Linear Algebraic Group
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In mathematics, a linear algebraic group is a
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 ∗. More precisely, ''H'' is a subgrou ...
of the group of
invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
n\times n matrices (under
matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the ...
) that is defined by
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
equations. An example is the
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
, defined by the relation M^TM = I_n where M^T is the
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
of M. Many
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...
s 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 In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symme ...
SL(''n'',R).) The simple Lie groups were classified by Wilhelm Killing and
Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry ...
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 Maurer is a German surname, translating in English to "bricklayer" or "wall builder." Notable people with the surname include: *Adrian Maurer (1901–1943), American football player *Alfred Maurer (politician) (1888–1954), Estonian politician * ...
,
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. He was a foundin ...
, and . In the 1950s,
Armand Borel Armand Borel (21 May 1923 – 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in ...
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.


Examples

For a
positive integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
n, the
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. This forms a group, because the product of two invertible matrices is again invertible ...
GL(n) over a field k, consisting of all invertible n\times n matrices, is a linear algebraic group over k. It contains the subgroups :U \subset B \subset GL(n) consisting of matrices of the form, resp., :\left ( \begin 1 & * & \dots & * \\ 0 & 1 & \ddots & \vdots \\ \vdots & \ddots & \ddots & * \\ 0 & \dots & 0 & 1\end \right ) and \left ( \begin * & * & \dots & * \\ 0 & * & \ddots & \vdots \\ \vdots & \ddots & \ddots & * \\ 0 & \dots & 0 & *\end \right ). The group U is an example of a
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 unipo ...
linear algebraic group, the group B is an example of a solvable algebraic group called the
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'' invertible matrices), the subgrou ...
of GL(n). It is a consequence of the Lie-Kolchin theorem that any connected solvable subgroup of \mathrm(n) is conjugated into B. Any unipotent subgroup can be conjugated into U. Another algebraic subgroup of \mathrm(n) is the
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. This is the normal subgroup of the gen ...
\mathrm(n) of matrices with determinant 1. The group \mathrm(1) is called the
multiplicative group In mathematics and group theory, the term multiplicative group refers to one of the following concepts: *the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
, usually denoted by \mathbf G_. The group of k-points \mathbf G_(k) is the multiplicative group k^* of nonzero elements of the field k. The additive group \mathbf G_, whose k-points are isomorphic to the additive group of k, can also be expressed as a matrix group, for example as the subgroup U in \mathrm(2) : :\begin 1 & * \\ 0 & 1 \end. 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). Every representation of the multiplicative group \mathbf G_ is a
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mo ...
of irreducible representations. (Its irreducible representations all have dimension 1, of the form x \mapsto x^n for an integer n.) By contrast, the only irreducible representation of the additive group \mathbf G_ is the trivial representation. So every representation of \mathbf G_ (such as the 2-dimensional representation above) is an iterated extension of trivial representations, not a direct sum (unless the representation is trivial). The structure theory of linear algebraic groups analyzes any linear algebraic group in terms of these two basic groups and their generalizations, tori and unipotent groups, as discussed below.


Definitions

For an algebraically closed field ''k'', much of the structure of an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers ...
''X'' over ''k'' is encoded in its set ''X''(''k'') of ''k''-
rational point In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the fie ...
s, which allows an elementary definition of a linear algebraic group. First, define a function from the abstract group ''GL''(''n'',''k'') to ''k'' to be regular if it can be written as a polynomial in the entries of an ''n''×''n'' matrix ''A'' and in 1/det(''A''), where det is the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if ...
. Then a linear algebraic group ''G'' over an algebraically closed field ''k'' is a subgroup ''G''(''k'') of the abstract group ''GL''(''n'',''k'') for some natural number ''n'' such that ''G''(''k'') is defined by the vanishing of some set of regular functions. For an arbitrary field ''k'', algebraic varieties over ''k'' are defined as a special case of schemes over ''k''. In that language, a linear algebraic group ''G'' over a field ''k'' is a smooth closed subgroup scheme of ''GL''(''n'') over ''k'' for some natural number ''n''. In particular, ''G'' is defined by the vanishing of some set of
regular function In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regul ...
s on ''GL''(''n'') over ''k'', and these functions must have the property that for every commutative ''k''-
algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
''R'', ''G''(''R'') is a subgroup of the abstract group ''GL''(''n'',''R''). (Thus an algebraic group ''G'' over ''k'' is not just the abstract group ''G''(''k''), but rather the whole family of groups ''G''(''R'') for commutative ''k''-algebras ''R''; this is the philosophy of describing a scheme by its
functor of points In algebraic geometry, a functor represented by a scheme ''X'' is a set-valued contravariant functor on the category of schemes such that the value of the functor at each scheme ''S'' is (up to natural bijections) the set of all morphisms S \to X. ...
.) In either language, one has the notion of a
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "sa ...
of linear algebraic groups. For example, when ''k'' is algebraically closed, a homomorphism from ''G'' ⊂ ''GL''(''m'') to ''H'' ⊂ ''GL''(''n'') is a homomorphism of abstract groups ''G''(''k'') → ''H''(''k'') which is defined by regular functions on ''G''. This makes the linear algebraic groups over ''k'' into a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
. In particular, this defines what it means for two linear algebraic groups to be isomorphic. In the language of schemes, a linear algebraic group ''G'' over a field ''k'' is in particular a
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, in the sense that all algebraic groups ha ...
over ''k'', meaning a scheme over ''k'' together with a ''k''-point 1 ∈ ''G''(''k'') and morphisms :m\colon G \times_k G \to G, \; i\colon G \to G over ''k'' which satisfy the usual axioms for the multiplication and inverse maps in a group (associativity, identity, inverses). A linear algebraic group is also smooth and of finite type over ''k'', and it is affine (as a scheme). Conversely, every affine group scheme ''G'' of finite type over a field ''k'' has a
faithful representation In mathematics, especially in an area of abstract algebra known as representation theory, a faithful representation ρ of a group on a vector space is a linear representation in which different elements of are represented by distinct linear ...
into ''GL''(''n'') over ''k'' for some ''n''. An example is the embedding of the additive group ''G''''a'' into ''GL''(2), as mentioned above. As a result, one can think of linear algebraic groups either as matrix groups or, more abstractly, as smooth affine group schemes over a field. (Some authors use "linear algebraic group" to mean any affine group scheme of finite type over a field.) For a full understanding of linear algebraic groups, one has to consider more general (non-smooth) group schemes. For example, let ''k'' be an algebraically closed field of characteristic ''p'' > 0. Then the homomorphism ''f'': ''G''''m'' → ''G''''m'' defined by ''x'' ↦ ''x''''p'' induces an isomorphism of abstract groups ''k''* → ''k''*, but ''f'' is not an isomorphism of algebraic groups (because ''x''1/''p'' is not a regular function). In the language of group schemes, there is a clearer reason why ''f'' is not an isomorphism: ''f'' is surjective, but it has nontrivial
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine lea ...
, namely the group scheme μ''p'' of ''p''th roots of unity. This issue does not arise in characteristic zero. Indeed, every group scheme of finite type over a field ''k'' of characteristic zero is smooth over ''k''. A group scheme of finite type over any field ''k'' is smooth over ''k'' if and only if it is geometrically reduced, meaning that the base change G_ is reduced, where \overline k is an
algebraic closure In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ...
of ''k''. Since an affine scheme ''X'' is determined by its ring ''O''(''X'') of regular functions, an affine group scheme ''G'' over a field ''k'' is determined by the ring ''O''(''G'') with its structure of a Hopf algebra (coming from the multiplication and inverse maps on ''G''). This gives an
equivalence of categories In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences ...
(reversing arrows) between affine group schemes over ''k'' and commutative Hopf algebras over ''k''. For example, the Hopf algebra corresponding to the multiplicative group ''G''''m'' = ''GL''(1) is the
Laurent polynomial In mathematics, a Laurent polynomial (named after Pierre Alphonse Laurent) in one variable over a field \mathbb is a linear combination of positive and negative powers of the variable with coefficients in \mathbb. Laurent polynomials in ''X'' f ...
ring ''k'' 'x'', ''x''−1 with comultiplication given by :x \mapsto x \otimes x.


Basic notions

For a linear algebraic group ''G'' over a field ''k'', the identity component ''G''o (the connected component containing the point 1) is a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
of finite
index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...
. So there is a group extension :1 \to G^\circ \to G \to F \to 1, where ''F'' is a finite algebraic group. (For ''k'' algebraically closed, ''F'' can be identified with an abstract finite group.) Because of this, the study of algebraic groups mostly focuses on connected groups. Various notions from abstract group theory can be extended to linear algebraic groups. It is straightforward to define what it means for a linear algebraic group to be
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
,
nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cl ...
, or solvable, by analogy with the definitions in abstract group theory. For example, a linear algebraic group is solvable if it has a
composition series In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many natu ...
of linear algebraic subgroups such that the quotient groups are commutative. Also, the
normalizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', ...
, the center, and the centralizer of a closed subgroup ''H'' of a linear algebraic group ''G'' are naturally viewed as closed subgroup schemes of ''G''. If they are smooth over ''k'', then they are linear algebraic groups as defined above. One may ask to what extent the properties of a connected linear algebraic group ''G'' over a field ''k'' are determined by the abstract group ''G''(''k''). A useful result in this direction is that if the field ''k'' is
perfect Perfect commonly refers to: * Perfection, completeness, excellence * Perfect (grammar), a grammatical category in some languages Perfect may also refer to: Film * Perfect (1985 film), ''Perfect'' (1985 film), a romantic drama * Perfect (2018 f ...
(for example, of characteristic zero), ''or'' if ''G'' is reductive (as defined below), then ''G'' is unirational over ''k''. Therefore, if in addition ''k'' is infinite, the group ''G''(''k'') is
Zariski dense In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is no ...
in ''G''. For example, under the assumptions mentioned, ''G'' is commutative, nilpotent, or solvable if and only if ''G''(''k'') has the corresponding property. The assumption of connectedness cannot be omitted in these results. For example, let ''G'' be the group μ''3'' ⊂ ''GL''(1) of cube roots of unity over the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s Q. Then ''G'' is a linear algebraic group over Q for which ''G''(Q) = 1 is not Zariski dense in ''G'', because G(\overline ) is a group of order 3. Over an algebraically closed field, there is a stronger result about algebraic groups as algebraic varieties: every connected linear algebraic group over an algebraically closed field is a rational variety.


The Lie algebra of an algebraic group

The
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...
\mathfrak g of an algebraic group ''G'' can be defined in several equivalent ways: as the tangent space ''T''1(''G'') at the identity element 1 ∈ ''G''(''k''), or as the space of left-invariant derivations. If ''k'' is algebraically closed, a derivation ''D'': ''O''(''G'') → ''O''(''G'') over ''k'' of the coordinate ring of ''G'' is left-invariant if :D \lambda_x = \lambda_x D for every ''x'' in ''G''(''k''), where λ''x'': ''O''(''G'') → ''O''(''G'') is induced by left multiplication by ''x''. For an arbitrary field ''k'', left invariance of a derivation is defined as an analogous equality of two linear maps ''O''(''G'') → ''O''(''G'') ⊗''O''(''G''). The Lie bracket of two derivations is defined by 'D''1, ''D''2=''D''1''D''2 − ''D''2''D''1. The passage from ''G'' to \mathfrak g is thus a process of differentiation. For an element ''x'' ∈ ''G''(''k''), the derivative at 1 ∈ ''G''(''k'') of the
conjugation Conjugation or conjugate may refer to: Linguistics *Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics *Complex conjugation, the change ...
map ''G'' → ''G'', ''g'' ↦ ''xgx''−1, is an automorphism of \mathfrak g, giving the adjoint representation: :\operatorname\colon G \to \operatorname(\mathfrak g). Over a field of characteristic zero, a connected subgroup ''H'' of a linear algebraic group ''G'' is uniquely determined by its Lie algebra \mathfrak h \subset \mathfrak g. But not every Lie subalgebra of \mathfrak g corresponds to an algebraic subgroup of ''G'', as one sees in the example of the torus ''G'' = (''G''''m'')2 over C. In positive characteristic, there can be many different connected subgroups of a group ''G'' with the same Lie algebra (again, the torus ''G'' = (''G''''m'')2 provides examples). For these reasons, although the Lie algebra of an algebraic group is important, the structure theory of algebraic groups requires more global tools.


Semisimple and unipotent elements

For an algebraically closed field ''k'', a matrix ''g'' in ''GL''(''n'',''k'') is called semisimple if it is
diagonalizable In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...
, and
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 unipo ...
if the matrix ''g'' − 1 is
nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cl ...
. Equivalently, ''g'' is unipotent if all
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...
s of ''g'' are equal to 1. The Jordan canonical form for matrices implies that every element ''g'' of ''GL''(''n'',''k'') can be written uniquely as a product ''g'' = ''g''ss''g''u such that ''g''ss is semisimple, ''g''u is unipotent, and ''g''''ss'' and ''g''u commute with each other. For any field ''k'', an element ''g'' of ''GL''(''n'',''k'') is said to be semisimple if it becomes diagonalizable over the algebraic closure of ''k''. If the field ''k'' is perfect, then the semisimple and unipotent parts of ''g'' also lie in ''GL''(''n'',''k''). Finally, for any linear algebraic group ''G'' ⊂ ''GL''(''n'') over a field ''k'', define a ''k''-point of ''G'' to be semisimple or unipotent if it is semisimple or unipotent in ''GL''(''n'',''k''). (These properties are in fact independent of the choice of a faithful representation of ''G''.) If the field ''k'' is perfect, then the semisimple and unipotent parts of a ''k''-point of ''G'' are automatically in ''G''. That is (the Jordan decomposition): every element ''g'' of ''G''(''k'') can be written uniquely as a product ''g'' = ''g''ss''g''u in ''G''(''k'') such that ''g''ss is semisimple, ''g''u is unipotent, and ''g''''ss'' and ''g''u commute with each other. This reduces the problem of describing the conjugacy classes in ''G''(''k'') to the semisimple and unipotent cases.


Tori

A torus over an algebraically closed field ''k'' means a group isomorphic to (''G''''m'')''n'', the product of ''n'' copies of the multiplicative group over ''k'', for some natural number ''n''. For a linear algebraic group ''G'', a maximal torus in ''G'' means a torus in ''G'' that is not contained in any bigger torus. For example, the group of diagonal matrices in ''GL''(''n'') over ''k'' is a maximal torus in ''GL''(''n''), isomorphic to (''G''''m'')''n''. A basic result of the theory is that any two maximal tori in a group ''G'' over an algebraically closed field ''k'' are conjugate by some element of ''G''(''k''). The rank of ''G'' means the dimension of any maximal torus. For an arbitrary field ''k'', a torus ''T'' over ''k'' means a linear algebraic group over ''k'' whose base change T_ to the algebraic closure of ''k'' is isomorphic to (''G''''m'')''n'' over \overline k, for some natural number ''n''. A split torus over ''k'' means a group isomorphic to (''G''''m'')''n'' over ''k'' for some ''n''. An example of a non-split torus over the real numbers R is :T=\, with group structure given by the formula for multiplying complex numbers ''x''+''iy''. Here ''T'' is a torus of dimension 1 over R. It is not split, because its group of real points ''T''(R) is the
circle group In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \ ...
, which is not isomorphic even as an abstract group to ''G''''m''(R) = R*. Every point of a torus over a field ''k'' is semisimple. Conversely, if ''G'' is a connected linear algebraic group such that every element of G(\overline k) is semisimple, then ''G'' is a torus. For a linear algebraic group ''G'' over a general field ''k'', one cannot expect all maximal tori in ''G'' over ''k'' to be conjugate by elements of ''G''(''k''). For example, both the multiplicative group ''G''''m'' and the circle group ''T'' above occur as maximal tori in ''SL''(2) over R. However, it is always true that any two maximal split tori in ''G'' over ''k'' (meaning split tori in ''G'' that are not contained in a bigger ''split'' torus) are conjugate by some element of ''G''(''k''). As a result, it makes sense to define the ''k''-rank or split rank of a group ''G'' over ''k'' as the dimension of any maximal split torus in ''G'' over ''k''. For any maximal torus ''T'' in a linear algebraic group ''G'' over a field ''k'', Grothendieck showed that T_ is a maximal torus in G_. It follows that any two maximal tori in ''G'' over a field ''k'' have the same dimension, although they need not be isomorphic.


Unipotent groups

Let ''U''''n'' be the group of upper-triangular matrices in ''GL''(''n'') with diagonal entries equal to 1, over a field ''k''. A group scheme over a field ''k'' (for example, a linear algebraic group) is called unipotent if it is isomorphic to a closed subgroup scheme of ''U''''n'' for some ''n''. It is straightforward to check that the group ''U''''n'' is nilpotent. As a result, every unipotent group scheme is nilpotent. A linear algebraic group ''G'' over a field ''k'' is unipotent if and only if every element of G(\overline) is unipotent. The group ''B''''n'' of upper-triangular matrices in ''GL''(''n'') is a semidirect product :B_n = T_n \ltimes U_n, where ''T''''n'' is the diagonal torus (''G''''m'')''n''. More generally, every connected solvable linear algebraic group is a semidirect product of a torus with a unipotent group, ''T'' ⋉ ''U''. A smooth connected unipotent group over a perfect field ''k'' (for example, an algebraically closed field) has a composition series with all quotient groups isomorphic to the additive group ''G''''a''.


Borel subgroups

The
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'' invertible matrices), the subgrou ...
s are important for the structure theory of linear algebraic groups. For a linear algebraic group ''G'' over an algebraically closed field ''k'', a Borel subgroup of ''G'' means a maximal smooth connected solvable subgroup. For example, one Borel subgroup of ''GL''(''n'') is the subgroup ''B'' of upper-triangular matrices (all entries below the diagonal are zero). A basic result of the theory is that any two Borel subgroups of a connected group ''G'' over an algebraically closed field ''k'' are conjugate by some element of ''G''(''k''). (A standard proof uses the Borel fixed-point theorem: for a connected solvable group ''G'' acting on a
proper variety In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces. Some authors call a proper variety over a field ''k'' a complete variety. For example, every projective variety over a field ' ...
''X'' over an algebraically closed field ''k'', there is a ''k''-point in ''X'' which is fixed by the action of ''G''.) The conjugacy of Borel subgroups in ''GL''(''n'') amounts to the Lie–Kolchin theorem: every smooth connected solvable subgroup of ''GL''(''n'') is conjugate to a subgroup of the upper-triangular subgroup in ''GL''(''n''). For an arbitrary field ''k'', a Borel subgroup ''B'' of ''G'' is defined to be a subgroup over ''k'' such that, over an algebraic closure \overline k of ''k'', B_is a Borel subgroup of G_. Thus ''G'' may or may not have a Borel subgroup over ''k''. For a closed subgroup scheme ''H'' of ''G'', the
quotient space Quotient space may refer to a quotient set when the sets under consideration are considered as spaces. In particular: *Quotient space (topology), in case of topological spaces * Quotient space (linear algebra), in case of vector spaces *Quotient ...
''G''/''H'' is a smooth quasi-projective scheme over ''k''. A smooth subgroup ''P'' of a connected group ''G'' is called parabolic if ''G''/''P'' is projective over ''k'' (or equivalently, proper over ''k''). An important property of Borel subgroups ''B'' is that ''G''/''B'' is a projective variety, called the flag variety of ''G''. That is, Borel subgroups are parabolic subgroups. More precisely, for ''k'' algebraically closed, the Borel subgroups are exactly the minimal parabolic subgroups of ''G''; conversely, every subgroup containing a Borel subgroup is parabolic. So one can list all parabolic subgroups of ''G'' (up to conjugation by ''G''(''k'')) by listing all the linear algebraic subgroups of ''G'' that contain a fixed Borel subgroup. For example, the subgroups ''P'' ⊂ ''GL''(3) over ''k'' that contain the Borel subgroup ''B'' of upper-triangular matrices are ''B'' itself, the whole group ''GL''(3), and the intermediate subgroups :\left \ and \left \. The corresponding
projective homogeneous varieties 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 smo ...
''GL''(3)/''P'' are (respectively): the flag manifold of all chains of linear subspaces :0\subset V_1\subset V_2\subset A^3_k with ''V''''i'' of dimension ''i''; a point; the projective space P2 of lines (1-dimensional linear subspaces) in ''A''3; and the dual projective space P2 of planes in ''A''3.


Semisimple and reductive groups

A connected linear algebraic group ''G'' over an algebraically closed field is called semisimple if every smooth connected solvable normal subgroup of ''G'' is trivial. More generally, a connected linear algebraic group ''G'' over an algebraically closed field is called reductive if every smooth connected unipotent normal subgroup of ''G'' is trivial. (Some authors do not require reductive groups to be connected.) A semisimple group is reductive. A group ''G'' over an arbitrary field ''k'' is called semisimple or reductive if G_ is semisimple or reductive. For example, the group ''SL''(''n'') of ''n'' × ''n'' matrices with determinant 1 over any field ''k'' is semisimple, whereas a nontrivial torus is reductive but not semisimple. Likewise, ''GL''(''n'') is reductive but not semisimple (because its center ''G''''m'' is a nontrivial smooth connected solvable normal subgroup). Every compact connected Lie group has a complexification, which is a complex reductive algebraic group. In fact, this construction gives a one-to-one correspondence between compact connected Lie groups and complex reductive groups, up to isomorphism. A linear algebraic group ''G'' over a field ''k'' is called simple (or ''k''-simple) if it is semisimple, nontrivial, and every smooth connected normal subgroup of ''G'' over ''k'' is trivial or equal to ''G''. (Some authors call this property "almost simple".) This differs slightly from the terminology for abstract groups, in that a simple algebraic group may have nontrivial center (although the center must be finite). For example, for any integer ''n'' at least 2 and any field ''k'', the group ''SL''(''n'') over ''k'' is simple, and its center is the group scheme μ''n'' of ''n''th roots of unity. Every connected linear algebraic group ''G'' over a perfect field ''k'' is (in a unique way) an extension of a reductive group ''R'' by a smooth connected unipotent group ''U'', called the unipotent radical of ''G'': :1\to U\to G\to R\to 1. If ''k'' has characteristic zero, then one has the more precise
Levi decomposition In Lie theory and representation theory, the Levi decomposition, conjectured by Wilhelm Killing and Élie Cartan and proved by , states that any finite-dimensional real Lie algebra ''g'' is the semidirect product of a solvable ideal and a ...
: every connected linear algebraic group ''G'' over ''k'' is a semidirect product R\ltimes U of a reductive group by a unipotent group.


Classification of reductive groups

Reductive groups include the most important linear algebraic groups in practice, such as the classical groups: ''GL''(''n''), ''SL''(''n''), the
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
s ''SO''(''n'') and the
symplectic group In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic g ...
s ''Sp''(2''n''). On the other hand, the definition of reductive groups is quite "negative", and it is not clear that one can expect to say much about them. Remarkably,
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. He was a foun ...
gave a complete classification of the reductive groups over an algebraically closed field: they are determined by
root data 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 introduced by Michel Demazure in SGA III, publis ...
. In particular, simple groups over an algebraically closed field ''k'' are classified (up to quotients by finite central subgroup schemes) by their
Dynkin diagram In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the classification of semisimple Lie algebr ...
s. It is striking that this classification is independent of the characteristic of ''k''. For example, the exceptional Lie groups ''G''2, ''F''4, ''E''6, ''E''7, and ''E''8 can be defined in any characteristic (and even as group schemes over Z). The
classification of finite simple groups In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or els ...
says that most finite simple groups arise as the group of ''k''-points of a simple algebraic group over a finite field ''k'', or as minor variants of that construction. Every reductive group over a field is the quotient by a finite central subgroup scheme of the product of a torus and some simple groups. For example, :GL(n)\cong (G_m\times SL(n))/\mu_n. For an arbitrary field ''k'', a reductive group ''G'' is called split if it contains a split maximal torus over ''k'' (that is, a split torus in ''G'' which remains maximal over an algebraic closure of ''k''). For example, ''GL''(''n'') is a split reductive group over any field ''k''. Chevalley showed that the classification of ''split'' reductive groups is the same over any field. By contrast, the classification of arbitrary reductive groups can be hard, depending on the base field. For example, every nondegenerate
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...
''q'' over a field ''k'' determines a reductive group ''SO''(''q''), and every
central simple algebra In ring theory and related areas of mathematics a central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'' which is simple, and for which the center is exactly ''K''. (Note that ''not'' every simp ...
''A'' over ''k'' determines a reductive group ''SL''1(''A''). As a result, the problem of classifying reductive groups over ''k'' essentially includes the problem of classifying all quadratic forms over ''k'' or all central simple algebras over ''k''. These problems are easy for ''k'' algebraically closed, and they are understood for some other fields such as number fields, but for arbitrary fields there are many open questions.


Applications


Representation theory

One reason for the importance of reductive groups comes from representation theory. Every irreducible representation of a unipotent group is trivial. More generally, for any linear algebraic group ''G'' written as an extension :1\to U\to G\to R\to 1 with ''U'' unipotent and ''R'' reductive, every irreducible representation of ''G'' factors through ''R''. This focuses attention on the representation theory of reductive groups. (To be clear, the representations considered here are representations of ''G'' ''as an algebraic group''. Thus, for a group ''G'' over a field ''k'', the representations are on ''k''-vector spaces, and the action of ''G'' is given by regular functions. It is an important but different problem to classify continuous representations of the group ''G''(R) for a real reductive group ''G'', or similar problems over other fields.) Chevalley showed that the irreducible representations of a split reductive group over a field ''k'' are finite-dimensional, and they are indexed by
dominant weight In the mathematical field of representation theory, a weight of an algebra ''A'' over a field F is an algebra homomorphism from ''A'' to F, or equivalently, a one-dimensional representation of ''A'' over F. It is the algebra analogue of a multiplica ...
s. This is the same as what happens in the representation theory of compact connected Lie groups, or the finite-dimensional representation theory of complex
semisimple Lie algebra In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is ...
s. For ''k'' of characteristic zero, all these theories are essentially equivalent. In particular, every representation of a reductive group ''G'' over a field of characteristic zero is a direct sum of irreducible representations, and if ''G'' is split, the characters of the irreducible representations are given by the Weyl character formula. The Borel–Weil theorem gives a geometric construction of the irreducible representations of a reductive group ''G'' in characteristic zero, as spaces of sections of line bundles over the flag manifold ''G''/''B''. The representation theory of reductive groups (other than tori) over a field of positive characteristic ''p'' is less well understood. In this situation, a representation need not be a direct sum of irreducible representations. And although irreducible representations are indexed by dominant weights, the dimensions and characters of the irreducible representations are known only in some cases. determined these characters (proving
Lusztig Lusztig is a surname. Notable people with the surname include: *George Lusztig George Lusztig (born ''Gheorghe Lusztig''; May 20, 1946) is an American-Romanian mathematician and Abdun Nur Professor at the Massachusetts Institute of Technology ...
's conjecture) when the characteristic ''p'' is sufficiently large compared to the
Coxeter number In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter. Definitions Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there a ...
of the group. For small primes ''p'', there is not even a precise conjecture.


Group actions and geometric invariant theory

An action of a linear algebraic group ''G'' on a variety (or scheme) ''X'' over a field ''k'' is a morphism :G \times_k X \to X that satisfies the axioms of a
group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphi ...
. As in other types of group theory, it is important to study group actions, since groups arise naturally as symmetries of geometric objects. Part of the theory of group actions is geometric invariant theory, which aims to construct a quotient variety ''X''/''G'', describing the set of
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such a ...
s of a linear algebraic group ''G'' on ''X'' as an algebraic variety. Various complications arise. For example, if ''X'' is an affine variety, then one can try to construct ''X''/''G'' as Spec of the ring of invariants ''O''(''X'')''G''. However, Masayoshi Nagata showed that the ring of invariants need not be finitely generated as a ''k''-algebra (and so Spec of the ring is a scheme but not a variety), a negative answer to
Hilbert's 14th problem In mathematics, Hilbert's fourteenth problem, that is, number 14 of Hilbert's problems proposed in 1900, asks whether certain algebras are finitely generated. The setting is as follows: Assume that ''k'' is a field and let ''K'' be a subfield of ...
. In the positive direction, the ring of invariants is finitely generated if ''G'' is reductive, by
Haboush's theorem In mathematics Haboush's theorem, often still referred to as the Mumford conjecture, states that for any semisimple algebraic group ''G'' over a field ''K'', and for any linear representation ρ of ''G'' on a ''K''- vector space ''V'', given ''v' ...
, proved in characteristic zero by Hilbert and Nagata. Geometric invariant theory involves further subtleties when a reductive group ''G'' acts on a projective variety ''X''. In particular, the theory defines open subsets of "stable" and "semistable" points in ''X'', with the quotient morphism only defined on the set of semistable points.


Related notions

Linear algebraic groups admit variants in several directions. Dropping the existence of the inverse map i\colon G \to G, one obtains the notion of a linear algebraic
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...
.


Lie groups

For a linear algebraic group ''G'' over the real numbers R, the group of real points ''G''(R) is a
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...
, essentially because real polynomials, which describe the multiplication on ''G'', are
smooth function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
s. Likewise, for a linear algebraic group ''G'' over C, ''G''(C) is a
complex Lie group In geometry, a complex Lie group is a Lie group over the complex numbers; i.e., it is a complex-analytic manifold that is also a group in such a way G \times G \to G, (x, y) \mapsto x y^ is holomorphic. Basic examples are \operatorname_n(\ma ...
. Much of the theory of algebraic groups was developed by analogy with Lie groups. There are several reasons why a Lie group may not have the structure of a linear algebraic group over R. *A Lie group with an infinite group of components G/Go cannot be realized as a linear algebraic group. *An algebraic group ''G'' over R may be connected as an algebraic group while the Lie group ''G''(R) is not connected, and likewise for
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...
groups. For example, the algebraic group ''SL''(2) is simply connected over any field, whereas the Lie group ''SL''(2,R) has
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
isomorphic to the integers Z. The double cover ''H'' of ''SL''(2,R), known as the metaplectic group, is a Lie group that cannot be viewed as a linear algebraic group over R. More strongly, ''H'' has no faithful finite-dimensional representation. * Anatoly Maltsev showed that every simply connected nilpotent Lie group can be viewed as a unipotent algebraic group ''G'' over R in a unique way. (As a variety, ''G'' is isomorphic to
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...
of some dimension over R.) By contrast, there are simply connected solvable Lie groups that cannot be viewed as real algebraic groups. For example, the
universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete ...
''H'' of the semidirect product ''S''1 ⋉ R2 has center isomorphic to Z, which is not a linear algebraic group, and so ''H'' cannot be viewed as a linear algebraic group over R.


Abelian varieties

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 both to algebraic geometry and group theory. ...
s which are not affine behave very differently. In particular, a smooth connected group scheme which is a projective variety over a field is called an
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 functi ...
. In contrast to linear algebraic groups, every abelian variety is commutative. Nonetheless, abelian varieties have a rich theory. Even the case of elliptic curves (abelian varieties of dimension 1) is central to
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
, with applications including the proof of
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have bee ...
.


Tannakian categories

The finite-dimensional representations of an algebraic group ''G'', together with the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...
of representations, form a
tannakian category 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 some sense, the category of linear ...
Rep''G''. In fact, tannakian categories with a "fiber functor" over a field are equivalent to affine group schemes. (Every affine group scheme over a field ''k'' is ''pro-algebraic'' in the sense that it is an
inverse limit In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits ...
of affine group schemes of finite type over ''k''.) For example, the Mumford–Tate group and the
motivic Galois group In algebraic geometry, motives (or sometimes motifs, following French usage) is a theory proposed by Alexander Grothendieck in the 1960s to unify the vast array of similarly behaved cohomology theories such as singular cohomology, de Rham cohom ...
are constructed using this formalism. Certain properties of a (pro-)algebraic group ''G'' can be read from its category of representations. For example, over a field of characteristic zero, Rep''G'' is a semisimple category if and only if the identity component of ''G'' is pro-reductive.Deligne & Milne (1982), Remark II.2.28.


See also

*The groups of Lie type are the finite simple groups constructed from simple algebraic groups over finite fields. * Lang's theorem * Generalized flag variety,
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 general expression of the princip ...
, BN pair,
Weyl group In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections ...
,
Cartan subgroup In algebraic geometry, a Cartan subgroup of a connected linear algebraic group over an algebraically closed field is the centralizer of a maximal torus (which turns out to be connected). Cartan subgroups are nilpotent and are all conjugate. Exam ...
, group of adjoint type,
parabolic induction In mathematics, parabolic induction is a method of constructing representations of a reductive group from representations of its parabolic subgroups. If ''G'' is a reductive algebraic group and P=MAN is the Langlands decomposition of a parabolic ...
*
Real form (Lie theory) In mathematics, the notion of a real form relates objects defined over the field of real and complex numbers. A real Lie algebra ''g''0 is called a real form of a complex Lie algebra ''g'' if ''g'' is the complexification of ''g''0: : \math ...
,
Satake diagram In the mathematical study of Lie algebras and Lie groups, a Satake diagram is a generalization of a Dynkin diagram introduced by whose configurations classify simple Lie algebras over the field of real numbers. The Satake diagrams associated to a ...
*
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'' = ''A''(''K'') of ''K''. It consists of the points of ''G'' having values in ''A'' ...
, Weil's conjecture on Tamagawa numbers * Langlands classification,
Langlands program In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic n ...
,
geometric Langlands program In mathematics, the geometric Langlands correspondence is a reformulation of the Langlands correspondence obtained by replacing the number fields appearing in the original number theoretic version by function fields and applying techniques from al ...
* Torsor, nonabelian cohomology, special group, cohomological invariant, essential dimension,
Kneser–Tits conjecture In mathematics, the Kneser–Tits problem, introduced by based on a suggestion by Martin Kneser, asks whether the Whitehead group ''W''(''G'',''K'') of a semisimple simply connected isotropic algebraic group ''G'' over a field ''K'' is trivial. Th ...
, Serre's conjecture II *
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 ...
* Differential Galois theory * Distribution on a linear algebraic group


Notes


References

* * * * * * * * * *


External links

* {{springer, title=Linear algebraic group, id=p/l059070