Classical group

In mathematics, the classical groups are defined as the special linear groups over the reals R, the complex numbers C and the quaternions H together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the complex classical groups. The finite analogues of the classical groups are the classical groups of Lie type. The term "classical group" was coined by Hermann Weyl, it being the title of his 1939 monograph The Classical Groups. The classical groups form the deepest and most useful part of the subject of linear Lie groups. Most types of classical groups find application in classical and modern physics. A fe

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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

The xi-stability on the affine grassmannian

Zongbin Chen

We introduce a notion of xi-stability on the affine grassmannian (SIC) for the classical groups, this is the local version of the xi-stability on the moduli space of Higgs bundles on a curve introduced by Chaudouard and Laumon. We prove that the quotient (SIC)(xi)/T of the stable part (SIC)(xi) by the maximal torus T exists as an ind-k-scheme, and we introduce a reduction process analogous to the Harder-Narasimhan reduction for vector bundles over an algebraic curve. For the group , we calculate the Poincar, series of the quotient (SIC)(xi)/T.

Springer Heidelberg2015

Embeddings of maximal tori in classical groups over local and global fields

Eva Bayer Fluckiger, Ting-Yu Lee

Embeddings of maximal tori in classical groups over fields of characteristic not 2 are the subject matter of several recent papers. The aim of the present paper is to give necessary and sufficient conditions for such an embedding to exist, when the base field is a local field, or the field of real numbers. This completes the results of [3], where a complete criterion is given for the Hasse principle to hold when the base field is a global field.

Turpion Ltd2016

Lie group

In mathematics, a Lie group (pronounced liː ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract conce

General linear group

In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a gr

Unitary group

In mathematics, the unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The

MATH-479: Linear algebraic groups

The aim of the course is to give an introduction to linear algebraic groups and to give an insight into a beautiful subject that combines algebraic geometry with group theory.