# Category: Algebraic groups

Rost invariant
In mathematics, the Rost invariant is a cohomological invariant of an absolutely simple simply connected algebraic group G over a field k, which associates an element of the Galois cohomology group H3
Tamagawa number
In mathematics, the Tamagawa number of a semisimple algebraic group defined over a global field k is the measure of , where is the adele ring of k. Tamagawa numbers were introduced by Tamagawa, and na
G2 (mathematics)
In mathematics, G2 is the name of three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras as well as some algebraic groups. They are the smallest of the
Geometric invariant theory
In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in
Fundamental lemma (Langlands program)
In the mathematical theory of automorphic forms, the fundamental lemma relates orbital integrals on a reductive group over a local field to stable orbital integrals on its endoscopic groups. It was co
Restricted Lie algebra
In mathematics, a restricted Lie algebra is a Lie algebra together with an additional "p operation."
F4 (mathematics)
In mathematics, F4 is the name of a Lie group and also its Lie algebra f4. It is one of the five exceptional simple Lie groups. F4 has rank 4 and dimension 52. The compact form is simply connected and
Lazard's universal ring
In mathematics, Lazard's universal ring is a ring introduced by Michel Lazard in over which the universal commutative one-dimensional formal group law is defined. There is a universal commutative one-
Lattice (discrete subgroup)
In Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure. In the special case
Borel–de Siebenthal theory
In mathematics, Borel–de Siebenthal theory describes the closed connected subgroups of a compact Lie group that have maximal rank, i.e. contain a maximal torus. It is named after the Swiss mathematici
Mumford–Tate group
In algebraic geometry, the Mumford–Tate group (or Hodge group) MT(F) constructed from a Hodge structure F is a certain algebraic group G. When F is given by a rational representation of an algebraic t
Severi–Brauer variety
In mathematics, a Severi–Brauer variety over a field K is an algebraic variety V which becomes isomorphic to a projective space over an algebraic closure of K. The varieties are associated to central
E7 (mathematics)
In mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133; the same notation E7 is used for the correspond
Bruhat decomposition
In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by Claude Chevalley in general) G = BWB of certain algebraic groups G into cells can be regarded as a g
Trace field of a representation
In mathematics, the trace field of a linear group is the field generated by the traces of its elements. It is mostly studied for Kleinian and Fuchsian groups, though related objects are used in the th
Root datum
In mathematical group theory, the root datum of a connected split reductive algebraic group over a field is a generalization of a root system that determines the group up to isomorphism. They were int
Étale group scheme
In mathematics, more precisely in algebra, an étale group scheme is a certain kind of group scheme.
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. For example, in the general linear group GLn (n x n i
Group of Lie type
In mathematics, specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic grou
Jordan–Chevalley decomposition
In mathematics, the Jordan–Chevalley decomposition, named after Camille Jordan and Claude Chevalley, expresses a linear operator as the sum of its commuting semisimple part and its nilpotent part. The
Kempf vanishing theorem
In algebraic geometry, the Kempf vanishing theorem, introduced by Kempf, states that the higher cohomology group Hi(G/B,L(λ)) (i > 0) vanishes whenever λ is a dominant weight of B. Here G is a reducti
The radical of an algebraic group is the identity component of its maximal normal solvable subgroup.For example, the radical of the general linear group (for a field K) is the subgroup consisting of s
Kostant polynomial
In mathematics, the Kostant polynomials, named after Bertram Kostant, provide an explicit basis of the ring of polynomials over the ring of polynomials invariant under the finite reflection group of a
Siegel parabolic subgroup
In mathematics, the Siegel parabolic subgroup, named after Carl Ludwig Siegel, is the parabolic subgroup of the symplectic group with abelian radical, given by the matrices of the symplectic group who
Verschiebung operator
In mathematics, the Verschiebung or Verschiebung operator V is a homomorphism between affine commutative group schemes over a field of nonzero characteristic p. For finite group schemes it is the Cart
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 qu
Jacobson–Morozov theorem
In mathematics, the Jacobson–Morozov theorem is the assertion that nilpotent elements in a semi-simple Lie algebra can be extended to sl2-triples. The theorem is named after , .
Weil's conjecture on Tamagawa numbers
In mathematics, the Weil conjecture on Tamagawa numbers is the statement that the Tamagawa number of a simply connected simple algebraic group defined over a number field is 1. In this case, simply co
Formal group law
In mathematics, a formal group law is (roughly speaking) a formal power series behaving as if it were the product of a Lie group. They were introduced by S. Bochner. The term formal group sometimes me
Dieudonné module
In mathematics, a Dieudonné module introduced by Jean Dieudonné , is a module over the non-commutative Dieudonné ring, which is generated over the ring of Witt vectors by two special endomorphisms and
Spaltenstein variety
In algebraic geometry, a Spaltenstein variety is a variety given by the fixed point set of a nilpotent transformation on a flag variety. They were introduced by Nicolas Spaltenstein . In the special c
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 = A(K) of K. It consists of the points of G having
Arason invariant
In mathematics, the Arason invariant is a cohomological invariant associated to a quadratic form of even rank and trivial discriminant and Clifford invariant over a field k of characteristic not 2, ta
Hochschild–Mostow group
In mathematics, the Hochschild–Mostow group, introduced by Hochschild and Mostow, is the universal pro-affine algebraic group generated by a group.
Springer resolution
In mathematics, the Springer resolution is a resolution of the variety of nilpotent elements in a semisimple Lie algebra, or the unipotent elements of a reductive algebraic group, introduced by Tonny
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
Approximation in algebraic groups
In algebraic group theory, approximation theorems are an extension of the Chinese remainder theorem to algebraic groups G over global fields k.
Differential algebraic group
In mathematics, a differential algebraic group is a differential algebraic variety with a compatible group structure. Differential algebraic groups were introduced by .
Differential Galois theory
In mathematics, differential Galois theory studies the Galois groups of differential equations.
Tannakian formalism
In mathematics, a Tannakian category is a particular kind of monoidal category C, equipped with some extra structure relative to a given field K. The role of such categories C is to approximate, in so
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 smoo
Superstrong approximation
Superstrong approximation is a generalisation of strong approximation in algebraic groups G, to provide spectral gap results. The spectrum in question is that of the Laplacian matrix associated to a f
Group scheme
In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups
Wonderful compactification
In algebraic group theory, a wonderful compactification of a variety acted on by an algebraic group is a -equivariant compactification such that the closure of each orbit is smooth. Corrado de Concini
Witt vector
In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt v
Cohomological invariant
In mathematics, a cohomological invariant of an algebraic group G over a field is an invariant of forms of G taking values in a Galois cohomology group.
Weyl module
In algebra, a Weyl module is a representation of a reductive algebraic group, introduced by Carter and Lusztig and named after Hermann Weyl. In characteristic 0 these representations are irreducible,
Serre group
In mathematics, the Serre group S is the pro-algebraic group whose representations correspond to CM-motives over the algebraic closure of the rationals, or to polarizable rational Hodge structures wit
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. They were introduced by Maxwe
Complexification (Lie group)
In mathematics, the complexification or universal complexification of a real Lie group is given by a continuous homomorphism of the group into a complex Lie group with the universal property that ever
Kazhdan–Margulis theorem
In Lie theory, an area of mathematics, the Kazhdan–Margulis theorem is a statement asserting that a discrete subgroup in semisimple Lie groups cannot be too dense in the group. More precisely, in any
Cartier duality
In mathematics,Cartier duality is an analogue of Pontryagin duality for commutative group schemes. It was introduced by Pierre Cartier.
Barsotti–Tate group
In algebraic geometry, Barsotti–Tate groups or p-divisible groups are similar to the points of order a power of p on an abelian variety in characteristic p. They were introduced by Barsotti under the
E6 (mathematics)
In mathematics, E6 is the name of some closely related Lie groups, linear algebraic groups or their Lie algebras , all of which have dimension 78; the same notation E6 is used for the corresponding ro
Langlands decomposition
In mathematics, the Langlands decomposition writes a parabolic subgroup P of a semisimple Lie group as a product of a reductive subgroup M, an abelian subgroup A, and a nilpotent subgroup N.
List of irreducible Tits indices
In the mathematical theory of linear algebraic groups, a Tits index (or index) is an object used to classify semisimple algebraic groups defined over a base field k, not assumed to be algebraically cl
Kneser–Tits conjecture
In mathematics, the Kneser–Tits problem, introduced by Tits based on a suggestion by Martin Kneser, asks whether the Whitehead group W(G,K) of a semisimple simply connected isotropic algebraic group G
Algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs bot
Arithmetic group
In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example They arise naturally in the study of arithmetic properties of quadratic forms and other
Lang's theorem
In algebraic geometry, Lang's theorem, introduced by Serge Lang, states: if G is a connected smooth algebraic group over a finite field , then, writing for the Frobenius, the morphism of varieties is
Taniyama group
In mathematics, the Taniyama group is a group that is an extension of the absolute Galois group of the rationals by the Serre group. It was introduced by Langlands using an observation by Deligne, and
Torus action
In algebraic geometry, a torus action on an algebraic variety is a group action of an algebraic torus on the variety. A variety equipped with an action of a torus T is called a T-variety. In different
Kazhdan–Lusztig polynomial
In the mathematical field of representation theory, a Kazhdan–Lusztig polynomial is a member of a family of integral polynomials introduced by David Kazhdan and George Lusztig. They are indexed by pai
Inner form
In mathematics, an inner form of an algebraic group over a field is another algebraic group such that there exists an isomorphism between and defined over (this means that is a -form of ) and in addit
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. A diagonalizable group defined over a field k is said to
(B, N) pair
In mathematics, a (B, N) pair is a structure on groups of Lie type that allows one to give uniform proofs of many results, instead of giving a large number of case-by-case proofs. Roughly speaking, it
Fixed-point subgroup
In algebra, the fixed-point subgroup of an automorphism f of a group G is the subgroup of G: More generally, if S is a set of automorphisms of G (i.e., a subset of the automorphism group of G), then t