilisées dans le calcul des règles de branchement impliquant ces sous-algèbres.

Casimirs and Lax operators from the structure of Lie algebras

This paper uses the structure of the Lie algebras to identify the Casimir invariant functions and Lax operators for matrix Lie groups. A novel mapping is found from the cotangent space to the dual Lie algebra which enables Lax operators to be found. The coordinate equations of motion are given in terms of the structure constants and the Hamiltonian.