Algebraic Groups: Difference between revisions
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In the field of [[Mathematics|mathematics]], algebraic groups are a central object of study in [[Algebraic Geometry|algebraic geometry]], with deep connections to [[Number Theory|number theory]], [[Representation Theory|representation theory]], and [[Homological Algebra|homological algebra]]. These groups are defined by [[Polynomial|polynomial]] equations and can be understood as a generalization of various types of classical groups such as [[Linear Algebraic Group|linear algebraic groups]], [[Abelian Variety|abelian varieties]], and [[Lie Group|Lie groups]]. | In the field of [[Mathematics|mathematics]], algebraic groups are a central object of study in [[Algebraic Geometry|algebraic geometry]], with deep connections to [[Number Theory|number theory]], [[Representation Theory|representation theory]], and [[Homological Algebra|homological algebra]]. These groups are defined by [[Polynomial|polynomial]] equations and can be understood as a generalization of various types of classical groups such as [[Linear Algebraic Group|linear algebraic groups]], [[Abelian Variety|abelian varieties]], and [[Lie Group|Lie groups]]. | ||
[[Image:Detail-77817.jpg|thumb|center|A group of mathematicians discussing and writing formulas related to algebraic groups on a chalkboard.]] | [[Image:Detail-77817.jpg|thumb|center|A group of mathematicians discussing and writing formulas related to algebraic groups on a chalkboard.|class=only_on_mobile]] | ||
[[Image:Detail-77818.jpg|thumb|center|A group of mathematicians discussing and writing formulas related to algebraic groups on a chalkboard.|class=only_on_desktop]] | |||
== Definition and Basic Properties == | == Definition and Basic Properties == | ||
Latest revision as of 22:49, 7 May 2024
Introduction
In the field of mathematics, algebraic groups are a central object of study in algebraic geometry, with deep connections to number theory, representation theory, and homological algebra. These groups are defined by polynomial equations and can be understood as a generalization of various types of classical groups such as linear algebraic groups, abelian varieties, and Lie groups.
Definition and Basic Properties
An algebraic group is a group that is an algebraic variety, such that the multiplication and inversion operations are given by polynomial maps. In other words, for an algebraic group G, the map G x G → G, (g, h) → gh, and the map G → G, g → g−1, are morphisms of algebraic varieties.
Structure of Algebraic Groups
The structure of algebraic groups can be studied using various approaches, including through their Lie algebras, tori, and root systems. These structures provide a rich and intricate theory, leading to classification results for certain types of algebraic groups.
Lie Algebras
The Lie algebra of an algebraic group is a fundamental tool in understanding its structure. It is a vector space equipped with a binary operation called the Lie bracket, which captures the local structure of the group near the identity element.
Tori
A torus in an algebraic group is a maximal commutative subgroup. The structure of a torus is relatively simple, and they play a crucial role in the study of the structure of algebraic groups.
Root Systems
Root systems are a tool used to study the structure of algebraic groups, particularly semisimple and reductive groups. They provide a combinatorial way to understand the group and its representations.
Classification of Algebraic Groups
The classification of algebraic groups is a major goal in the theory of these objects. The most complete results are for linear algebraic groups, which can be classified up to isomorphism by their root data.
Linear Algebraic Groups
Linear algebraic groups are those that can be embedded into the general linear group GL(n) for some n. These groups have been completely classified, with the main families being the classical groups, the exceptional groups, and the tori.
Abelian Varieties
Abelian varieties are a special class of algebraic groups, which are projective and commutative. They have been classified in terms of their dimension and polarization.
Lie Groups
Lie groups are a class of algebraic groups that are also smooth manifolds. They have been classified in terms of their Lie algebras.
Applications of Algebraic Groups
Algebraic groups have found numerous applications in various areas of mathematics. They play a central role in the theory of algebraic numbers, algebraic geometry, and representation theory.