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Concept

Simple Lie group

Summary

In mathematics, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symmetric spaces. Together with the commutative Lie group of the real numbers, \mathbb{R}, and that of the unit-magnitude complex numbers, U(1) (the unit circle), simple Lie groups give the atomic "blocks" that make up all (finite-dimensional) connected Lie groups via the operation of group extension. Many commonly encountered Lie groups are either simple or 'close' to being simple: for example, the so-called "special linear group" SL(n) of n by n matrices with determinant equal to 1 is simple for all n > 1. The first classification of simple Lie groups was by Wilhelm Killing, and this work was later perfected by Élie Cartan. The final classification is often referred to as Killing-Cartan classification. Definition Unfortunat

Official source

https://en.wikipedia.org/wiki/Simple_Lie_group

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The goal of the course is to introduce relativistic quantum field theory as the conceptual and mathematical framework describing fundamental interactions.

We will establish the major results in the representation theory of semisimple Lie algebras over the field of complex numbers, and that of the related algebraic groups.

On introduit les algèbres de Lie semisimples de dimension finie sur les nombres complexes et démontre le théorème de classification de celles-ci.

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Irreducibility of disconnected subgroups of exceptional algebraic groups

Soumaïa Nadia Ghandour

This dissertation is concerned with the study of irreducible embeddings of simple algebraic groups of exceptional type. It is motivated by the role of such embeddings in the study of positive dimensional closed subgroups of classical algebraic groups. The classification of the maximal closed connected subgroups of simple algebraic groups was carried out by E. B. Dynkin, G. M. Seitz and D. M. Testerman. Their analysis for the classical groups was based primarily on a striking result: if G is a simple algebraic group and ø : G → SL(V ) is a tensor indecomposable irreducible rational representation then, with specified exceptions, the image of G is maximal among closed connected subgroups of one of the classical groups SL(V), Sp(V ) or SO(V ). In the case of closed, not necessarily connected, subgroups of the classical groups, one is interested in considering irreducible embeddings of simple algebraic groups and their automorphism groups: given a simple algebraic group Y defined over an algebraically closed field K, one is led to study the embeddings G < Aut(Y ), where G and Aut(Y ) are closed subgroups of SL(V ) and V is an irreducible rational KY -module on which G acts irreducibly. A partial analysis of such embeddings in the case of classical algebraic groups Y was carried out by B. Ford. We purpose to classify all triples (G, Y, V ) where Y is a simple algebraic group of exceptional type, defined over an algebraically closed field K of characteristic p > 0, G is a closed non-connected positive dimensional subgroup of Y and V is a nontrivial irreducible rational KY -module such that V|G is irreducible. We obtain a precise description of such triples (G, Y, V ).

EPFL2008

This paper studies the irreducible embeddings of simple algebraic groups of exceptional type. It is motivated by the role of such embeddings in the study of positive dimensional closed subgroups of classical algebraic groups. In this paper we give a classification of all triples where Y is a simple algebraic group of exceptional type, defined over an algebraically closed field K of characteristic , G is a closed disconnected positive dimensional subgroup of Y and V is a nontrivial rational irreducible KY-module on which G acts irreducibly.

2010

Maximal subgroups acting with two composition factors on irreducible representations of exceptional algebraic groups

Nathan Scheinmann

Let

Y

be a simply connected simple algebraic group over an algebraically closed field

k

of characteristic

p

and let

X

be a maximal closed connected simple subgroup of

Y

. Excluding some small primes in specific cases, we classify the

p

-restricted irreducible representations of

Y

on which

X

acts with exactly two composition factors. This work follows on naturally from the classification of irreducible subgroups of exceptional algebraic groups given by Testerman.

EPFL2019