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Schémas en groupes et immeubles des groupes exceptionnels sur un corps local. Première partie : le groupe G 2

Wee Teck Gan, Jiu-Kang Yu (2003)

Bulletin de la Société Mathématique de France

Nous obtenons une version explicite de la théorie de Bruhat-Tits pour les groupes exceptionnels de type G 2 sur un corps local. Nous décrivons chaque construction concrètement en termes de réseaux : l’immeuble, les appartements, la structure simpliciale, les schémas en groupes associés. Les appendices traitent de l’analogie avec les espaces symétriques réels et des espaces symétriques associés à G 2 réel et complexe.

Schémas en groupes et immeubles des groupes exceptionnels sur un corps local. Deuxième partie : les groupes F 4 et E 6

Wee Teck Gan, Jiu-Kang Yu (2005)

Bulletin de la Société Mathématique de France

Nous obtenons une version explicite de la théorie de Bruhat-Tits pour les groupes exceptionnels des type F 4 ou E 6 sur un corps local. Nous décrivons chaque construction concrètement en termes de réseaux : l’immeuble, les appartements, la structure simpliciale, les schémas en groupes associés.

Seeable matter; unseeable antimatter

Geoffrey Dixon (2014)

Commentationes Mathematicae Universitatis Carolinae

The universe we see gives every sign of being composed of matter. This is considered a major unsolved problem in theoretical physics. Using the mathematical modeling based on the algebra 𝐓 : = 𝐂 ⊗ 𝐇 ⊗ 𝐎 , an interpretation is developed that suggests that this seeable universe is not the whole universe; there is an unseeable part of the universe composed of antimatter galaxies and stuff, and an extra 6 dimensions of space (also unseeable) linking the matter side to the antimatter—at the very least.

Sets with two associative operations

Teimuraz Pirashvili (2003)

Open Mathematics

In this paper we consider duplexes, which are sets with two associative binary operations. Dimonoids in the sense of Loday are examples of duplexes. The set of all permutations carries a structure of a duplex. Our main result asserts that it is a free duplex with an explicitly described set of generators. The proof uses a construction of the free duplex with one generator by planary trees.

Shuffle bialgebras

María Ronco (2011)

Annales de l’institut Fourier

The goal of our work is to study the spaces of primitive elements of some combinatorial Hopf algebras, whose underlying vector spaces admit linear basis labelled by subsets of the set of maps between finite sets. In order to deal with these objects we introduce the notion of shuffle algebras, which are coloured algebras where composition is not always defined. We define bialgebras in this framework and compute the subpaces of primitive elements associated to them. These spaces of primitive elements...

Simple multilinear algebras and hermitian operators

T. S. R. Fuad, Jon D. Phillips, Xiaorong Shen, Jonathan D. H. Smith (2000)

Commentationes Mathematicae Universitatis Carolinae

The paper studies multilinear algebras, known as comtrans algebras, that are determined by so-called T -Hermitian matrices over an arbitrary field. The main result of this paper shows that the comtrans algebra of n -dimensional T -Hermitian matrices furnishes a simple comtrans algebra.

Some necessary and sufficient conditions for nilpotent n -Lie superalgebras

Baoling Guan, Liangyun Chen, Yao Ma (2014)

Czechoslovak Mathematical Journal

The paper studies nilpotent n -Lie superalgebras over a field of characteristic zero. More specifically speaking, we prove Engel’s theorem for n -Lie superalgebras which is a generalization of those for n -Lie algebras and Lie superalgebras. In addition, as an application of Engel’s theorem, we give some properties of nilpotent n -Lie superalgebras and obtain several sufficient conditions for an n -Lie superalgebra to be nilpotent by using the notions of the maximal subalgebra, the weak ideal and the...

Some remarks on the Akivis algebras and the Pre-Lie algebras

Yuqun Chen, Yu Li (2011)

Czechoslovak Mathematical Journal

In this paper, by using the Composition-Diamond lemma for non-associative algebras invented by A. I. Shirshov in 1962, we give Gröbner-Shirshov bases for free Pre-Lie algebras and the universal enveloping non-associative algebra of an Akivis algebra, respectively. As applications, we show I. P. Shestakov’s result that any Akivis algebra is linear and D. Segal’s result that the set of all good words in X * * forms a linear basis of the free Pre-Lie algebra PLie ( X ) generated by the set X . For completeness,...

Spectral sequences for commutative Lie algebras

Friedrich Wagemann (2020)

Communications in Mathematics

We construct some spectral sequences as tools for computing commutative cohomology of commutative Lie algebras in characteristic 2 . In a first part, we focus on a Hochschild-Serre-type spectral sequence, while in a second part we obtain spectral sequences which compare Chevalley-Eilenberg-, commutative- and Leibniz cohomology. These methods are illustrated by a few computations.

Square subgroups of rank two abelian groups

A. M. Aghdam, A. Najafizadeh (2009)

Colloquium Mathematicae

Let G be an abelian group and ◻ G its square subgroup as defined in the introduction. We show that the square subgroup of a non-homogeneous and indecomposable torsion-free group G of rank two is a pure subgroup of G and that G/◻ G is a nil group.

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