Reductive group | EPFL Graph Search

Are you an EPFL student looking for a semester project?

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a perfect field is reductive if it has a representation that has a finite kernel and is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group GL(n) of invertible matrices, the special orthogonal group SO(n), and the symplectic group Sp(2n). Simple algebraic groups and (more generally) semisimple algebraic groups are reductive. Claude Chevalley showed that the classification of reductive groups is the same over any algebraically closed field. In particular, the simple algebraic groups are classified by Dynkin diagrams, as in the theory of compact Lie groups or complex semisimple Lie algebras. Reductive groups over an arbitrary field are harder to classify, but for many fields such as the real numbers R or a number field, the classification is well understood. The classification of finite simple groups says that most finite simple groups arise as the group G(k) of k-rational points of a simple algebraic group G over a finite field k, or as minor variants of that construction. Reductive groups have a rich representation theory in various contexts. First, one can study the representations of a reductive group G over a field k as an algebraic group, which are actions of G on k-vector spaces. But also, one can study the complex representations of the group G(k) when k is a finite field, or the infinite-dimensional unitary representations of a real reductive group, or the automorphic representations of an adelic algebraic group. The structure theory of reductive groups is used in all these areas. Linear algebraic group A linear algebraic group over a field k is defined as a smooth closed subgroup scheme of GL(n) over k, for some positive integer n. Equivalently, a linear algebraic group over k is a smooth affine group scheme over k.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (37)

Related people (3)

Related concepts (69)

Related courses (129)

Related lectures (939)

Related MOOCs (13)

MATH-506: Topology IV.b - cohomology rings

Singular cohomology is defined by dualizing the singular chain complex for spaces. We will study its basic properties, see how it acquires a multiplicative structure and becomes a graded commutative a

MATH-616: Numerical methods for random PDEs and uncertainty

The course focuses on mathematical models based on PDEs with random parameters, and presents numerical techniques for forward uncertainty propagation, inverse uncertainty analysis in a Bayesian framew

PHYS-314: Quantum physics II

L'objectif de ce cours est de familiariser l'étudiant avec les concepts, les méthodes et les conséquences de la physique quantique. En particulier, le moment cinétique, la théorie de perturbation, les

Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication).

Algebraic torus

In mathematics, an algebraic torus, where a one dimensional torus is typically denoted by , , or , is a type of commutative affine algebraic group commonly found in projective algebraic geometry and toric geometry. Higher dimensional algebraic tori can be modelled as a product of algebraic groups . These groups were named by analogy with the theory of tori in Lie group theory (see Cartan subgroup). For example, over the complex numbers the algebraic torus is isomorphic to the group scheme , which is the scheme theoretic analogue of the Lie group .

Reductive group

Coxeter Combinatorics For Sum Formulas In The Representation Theory Of Algebraic Groups

Jonathan Gruber

Let G be a simple algebraic group over an algebraically closed field F of characteristic p >= h, the Coxeter number of G. We observe an easy 'recursion formula' for computing the Jantzen sum formula o

AMER MATHEMATICAL SOC2022

Multiplicity-free representations of algebraic groups II

Donna Testerman, Martin W. Liebeck

We continue our work, started in [9], on the program of classifying triples (X, Y, V), where X, Yare simple algebraic groups over an algebraically closed field of characteristic zero with X < Y, and V

ACADEMIC PRESS INC ELSEVIER SCIENCE2022

Generic direct summands of tensor products for simple algebraic groups and quantum groups at roots of unity

Jonathan Gruber

Let G be either a simple linear algebraic group over an algebraically closed field of characteristic l>0 or a quantum group at an l-th root of unity. The category Rep(G) of finite-dimensional G-module

EPFL2022

Explores the classification of extensions in group theory, emphasizing split extensions and semi-direct products.

Explores irreducible representations of the rotations group and their mathematical properties.

Explores the support services and environment of Moodle at EPFL, including group modes, admin tools, course management, and user interactions.

Un MOOC francophone d'algèbre linéaire accessible à tous, enseigné de manière rigoureuse et ne nécessitant aucun prérequis.