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Classical group

Classical group

Overview
The classical Lie groups are four infinite families of 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 closely related to the symmetries of Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions higher dimensions...

s. There is a certain leeway in using the term classical group depending on the context. The term seems to have been coined by Hermann Weyl
Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His research has had major...

 (as seen in the title of his 1940 monograph). It probably reflects their relation to "classical" geometry, in the spirit of Felix Klein
Felix Klein
Felix Christian Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...

's Erlangen program
Erlangen program
An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen...

.

Contrasting with the classical Lie groups are the exceptional Lie groups, which share their abstract properties, but not their familiarity.

Sometimes classical groups are discussed in the restricted setting of compact group
Compact group
In mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...

s, a formulation which makes their representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. In essence, a representation makes an abstract algebraic object concrete by describing its elements by matrices and the algebraic...

 and algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism...

 easiest to handle.
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Encyclopedia
The classical Lie groups are four infinite families of 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 closely related to the symmetries of Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions higher dimensions...

s. There is a certain leeway in using the term classical group depending on the context. The term seems to have been coined by Hermann Weyl
Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His research has had major...

 (as seen in the title of his 1940 monograph). It probably reflects their relation to "classical" geometry, in the spirit of Felix Klein
Felix Klein
Felix Christian Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...

's Erlangen program
Erlangen program
An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen...

.

Contrasting with the classical Lie groups are the exceptional Lie groups, which share their abstract properties, but not their familiarity.

Sometimes classical groups are discussed in the restricted setting of compact group
Compact group
In mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...

s, a formulation which makes their representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. In essence, a representation makes an abstract algebraic object concrete by describing its elements by matrices and the algebraic...

 and algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism...

 easiest to handle. It does however exclude the general linear group
General linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is...

, which in contemporary thinking is taken to be the most classical group of all.

Relationship with bilinear forms


The unifying feature of classical Lie groups is that they are close to the isometry group
Isometry group
In mathematics, the isometry group of a metric space is the set of all isometries from the metric space onto itself, with the function composition as group operation...

s of a certain bilinear
Bilinear form
In mathematics, a bilinear form on a vector space V is a bilinear mapping V × V → F, where F is the field of scalars...

 or sesquilinear
Sesquilinear form
In mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and antilinear in the other. The name originates from the numerical prefix sesqui- meaning "one and a half"...

 forms. The four series are labelled by the Dynkin diagram attached to it, with subscript n ≥ 1. The families may be represented as follows:
  • An = SU(n+1), the special unitary group
    Special unitary group
    In mathematics, the special unitary group of degree n, denoted SU, is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication...

     of unitary n+1-by-n+1 complex matrices with determinant 1.
  • Bn = SO(2n+1), the special orthogonal group of orthogonal 2n+1-by-2n+1 real matrices with determinant 1.
  • Cn = Sp(n), the symplectic group
    Symplectic group
    In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups. In this article, we shall denote these two groups Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former...

     of n-by-n quaternion
    Quaternion
    In mathematics, quaternions are a noncommutative number system that extends the complex numbers. The quaternions were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...

    ic matrices that preserve the usual inner product on Hn.
  • Dn = SO(2n), the special orthogonal group of orthogonal 2n-by-2n real matrices with determinant 1.


For certain purposes it is also natural to drop the condition that the determinant be 1 and consider unitary groups and (disconnected) orthogonal groups. The table lists the so-called connected compact real forms of the groups; they have closely-related complex analogues and various non-compact forms, for example, together with compact orthogonal groups one considers indefinite orthogonal groups. 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...

s corresponding to these groups are known as the classical Lie algebras.

Classical groups over general fields or rings


Classical groups, more broadly considered in algebra, provide particularly interesting matrix group
Matrix group
In mathematics, a matrix group is a group G consisting of invertible matrices over some field K, usually fixed in advance, with operations of matrix multiplication and inversion. More generally, one can consider n × n matrices over a commutative ring R...

s. When the ring of coefficients of the matrix group is the real number or complex number field, these groups are just certain of the classical Lie groups.

When the underlying ring is a finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains only finitely many elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

 the classical groups are groups of Lie type. These groups play an important role in the classification of finite simple groups
Classification of finite simple groups
The classification of the finite simple groups, also called the enormous theorem, is believed to classify all finite simple groups. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic building blocks of the natural...

. Considering their abstract group theory, many linear groups have a "special" subgroup, usually consisting of the elements of determinant 1 (for orthogonal groups in characteristic 2 it consists of the elements of Dickson invariant 0),
and most of them have associated "projective" quotients, which are the quotients by the center of the group.

The word "general" in front of a group name usually means that the group is allowed to multiply some sort of form by a constant, rather than leaving it fixed. The subscript n usually indicates the dimension of the module on which the group is acting. Caveat: this notation clashes somewhat with the n of Dynkin diagrams, which is the rank.

General and special linear groups


The general linear group
General linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is...

 GLn(R) is the group of all automorphisms of some module. There is a subgroup: the special linear group
Special linear group
In mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....

 SLn(R), and their quotients: the projective general linear group PGLn(R) = GLn(R)/Z(GLn(R)) and the projective special linear group PSLn(R) = SLn(R)/Z(SLn(R)). The projective special linear group PSLn(R) over a field R is simple for n≥2, except for the 2 cases when n=2 and the field has order 2 or 3.

Unitary groups


The unitary group
Unitary group
In mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrices, with the group operation that of matrix multiplication...

 Un(R) is a group preserving a sesquilinear form
Sesquilinear form
In mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and antilinear in the other. The name originates from the numerical prefix sesqui- meaning "one and a half"...

 on a module. There is a subgroup, the special unitary group
Special unitary group
In mathematics, the special unitary group of degree n, denoted SU, is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication...

 SUn(R) and their quotients the projective unitary group
Projective unitary group
In mathematics, the projective unitary group PU is the quotient of the unitary group U by the right multiplication of its center, U, embedded as scalars....

 PUn(R) = Un(R)/Z(Un(R)) and the projective special unitary group PSUn(R) = SUn(R)/Z(SUn(R))

Symplectic groups


The symplectic group
Symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups. In this article, we shall denote these two groups Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former...

 Sp2n(R) preserves a skew symmetric form on a module. It has a quotient, the projective symplectic group PSp2n(R). The general symplectic group GSp2n(R) consists of the automorphisms of a module multiplying a skew symmetric form by some invertible scalar. The projective symplectic group PSp2n(R) over a field R is simple for n≥1, except for the 2 cases when n=1 and the field has order 2 or 3.

Orthogonal groups


The orthogonal group
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. This is a subgroup of the general linear group GL given bywhere QT is the transpose of Q...

 On(R) preserves a non-degenerate quadratic form on a module. There is a subgroup, the special orthogonal group SOn(R) and quotients, the projective orthogonal group POn(R), and the projective special orthogonal group PSOn(R). (In characteristic 2 the determinant is always 1, so the special orthogonal group is often defined as the subgroup of elements of Dickson invariant 1.)

There is a nameless group often denoted by Ωn(R) consisting of the elements of the orthogonal group of elements of spinor norm 1, with corresponding subgroup and quotient groups SΩn(R), PΩn(R), PSΩn(R). (For positive definite quadratic forms over the reals, the group Ω happens to be the same as the orthogonal group, but in general it is smaller.) There is also a double cover of Ωn(R), called the pin group
Pin group
In mathematics, the pin group is a certain subgroup of the Clifford algebra associated to a quadratic space. It maps 2-to-1 to the orthogonal group, just as the spin group maps 2-to-1 to the special orthogonal group....

  Pinn(R), and it has a subgroup called the spin group
Spin group
In mathematics the spin group Spin is the double cover of the special orthogonal group SO, such that there exists a short exact sequence of Lie groups...

 Spinn(R). The general orthogonal group GOn(R) consists of the automorphisms of a module multiplying a quadratic form by some invertible scalar.