Linear algebraic group
Encyclopedia
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. An example is the orthogonal group
, defined by the relation MTM = I where MT is the transpose
of M.
The main examples of linear algebraic groups are certain Lie group
s, where the underlying field
is the real
or complex
field. (For example, every compact Lie group can be regarded as the group of points of a real linear algebraic group, essentially by the Peter-Weyl theorem.)
These were the first algebraic groups to be extensively studied. Such groups were known for a long time before their abstract algebraic theory was developed according to the needs of major applications. Compact Lie groups were considered by Élie Cartan
, Ludwig Maurer
, Wilhelm Killing
, and Sophus Lie
in the 1880s and 1890s in the context of differential equation
s and Galois theory
. However, a purely algebraic theory was first developed by , with Armand Borel
as one of its pioneers. The Picard-Vessiot theory did lead to algebraic groups.
The first basic theorem of the subject is that any affine algebraic group
is a linear algebraic group: that is, any affine variety V that has an algebraic group law has a faithful linear representation, over the same field. For example the additive group of an n-dimensional vector space
has a faithful representation as n+1×n+1 matrices.
One can define the Lie algebra
of an algebraic group purely algebraically (it consists of the dual number
points based at the identity element); and this theorem shows that we get a matrix Lie algebra. A linear algebraic group G consists of a finite number of irreducible components, that are in fact also the connected components: the one Go containing the identity will be a normal subgroup
of G.
One of the first uses for the theory was to define the Chevalley groups.
The deeper structure theory applies to connected linear algebraic groups G, and begins with the definition of Borel subgroup
s B. These turn out to be maximal as connected solvable subgroups (i.e., subgroups with composition series
having as factors one-dimensional subgroups, all of which are groups of additive or multiplicative type); and also minimal such that G/B is a projective variety.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a linear algebraic group is a subgroup
Subgroup
In group theory, 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 subgroup of G if the restriction of * to H x H is a group operation on H...
of the group of invertible n×n matrices (under matrix multiplication
Matrix multiplication
In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...
) that is defined by polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
equations. An example is the orthogonal group
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
, defined by the relation MTM = I where MT is the transpose
Transpose
In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...
of M.
The main examples of linear algebraic groups are certain Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s, where the underlying field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
is the real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
or complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
field. (For example, every compact Lie group can be regarded as the group of points of a real linear algebraic group, essentially by the Peter-Weyl theorem.)
These were the first algebraic groups to be extensively studied. Such groups were known for a long time before their abstract algebraic theory was developed according to the needs of major applications. Compact Lie groups were considered by Élie Cartan
Élie Cartan
Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...
, Ludwig Maurer
Ludwig Maurer
Ludwig Maurer was a German mathematician professor on the Tübingen University. He was the eldest son of Konrad von Maurer and Valerie von Faulhaber . His 1887 dissertation at the University of Strassburg was on the theory of linear substitutions, known today as matrix groups...
, Wilhelm Killing
Wilhelm Killing
Wilhelm Karl Joseph Killing was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry....
, and Sophus Lie
Sophus Lie
Marius Sophus Lie was a Norwegian mathematician. He largely created the theory of continuous symmetry, and applied it to the study of geometry and differential equations.- Biography :...
in the 1880s and 1890s in the context of differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s and Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
. However, a purely algebraic theory was first developed by , with Armand Borel
Armand Borel
Armand Borel 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...
as one of its pioneers. The Picard-Vessiot theory did lead to algebraic groups.
The first basic theorem of the subject is that any affine algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
is a linear algebraic group: that is, any affine variety V that has an algebraic group law has a faithful linear representation, over the same field. For example the additive group of an n-dimensional vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
has a faithful representation as n+1×n+1 matrices.
One can define the Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
of an algebraic group purely algebraically (it consists of the dual number
Dual number
In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε2 = 0 . The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers. Every dual number has the form z = a + bε with a and...
points based at the identity element); and this theorem shows that we get a matrix Lie algebra. A linear algebraic group G consists of a finite number of irreducible components, that are in fact also the connected components: the one Go containing the identity will be a normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
of G.
One of the first uses for the theory was to define the Chevalley groups.
The deeper structure theory applies to connected linear algebraic groups G, and begins with the definition of Borel subgroup
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 group GLn ,...
s B. These turn out to be maximal as connected solvable subgroups (i.e., subgroups with composition series
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 naturally occurring modules are not semisimple, hence...
having as factors one-dimensional subgroups, all of which are groups of additive or multiplicative type); and also minimal such that G/B is a projective variety.
Non-algebraic Lie groups
There are several classes of examples of Lie groups that aren't the real or complex points of an algebraic group.- Any Lie group with an infinite group of components G/Go cannot be realized as an algebraic group (see identity componentIdentity componentIn mathematics, the identity component of a topological group G is the connected component G0 of G that contains the identity element of the group...
). - The center of a linear algebraic group is again a linear algebraic group. Thus, any group whose center has infinitely many components is not a linear algebraic group. An interesting example is the universal cover of SL2(R). This is a Lie group that maps infinite-to-one to SL2(R), since the fundamental groupFundamental groupIn mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
is here infinite cyclic - and in fact the cover has no faithful matrix representation. - The general solvable Lie group need not have a group law expressible by polynomials.
See also
- Differential Galois theoryDifferential Galois theoryIn mathematics, differential Galois theory studies the Galois groups of differential equations.Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential fields, i.e. fields that are equipped with a derivation, D. Much of...
- Group of Lie typeGroup of Lie typeIn mathematics, a group of Lie type G is a group of rational points of a reductive linear algebraic group G with values in the field k. Finite groups of Lie type form the bulk of nonabelian finite simple groups...
is a group of rational points of a linear algebraic group.