Faster access than browser!
66 relations: Abelian group, Abelian variety, Adelic algebraic group, Affine variety, Algebraic closure, Algebraic geometry, Algebraic topology (object), Algebraic torus, Algebraic variety, Borel subgroup, Category (mathematics), Category theory, Character group, Chevalley's structure theorem, Claude Chevalley, Complete variety, Connected space, Covering space, Coxeter group, Derivation (differential algebra), Diagonal matrix, Diagonalizable group, Differential of the first kind, Elliptic curve, Faithful representation, Field (mathematics), Field with one element, Finite group, General linear group, Generalized Jacobian, Glossary of algebraic geometry, Graduate Texts in Mathematics, Group (mathematics), Group object, Group scheme, Hopf algebra, Identity component, Index of a subgroup, Invertible matrix, Jet group, Lie algebra, Lie group, Linear algebraic group, Mathematics, Matrix group, Morley rank, Morphism of algebraic varieties, Multiplicative group, Nilpotent group, Oxford University Press, ..., Perfect field, Pseudo-reductive group, Q-Pochhammer symbol, Radical of an algebraic group, Reductive group, Representation theory, Scheme (mathematics), Special linear group, Springer Science+Business Media, Stable group, Subgroup, Tame group, Tangent space, Unipotent, Weierstrass functions, Zariski topology. Expand index (16 more) »
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.
New!!: Algebraic group and Abelian group · See more »
Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions.
New!!: Algebraic group and Abelian variety · See more »
Adelic algebraic group
In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group G over a number field K, and the adele ring A.
New!!: Algebraic group and Adelic algebraic group · See more »
Affine variety
In algebraic geometry, an affine variety over an algebraically closed field k is the zero-locus in the affine ''n''-space k^n of some finite family of polynomials of n variables with coefficients in k that generate a prime ideal.
New!!: Algebraic group and Affine variety · See more »
Algebraic closure
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed.
New!!: Algebraic group and Algebraic closure · See more »
Algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.
New!!: Algebraic group and Algebraic geometry · See more »
Algebraic topology (object)
In mathematics, the algebraic topology on the set of group representations from G to a topological group H is the topology of pointwise convergence, i.e. pi converges to p if the limit of pi(g).
New!!: Algebraic group and Algebraic topology (object) · See more »
Algebraic torus
In mathematics, an algebraic torus is a type of commutative affine algebraic group.
New!!: Algebraic group and Algebraic torus · See more »
Algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry.
New!!: Algebraic group and Algebraic variety · See more »
Borel subgroup
In the theory of algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup.
New!!: Algebraic group and Borel subgroup · See more »
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is an algebraic structure similar to a group but without requiring inverse or closure properties.
New!!: Algebraic group and Category (mathematics) · See more »
Category theory
Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).
New!!: Algebraic group and Category theory · See more »
Character group
In mathematics, a character group is the group of representations of a group by complex-valued functions.
New!!: Algebraic group and Character group · See more »
Chevalley's structure theorem
In algebraic geometry, Chevalley's structure theorem states that a smooth connected algebraic group over a perfect field has a unique normal smooth connected affine algebraic subgroup such that the quotient is an abelian variety.
New!!: Algebraic group and Chevalley's structure theorem · See more »
Claude Chevalley
Claude Chevalley (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.
New!!: Algebraic group and Claude Chevalley · See more »
Complete variety
In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety X, such that for any variety Y the projection morphism is a closed map, i.e. maps closed sets onto closed sets.
New!!: Algebraic group and Complete variety · See more »
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets.
New!!: Algebraic group and Connected space · See more »
Covering space
In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below.
New!!: Algebraic group and Covering space · See more »
Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors).
New!!: Algebraic group and Coxeter group · See more »
Derivation (differential algebra)
In mathematics, a derivation is a function on an algebra which generalizes certain features of the derivative operator.
New!!: Algebraic group and Derivation (differential algebra) · See more »
Diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero.
New!!: Algebraic group and Diagonal matrix · See more »
Diagonalizable group
In mathematics, an affine algebraic group is said to be diagonalizable if it is isomorphic to a subgroup of Dn, the group of diagonal matrices.
New!!: Algebraic group and Diagonalizable group · See more »
Differential of the first kind
In mathematics, differential of the first kind is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic curves (more generally, algebraic varieties), for everywhere-regular differential 1-forms.
New!!: Algebraic group and Differential of the first kind · See more »
Elliptic curve
In mathematics, an elliptic curve is a plane algebraic curve defined by an equation of the form which is non-singular; that is, the curve has no cusps or self-intersections.
New!!: Algebraic group and Elliptic curve · See more »
Faithful representation
In mathematics, especially in an area of abstract algebra known as representation theory, a faithful representation ρ of a group G on a vector space V is a linear representation in which different elements g of G are represented by distinct linear mappings \rho(g).
New!!: Algebraic group and Faithful representation · See more »
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.
New!!: Algebraic group and Field (mathematics) · See more »
Field with one element
In mathematics, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist.
New!!: Algebraic group and Field with one element · See more »
Finite group
In abstract algebra, a finite group is a mathematical group with a finite number of elements.
New!!: Algebraic group and Finite group · See more »
General linear group
In mathematics, the general linear group of degree n is the set of invertible matrices, together with the operation of ordinary matrix multiplication.
New!!: Algebraic group and General linear group · See more »
Generalized Jacobian
In algebraic geometry a generalized Jacobian is a commutative algebraic group associated to a curve with a divisor, generalizing the Jacobian variety of a complete curve.
New!!: Algebraic group and Generalized Jacobian · See more »
Glossary of algebraic geometry
This is a glossary of algebraic geometry.
New!!: Algebraic group and Glossary of algebraic geometry · See more »
Graduate Texts in Mathematics
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag.
New!!: Algebraic group and Graduate Texts in Mathematics · See more »
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.
New!!: Algebraic group and Group (mathematics) · See more »
Group object
In category theory, a branch of mathematics, group objects are certain generalizations of groups which are built on more complicated structures than sets.
New!!: Algebraic group and Group object · See more »
Group scheme
In mathematics, a group scheme is a type of algebro-geometric object equipped with a composition law.
New!!: Algebraic group and Group scheme · See more »
Hopf algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an (unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property.
New!!: Algebraic group and Hopf algebra · See more »
Identity component
In mathematics, the identity component of a topological group G is the connected component G0 of G that contains the identity element of the group.
New!!: Algebraic group and Identity component · See more »
Index of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" (cosets) of H that fill up G. For example, if H has index 2 in G, then intuitively half of the elements of G lie in H. The index of H in G is usually denoted |G: H| or or (G:H).
New!!: Algebraic group and Index of a subgroup · See more »
Invertible matrix
In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular or nondegenerate) if there exists an n-by-n square matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication.
New!!: Algebraic group and Invertible matrix · See more »
Jet group
In mathematics, a jet group is a generalization of the general linear group which applies to Taylor polynomials instead of vectors at a point.
New!!: Algebraic group and Jet group · See more »
Lie algebra
In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.
New!!: Algebraic group and Lie algebra · See more »
Lie group
In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.
New!!: Algebraic group and Lie group · See more »
Linear algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices (under matrix multiplication) that is defined by polynomial equations.
New!!: Algebraic group and Linear algebraic group · See more »
Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
New!!: Algebraic group and Mathematics · See more »
Matrix group
In mathematics, a matrix group is a group G consisting of invertible matrices over a specified field K, with the operation of matrix multiplication.
New!!: Algebraic group and Matrix group · See more »
Morley rank
In mathematical logic, Morley rank, introduced by, is a means of measuring the size of a subset of a model of a theory, generalizing the notion of dimension in algebraic geometry.
New!!: Algebraic group and Morley rank · See more »
Morphism of algebraic varieties
In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials.
New!!: Algebraic group and Morphism of algebraic varieties · See more »
Multiplicative group
In mathematics and group theory, the term multiplicative group refers to one of the following concepts.
New!!: Algebraic group and Multiplicative group · See more »
Nilpotent group
A nilpotent group G is a group that has an upper central series that terminates with G. Provably equivalent definitions include a group that has a central series of finite length or a lower central series that terminates with.
New!!: Algebraic group and Nilpotent group · See more »
Oxford University Press
Oxford University Press (OUP) is the largest university press in the world, and the second oldest after Cambridge University Press.
New!!: Algebraic group and Oxford University Press · See more »
Perfect field
In algebra, a field k is said to be perfect if any one of the following equivalent conditions holds.
New!!: Algebraic group and Perfect field · See more »
Pseudo-reductive group
In mathematics, a pseudo-reductive group over a field k (sometimes called a k-reductive group) is a smooth connected affine algebraic group defined over k whose k-unipotent radical (i.e., largest smooth connected unipotent normal k-subgroup) is trivial.
New!!: Algebraic group and Pseudo-reductive group · See more »
Q-Pochhammer symbol
In mathematics, in the area of combinatorics, a q-Pochhammer symbol, also called a q-shifted factorial, is a ''q''-analog of the Pochhammer symbol.
New!!: Algebraic group and Q-Pochhammer symbol · See more »
Radical of an algebraic group
The radical of an algebraic group is the identity component of its maximal normal solvable subgroup.
New!!: Algebraic group and Radical of an algebraic group · See more »
Reductive group
In mathematics, a reductive group is a type of linear algebraic group over a field.
New!!: Algebraic group and Reductive group · See more »
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.
New!!: Algebraic group and Representation theory · See more »
Scheme (mathematics)
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x.
New!!: Algebraic group and Scheme (mathematics) · See more »
Special linear group
In mathematics, the special linear group of degree n over a field F is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion.
New!!: Algebraic group and Special linear group · See more »
Springer Science+Business Media
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
New!!: Algebraic group and Springer Science+Business Media · See more »
Stable group
In model theory, a stable group is a group that is stable in the sense of stability theory.
New!!: Algebraic group and Stable group · See more »
Subgroup
In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.
New!!: Algebraic group and Subgroup · See more »
Tame group
In mathematical group theory, a tame group is a certain kind of group defined in model theory.
New!!: Algebraic group and Tame group · See more »
Tangent space
In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector that gives the displacement of the one point from the other.
New!!: Algebraic group and Tangent space · See more »
Unipotent
In mathematics, a unipotent element r of a ring R is one such that r − 1 is a nilpotent element; in other words, (r − 1)n is zero for some n. In particular, a square matrix, M, is a unipotent matrix, if and only if its characteristic polynomial, P(t), is a power of t − 1.
New!!: Algebraic group and Unipotent · See more »
Weierstrass functions
In mathematics, the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function.
New!!: Algebraic group and Weierstrass functions · See more »
Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology on algebraic varieties, introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring a topological space, called the spectrum of the ring.
New!!: Algebraic group and Zariski topology · See more »
Redirects here:
Algebraic Group, Algebraic groups, Algebraic subgroup, Glossary of algebraic groups, Group variety.
References
[1] https://en.wikipedia.org/wiki/Algebraic_group
