Algebraic groups

Algebraic groups are a central concept in an area of mathematics that blends algebra, geometry, and number theory. An algebraic group is defined as a group that is also an algebraic variety, meaning that its group operations (multiplication and inversion) can be described by polynomial equations. More formally, an algebraic group is a set that satisfies the group axioms (associativity, identity, and inverses) and is also equipped with a structure of an algebraic variety.

An Abelian variety is a special type of algebraic variety that is defined over a field, typically the field of complex numbers or a finite field. They have a number of important properties that make them central to the study of algebraic geometry and number theory. Here are some key characteristics and definitions related to Abelian varieties: 1. **Group Structure**: An Abelian variety is not just a geometric object; it has a natural structure that turns it into a group.

Algebraic homogeneous spaces are mathematical structures that arise in the context of algebraic geometry and representation theory. More specifically, they are typically associated with algebraic groups and their actions on varieties. ### Definition An **algebraic homogeneous space** can be defined in the following way: 1. **Algebraic Group**: Let \( G \) be an algebraic group defined over an algebraically closed field (like the field of complex numbers).

E8 is a highly complex and deeply interesting mathematical structure that appears in various areas, including geometry, algebra, and theoretical physics. It is most commonly referred to in the context of group theory and is one of the five exceptional simple Lie groups. Here are some key points about E8: 1. **Lie Group**: E8 is one of the simplest types of continuous symmetry groups, known as a Lie group. Simple Lie groups are those that cannot be decomposed into smaller, simpler groups.

Exceptional Lie algebras are a special class of Lie algebras that are neither classical nor affine. They are characterized by their exceptional properties, most notably their dimension and the structure of their root systems. Unlike the classical Lie algebras (which include types A, B, C, D corresponding to the classical groups, and E, F, G corresponding to exceptional types), the exceptional Lie algebras cannot be directly described in terms of standard matrix groups.

Linear algebraic groups are a fundamental concept in the field of algebraic geometry that connect algebraic groups and linear algebra. More specifically, a linear algebraic group is a group that is also an algebraic variety, where the group operations (multiplication and inversion) are given by polynomial functions.

Representation theory of algebraic groups is a branch of mathematics that studies how algebraic groups can act on vector spaces through linear transformations. More specifically, it examines the ways in which algebraic groups can be represented as groups of matrices, and how these representations can be understood and classified. ### Key Concepts: 1. **Algebraic Groups**: These are groups that have a structure of algebraic varieties.

An **adelic algebraic group** is a concept that arises in the context of algebraic groups and number theory, particularly in the study of rational points and arithmetic geometry. To explain it more precisely, we first need to understand what an algebraic group is and then what "adelic" means in this context. ### Algebraic Groups An **algebraic group** is a group that is also an algebraic variety.

An **algebraic group** is a group that is also an algebraic variety, where the group operations (multiplication and taking inverses) are given by polynomial functions. More formally, an algebraic group is a set equipped with a group structure and additional structure that satisfies certain properties of being defined over algebraically closed fields. ### Key Concepts 1. **Algebraic Variety**: An algebraic variety is a geometric object defined as the solution set of a system of polynomial equations.

In the context of algebraic groups, approximation often refers to various ways to understand and study algebraic structures through simpler or more manageable models. The term could encompass different specific concepts depending on the branch of mathematics or the particular problems being addressed.

A Barsotti–Tate group is an important concept in the area of algebraic geometry and representation theory, particularly in the study of p-adic representations and finite field extensions. Named after mathematicians Francesco Barsotti and John Tate, these groups are essentially a kind of p-divisible group that has additional structure, allowing them to be classified and understood in terms of their representation theory.

In the context of algebraic groups and group theory, a **Borel subgroup** is a specific type of subgroup that is particularly important in the study of linear algebraic groups. Here are the key points regarding Borel subgroups: 1. **Definition**: A Borel subgroup of an algebraic group \( G \) is a maximal connected solvable subgroup of \( G \). This means that it cannot be contained in any larger connected solvable subgroup of \( G \).

Borel–de Siebenthal theory is a mathematical framework primarily associated with the study of compact Lie groups and their representations, particularly in the context of algebraic groups and symmetric spaces. The theory deals with the classification of maximal connected solvable subgroups, or Borel subgroups, in the context of semisimple Lie groups. It extends concepts of Borel subgroups from the language of algebraic groups to that of Lie groups.

Bruhat decomposition is a fundamental concept in the theory of Lie groups and algebraic groups, particularly in the study of algebraic varieties and symmetric spaces. It provides a way to decompose a group into pieces that can be analyzed more easily.

Cartier duality is a concept in the field of algebraic geometry and representation theory, particularly related to schemes and étale cohomology. It is named after the mathematician Pierre Cartier. At its core, Cartier duality establishes a relationship between a finite commutative group scheme over a field and its dual group scheme.

Chevalley's structure theorem is a fundamental result in the theory of algebraic groups and linear algebraic groups over algebraically closed fields. It provides a classification of connected algebraic groups over algebraically closed fields in terms of their semi-simple and unipotent parts.

Cohomological invariants are tools used in algebraic topology, algebraic geometry, and related fields to study the properties of topological spaces, algebraic varieties, or other mathematical structures through their cohomology groups. Cohomology provides a way to classify and distinguish topological spaces by associating algebraic invariants to them.

Complexification of a Lie group is a process that involves taking a real Lie group and extending it to a complex Lie group. This technique is useful in many areas of mathematics and theoretical physics because it allows for the application of complex analysis techniques to problems originally framed in the context of real manifolds.

A group is said to be diagonalizable if it can be represented in a certain way with respect to its action on a vector space, particularly in the context of linear algebra. More specifically, in the context of linear representations, a group is diagonalizable when its representation can be expressed in a diagonal form. In this context, consider a group \( G \) acting on a vector space \( V \) over some field, typically the complex numbers.

The Dieudonné module is an important concept in the field of arithmetic geometry, particularly in the study of the formal geometry over fields of positive characteristic, like finite fields. It arises within the context of formal schemes and is closely tied to the theory of p-divisible groups and formal groups.

Differential algebraic groups are mathematical structures that arise in the study of algebraic groups and differential equations. They combine concepts from algebraic geometry and differential geometry, specifically the theory of algebraic groups over differential fields. Here’s a more detailed breakdown of the concept: ### Algebraic Groups An algebraic group is a group that is also an algebraic variety, meaning it can be defined by polynomial equations. The group operations (multiplication and inversion) are also given by regular (i.

In mathematics, E6 refers to a specific complex Lie group, which is part of a classification of simple Lie groups. The E6 group is one of the five exceptional simple Lie groups, and it has applications in various fields, including theoretical physics, particularly in string theory and particle physics. The E6 group is often represented in terms of its root system, which consists of 72 roots in an 8-dimensional vector space.

In mathematics, "E7" typically refers to one of the exceptional Lie groups, which are important in various fields, including algebra, geometry, and theoretical physics. Specifically, E7 is a complex, simple Lie group of rank 7 that can be understood in terms of its root system and algebraic structure.

In mathematics, "F4" can refer to different concepts depending on the context. Here are a couple of potential interpretations: 1. **F_4 (Lie Algebra)**: In the context of Lie algebras, \( \mathfrak{f}_4 \) is one of the five exceptional simple Lie algebras.

In the context of group theory, a fixed-point subgroup refers to the set of elements in a group that remain unchanged under the action of a particular element or a group of elements, typically in the context of a group acting on a set. It's related to the idea of certain symmetries or invariances in that action. More formally, consider a group \( G \) acting on a set \( X \).

In mathematics, "G2" can refer to several concepts depending on the context. Here are a couple of prominent interpretations: 1. **Lie Group G2**: In the context of algebraic and geometric structures, G2 is one of the five exceptional simple Lie groups. It has a dimension of 14 and is associated with a specific type of symmetry.

The Generalized Jacobian is a mathematical concept that extends the idea of the Jacobian matrix, which is primarily used in calculus to describe how a function's output changes in response to small changes in its input. While the traditional Jacobian is applicable to smooth functions, the Generalized Jacobian is particularly useful in the context of nonsmooth analysis and optimization.

Geometric Invariant Theory (GIT) is a branch of algebraic geometry that studies the action of group actions on algebraic varieties, particularly focusing on understanding the properties of orbits and established notions of stability. It was developed primarily in the 1950s by mathematician David Mumford, building on ideas from group theory, algebraic geometry, and representation theory.

The Jacobson–Morozov theorem is a result in the representation theory of Lie algebras, specifically concerning the existence of certain embeddings of semisimple Lie algebras.

The Kazhdan–Margulis theorem is a result in the field of geometry and group theory, specifically concerning the behavior of discrete groups of isometries in the context of hyperbolic geometry. It was formulated by mathematicians David Kazhdan and Gregory Margulis in the 1970s. The theorem primarily addresses the structure of lattices in semi-simple Lie groups, particularly focusing on the behavior of certain types of actions of these groups on homogeneous spaces.

The Kempf vanishing theorem is a result in algebraic geometry that deals with the behavior of sections of certain vector bundles on algebraic varieties, particularly in the context of ample line bundles. Named after G. R. Kempf, the theorem addresses the vanishing of global sections of certain sheaves associated with a variety.

The Kneser–Tits conjecture is a statement in the field of algebraic groups and the theory of group actions, particularly concerning the structure of algebraic groups and their associated buildings. It was proposed by mathematicians Max Kneser and Jacques Tits. The conjecture pertains to the relationship between a certain class of algebraic groups defined over a field and their maximal compact subgroups.

Kostant polynomials are a class of polynomials that arise in the study of Lie algebras, representation theory, and several areas of algebraic geometry. They were introduced by Bertram Kostant in his work on the structure of semisimple Lie algebras and their representations. In particular, Kostant polynomials are closely associated with the weights of representations of a Lie algebra and its root system.

Lang's theorem is a result in the field of algebraic geometry, specifically related to the properties of algebraic curves. It is named after the mathematician Serge Lang. The theorem primarily concerns algebraic curves and their points over various fields, specifically in the context of rational points and rational functions. One important version of Lang's theorem states that a smooth projective curve over a number field has only finitely many rational points unless the curve is of genus zero.

Langlands decomposition is a concept in the context of representation theory of Lie groups, specifically related to the structure of semisimple Lie algebras and their representations.

Tits indices, named after the mathematician Jacques Tits, are a concept in the area of group theory and algebraic groups, particularly in the study of algebraic group representations and the structure of certain algebraic objects. Irreducible Tits indices are used to classify the irreducible representations of a group in relation to the structure of the group and its associated algebraic objects.

The Mumford–Tate group is a concept from algebraic geometry and number theory that arises in the study of abelian varieties and the associated Hodge structures. It is named after mathematicians David Mumford and John Tate. In the context of algebraic geometry, an abelian variety is a projective algebraic variety that has a group structure.

In the context of algebraic groups and representation theory, a pseudo-reductive group is a certain type of algebraic group that generalizes the notion of reductive groups. While reductive groups are well-studied and have nice properties, pseudo-reductive groups allow for a more general framework that still retains many desirable features.

In the context of algebraic groups, the **radical** refers to a specific type of subgroup that is closely related to the structure of the group itself. More formally, the radical of an algebraic group \( G \) is defined as the largest normal solvable subgroup of \( G \). ### Key Concepts: 1. **Algebraic Group**: An algebraic group is a group that is also an algebraic variety, meaning it can be defined by polynomial equations.

A **restricted Lie algebra** is a special type of Lie algebra that comes equipped with a unary operation called the "p-th power" which generalizes the notion of taking powers of elements in the context of Lie algebras. This concept is particularly important in the study of Lie algebras over fields of characteristic \( p \), where \( p \) is a prime number.

In the context of algebraic groups and Lie algebras, a **root datum** is a structured way of encoding certain aspects of the symmetries and properties of these mathematical objects. Specifically, a root datum consists of the following components: 1. **A finite set of roots**: These are usually vectors in a Euclidean space, which can be thought of as directions that reflect the symmetries of the system.

The Rost invariant is an important concept in the field of algebraic groups, particularly in relation to the study of quadratic forms and the theory of \(K\)-theory. It is named after the mathematician Ulrich Rost. The Rost invariant is defined in the context of central simple algebras, specifically those over a field. More concretely, it deals with the classification of certain types of quadratic forms and their behavior under various operations.

In mathematics, particularly in the area of algebraic geometry and number theory, a Serre group generally refers to a certain type of group that is associated with the work of Jean-Pierre Serre, a prominent French mathematician. There are different contexts in which "Serre group" may be used, but one of the more common references involves the concept related to *Serre's conjectures* in the theory of abelian varieties and algebraic groups.

A Severi–Brauer variety, named after the mathematicians Francesco Severi and Hans von Brauer, is a specific type of algebraic variety that is related to the study of division algebras and central simple algebras in algebraic geometry.

In the context of the theory of algebraic groups, particularly in the study of the general linear group \( GL(n, \mathbb{C}) \) or similar groups over other fields, a **Siegel parabolic subgroup** is a particular type of parabolic subgroup that is associated with a certain block upper triangular structure.

A Spaltenstein variety is a specific type of algebraic variety that is studied in the context of representation theory and algebraic geometry, particularly in relation to the study of finite dimensional representations of algebraic groups or algebraic varieties. Spaltenstein varieties arise in the context of the so-called "nilpotent cones." More specifically, they can be associated with certain types of objects called "nilpotent elements" in the representation theory of Lie algebras or algebraic groups.

The Tamagawa number is a concept in the field of number theory, specifically in the study of algebraic groups and arithmetic geometry. It is associated with a connected reductive algebraic group defined over a global field, such as a number field or a function field.

The Taniyama group, named after mathematician Yutaka Taniyama, is a group in the context of number theory that is closely related to the study of elliptic curves and modular forms. It is particularly famous for its connection to the Taniyama-Shimura-Weil conjecture, which posited that every elliptic curve over the rational numbers is associated with a modular form.

A **Torus action** refers to the action of a torus (typically a compact, connected Lie group isomorphic to the product of several circles, denoted as \(T^n\), where \(n\) is the number of circles) on some space, often a manifold. This concept arises in various areas of mathematics, including differential geometry, algebraic geometry, and symplectic geometry.

In the context of representation theory, the "trace field" of a representation typically refers to the field over which the representations of a group or algebra are defined, particularly when considering the trace of endomorphisms associated with the representation.

The Verschiebung operator, also known as the shift operator, is a mathematical operator used in various fields, including quantum mechanics and functional analysis. The term "Verschiebung" is German for "shift," and the operator is typically denoted by \( S \). In the context of quantum mechanics, for example, the shift operator can shift states in a Hilbert space.

Weil's conjecture on Tamagawa numbers is a part of the broader framework concerning algebraic groups and number theory, and specifically relates to the study of algebraic groups over global fields (like number fields or function fields). The conjecture connects the structure of algebraic groups to certain arithmetic invariants known as Tamagawa numbers.

Weyl modules are a family of representations associated with Lie algebras and are particularly important in the representation theory of semisimple Lie algebras. They are named after Hermann Weyl, who made significant contributions to the field of representation theory. ### Definition For a semisimple Lie algebra \(\mathfrak{g}\) over a field, a Weyl module \(V_\lambda\) is constructed for a given dominant integral weight \(\lambda\).

An étale group scheme is a concept from algebraic geometry and the theory of group schemes. It can be understood in the context of scheme theory, which is a branch of mathematics that deals with geometric objects defined by polynomial equations, among other things. ### Group Schemes First, let's break down the term "group scheme." A group scheme is a scheme equipped with a group structure.