In mathematics, a linear algebraic group is a subgroup of the group of invertible matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation where is the transpose of . Many Lie groups can be viewed as linear algebraic groups over the field of real or complex numbers. (For example, every compact Lie group can be regarded as a linear algebraic group over R (necessarily R-anisotropic and reductive), as can many noncompact groups such as the simple Lie group SL(n,R).) The simple Lie groups were classified by Wilhelm Killing and Élie Cartan in the 1880s and 1890s. At that time, no special use was made of the fact that the group structure can be defined by polynomials, that is, that these are algebraic groups. The founders of the theory of algebraic groups include Maurer, Chevalley, and . In the 1950s, Armand Borel constructed much of the theory of algebraic groups as it exists today. One of the first uses for the theory was to define the Chevalley groups. For a positive integer , the general linear group over a field , consisting of all invertible matrices, is a linear algebraic group over . It contains the subgroups consisting of matrices of the form, resp., and . The group is an example of a unipotent linear algebraic group, the group is an example of a solvable algebraic group called the Borel subgroup of . It is a consequence of the Lie-Kolchin theorem that any connected solvable subgroup of is conjugated into . Any unipotent subgroup can be conjugated into . Another algebraic subgroup of is the special linear group of matrices with determinant 1. The group is called the multiplicative group, usually denoted by . The group of -points is the multiplicative group of nonzero elements of the field . The additive group , whose -points are isomorphic to the additive group of , can also be expressed as a matrix group, for example as the subgroup in : These two basic examples of commutative linear algebraic groups, the multiplicative and additive groups, behave very differently in terms of their linear representations (as algebraic groups).

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.
Ontological neighbourhood
Related courses (33)
MATH-492: Representation theory of semisimple lie algebras
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.
MATH-111(a): Linear Algebra
L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et ses applications.
MATH-314: Representation theory I - finite groups
This is a standard course in representation theory of finite groups.
Show more
Related lectures (357)
Convex Optimization: Convex Functions
Covers the concept of convex functions and their applications in optimization problems.
Linear Algebra: Course Information
Provides details about Linear Algebra course logistics, exercise sessions, and exam information.
Determinants: Alternate Forms and Symmetrization
Explores alternate forms, symmetric forms, and determinants in linear algebra.
Show more
Related publications (126)

Multiplicity-free Representations of Algebraic Groups

Donna Testerman, Martin W. Liebeck

Let K be an algebraically closed field of characteristic zero, and let G be a connected reductive algebraic group over K. We address the problem of classifying triples (G, H, V ), where H is a proper connected subgroup of G, and V is a finitedimensional ir ...
Amer Mathematical Soc2024

The ABCD of topological recursion

Nicolas Gerson Orantin

Kontsevich and Soibelman reformulated and slightly generalised the topological recursion of [43], seeing it as a quantisation of certain quadratic Lagrangians in T*V for some vector space V. KS topological recursion is a procedure which takes as initial da ...
San Diego2024

Unlabeled Principal Component Analysis and Matrix Completion

Yunzhen Yao, Liangzu Peng

We introduce robust principal component analysis from a data matrix in which the entries of its columns have been corrupted by permutations, termed Unlabeled Principal Component Analysis (UPCA). Using algebraic geometry, we establish that UPCA is a well-de ...
Microtome Publ2024
Show more
Related concepts (28)
Reductive group
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).
Claude Chevalley
Claude Chevalley (ʃəvalɛ; 11 February 1909 – 28 June 1984) was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory and the theory of algebraic groups. He was a founding member of the Bourbaki group. His father, Abel Chevalley, was a French diplomat who, jointly with his wife Marguerite Chevalley née Sabatier, wrote The Concise Oxford French Dictionary. Chevalley graduated from the École Normale Supérieure in 1929, where he studied under Émile Picard.
Generalized flag variety
In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space V over a field F. When F is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a real or complex flag manifold. Flag varieties are naturally projective varieties. Flag varieties can be defined in various degrees of generality. A prototype is the variety of complete flags in a vector space V over a field F, which is a flag variety for the special linear group over F.
Show more
Related MOOCs (9)
Algebra (part 1)
Un MOOC francophone d'algèbre linéaire accessible à tous, enseigné de manière rigoureuse et ne nécessitant aucun prérequis.
Algebra (part 1)
Un MOOC francophone d'algèbre linéaire accessible à tous, enseigné de manière rigoureuse et ne nécessitant aucun prérequis.
Algebra (part 2)
Un MOOC francophone d'algèbre linéaire accessible à tous, enseigné de manière rigoureuse et ne nécessitant aucun prérequis.
Show more

Graph Chatbot

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.