Nous démontrons que la catégorie de von Neumann est équivalente à la catégorie des cônes autopolaires, facialement homogènes, complexes. Un cône $\vee$ dans un espace hilbertien réel est dit : 1) facialement homogène quand pour toute face $F$ de $\vee$ l’opérateur $\delta =$ (Projection sur $F-F$) $-$ (Projection sur $F^{\perp }-F^{\perp }$) est une dérivation de $\vee$ (i.e. $e^{t\delta }\vee =\vee \forall t\in R$) ; 2) complexe quand on s’est donné une structure d’algèbre de Lie complexe sur l’algèbre de Lie réelle des dérivations de $\vee$, modulo son centre. Nous caractérisons les espaces...

In this work the properties of Cartan subalgebras and weight spaces of finite dimensional Lie algebras are extended to the case of Leibniz algebras. Namely, the relation between Cartan subalgebras and regular elements are described, also an analogue of Cartan s criterion of solvability is proved.

In this paper we study the BGG-categories ${𝒪}_q$ associated to quantum groups. We prove that many properties of the ordinary BGG-category $𝒪$ for a semisimple complex Lie algebra carry over to the quantum case. Of particular interest is the case when $q$ is a complex root of unity. Here we prove a tensor decomposition for both simple modules, projective modules, and indecomposable tilting modules. Using the known Kazhdan-Lusztig conjectures for $𝒪$ and for finite dimensional $U_q$-modules we are able to determine...

Cayley-Dickson construction produces a sequence of normed algebras over real numbers. Its consequent applications result in complex numbers, quaternions, octonions, etc. In this paper we formalize the construction and prove its basic properties.

[For the entire collection see Zbl 0742.00067.]We are interested in partial differential equations on domains in ${𝒞}^n$. One of the most natural questions is that of analytic continuation of solutions and domains of holomorphy. Our aim is to describe the domains of holomorphy for solutions of the complex Laplace and Dirac equations. We call them cells of harmonicity. We deduce their properties mostly by examining geometrical properties of the characteristic surface (which is the same for both equations),...

Let $(𝔄,{ϵ}_u)$ and $(𝔅,{ϵ}_b)$ be two pointed sets. Given a family of three maps $ℱ=\{f_1:𝔄\to 𝔄;f_2:𝔄\times 𝔄\to 𝔄;f_3:𝔄\times 𝔄\to 𝔅\}$, this family provides an adequate decomposition of $𝔄\setminus \left\{{ϵ}_u\right\}$ as the orthogonal disjoint union of well-described $ℱ$-invariant subsets. This decomposition is applied to the structure theory of graded involutive algebras, graded quadratic algebras and graded weak $H^{*}$-algebras.

Let $D$ be a division ring finite dimensional over its center $F$. The goal of this paper is to prove that for any positive integer $n$ there exists $a\in D^{\left(n\right)},$ the $n$th multiplicative derived subgroup such that $F\left(a\right)$ is a maximal subfield of $D$. We also show that a single depth-$n$ iterated additive commutator would generate a maximal subfield of $D.$

In this paper a construction of characteristic classes for a subfoliation $(F_1,F_2)$ is given by using Kamber-Tondeur’s techniques. For this purpose, the notion of $(F_1,F_2)$-foliated principal bundle, and the definition of its associated characteristic homomorphism, are introduced. The relation with the characteristic homomorphism of $F_i$-foliated bundles, $i=1,2$, the results of Kamber-Tondeur on the cohomology of $g$-$DG$-algebras. Finally, Goldman’s results on the restriction of foliated bundles to the leaves of a foliation...

Given a principal ideal domain $R$ of characteristic zero, containing 1/2, and a two-cone $X$ of appropriate connectedness and dimension, we present a sufficient algebraic condition, in terms of Adams-Hilton models, for the Hopf algebra $FH(\Omega X;R)$ to be isomorphic with the universal enveloping algebra of some $R$-free graded Lie algebra; as usual, $F$ stands for free part, $H$ for homology, and $\Omega$ for the Moore loop space functor.