classical groups

1 Classical Groups

It is commonplace to express the classical groupsMathworldPlanetmath with explicit matrices; however, the theory and classification of classical groups can benefit from a basis free consideration.

Given a finite dimensional vector spaceMathworldPlanetmath V over any field k, the set of all linear transformations on V is denoted GL(V) and called the general linear groupMathworldPlanetmath. We now define several significant related groups of GL(V).

Theorem 1 (Birkhoff-von Neumann).

Given a reflexive non-degenerate sesquilinear form b:V×Vk, then up to a constant b is one of the following:

  • Alternating: that is b(v,v)=0.

  • Hermitian: so b(v,w)=b(w,v)σ where σ is a field automorphism of k of order 2.

  • SymmetricMathworldPlanetmathPlanetmathPlanetmathPlanetmath: so b(v,w)=b(w,v).

(Refer to [3, TheoremMathworldPlanetmath 7.1] and [1, Chapter V].) We prefer the definition b(v,v)=0 over b(v,w)=-b(w,v) so that we can accommodate the fields of characteristicPlanetmathPlanetmath 2. In all other characteristics these two properties are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath.

In keeping with tradition for group theory, we let GL(V) act on the vector space V on the right hand side. This means given vV and fGL(V), vf corresponds the vector in V which f sends v to. If one thinks of f as a matrix this requires v to be a row vectorMathworldPlanetmath. It is also common to consider f as a functionMathworldPlanetmath and use the notation f(v).

Definition 2.

Given a reflexive non-degenerate sesquilinear form b:V×Vk we define


This is called the isometry group of b in GL(V).

Proposition 3.

Isom(b) is a subgroupMathworldPlanetmathPlanetmath of GL(V).


Given f,gIsom(b), v,wV then


Hence fgIsom(b). Clearly 1Isom(b) as well. Finally,


So f-1Isom(b) and Isom(b) is a subgroup of GL(V). ∎

Now if we return to Theorem 1 we find that there are only three isometry group types, as there are only three types of reflexive non-degenerate sesquilinear forms. These receive the well-known names:

A vector space V equipped with a reflexiveMathworldPlanetmathPlanetmath non-degenerated sesquilinear formPlanetmathPlanetmath b is also given the designation symplectic, unitaryMathworldPlanetmath, and orthogonalMathworldPlanetmathPlanetmathPlanetmath based on the classification of the form.

Because symplectic spaces have a standard hyperbolic basis it follows every symplectic group over a vector space of the same dimensionPlanetmathPlanetmath is isometric, meaning isomorphicPlanetmathPlanetmathPlanetmath as vector spaces but with an isomorphismPlanetmathPlanetmathPlanetmath which respects the forms. Thus we can write Sp(V) instead of Sp(V,b). For unitary and orthogonal groups more care is required.

Definition 4.

A classical group is any one of the family of groups derived from these three and the general linear group.

2 Classical Groups as Matrices

When expressing these groups with matrices it becomes necessary to establish the bilinear formsPlanetmathPlanetmath with matrices. Given any n×n-matrix B over some field k, and row vectors v,wkn we have a reflexive bilinear form defined by


Whence b is non-degenerate if and only if detB0.

The most common example is the identity matrixMathworldPlanetmath B=In. For then


is the usual dot productMathworldPlanetmath, only perhaps without the positive definitePlanetmathPlanetmath axiom which makes sense only for ordered fields like the rationals and reals .

The isometry group of I is nothing more than the invertible matrices A where


Thus it is common for O(n) to denote the orthogonal group over and be given by


For symplectic groups the form is the typical J=[0I-I0] matrix found in the definition of symplectic matrices. Hence the isometry condition for an alternating form


show that AJAt=J. Thus it is common to define


Thus the symplectic matrices form a group of isometries.

3 Special subgroups

The commutator of GL(V) is the special linear groupMathworldPlanetmath SL(V) composed of all invertible linear transformations of determinantDlmfMathworldPlanetmath 1. Given a reflexive non-degenerate sesquilinear form b on V, we can create the groups


These get the names

Sp(V),SU(V,b), and SO(V,b).

Notice that Sp(V)SL(V) so we do not require a new name.

4 Projective groups

The projective geometryMathworldPlanetmath of a vector space V, denoted PG(V) is its latticeMathworldPlanetmathPlanetmath of subspacesPlanetmathPlanetmathPlanetmath. Clearly invertible linear mapsMathworldPlanetmath act on the projective geometry because they send points (1-dimensional subspaces) to points, and lines (2-dimensional subspaces) to lines, and in general m-dimensional subspaces to other m-dimensional subspaces.

However, the scalar transformationsMathworldPlanetmath, i.e.: those fGL(V) such that vf=vλ for some fixed λk, do not move any of the subspaces of V – they fix PG(V). Therefore when we consider the action of GL(V) on PG(V) we factor out the kernel of the action – that is the scalar transforms (matrices.) We denote this group by PGL(V). Because scalar matrices commute with all other, and not other matrices do, we notice this is the same as factoring by the center Z(GL(V)).

Immediately this gives rise the projective versions of each of the classical groups: Let Z=Z(GL(V)) – the set of scalar transformations (a group isomorphic to k×.)

  • PGL(V)=GL(V)/Z, PSL(V)=SL(V)/(ZSL(V))

  • PSp(V)=Sp(V)/(ZSp(V))

  • PU(V)=U(V,b)/(ZU(V,b)), PSU(V,b)=SU(V,b)/(ZSU(V,b))

  • PO(V)=O(V,b)/(ZO(V,b)), PSO(V,b)=SO(V,b)/(ZSO(V,b))

Most of the time the projective special isometry groups are simple groupsMathworldPlanetmathPlanetmath. The exceptions arise for small dimensional vector spaces and/or small fields, or with the orthogonal groups.


  • 1 Gruenberg, K. W. and Weir, A.J. Linear GeometryMathworldPlanetmathPlanetmathPlanetmath 2nd Ed. (English) [B] Graduate Texts in Mathematics. 49. New York - Heidelberg - Berlin: Springer-Verlag. (1977), pp. x-198.
  • 2 Kantor, W. M. Lectures notes on Classical Groups.
  • 3 Taylor, Donald E. The geometry of the classical groups Sigma Series in Pure Mathematics. 9. Heldermann Verlag, Berlin, xii+229, (1992), ISBN 3-88538-009-9.
Title classical groups
Canonical name ClassicalGroups
Date of creation 2013-03-22 15:50:13
Last modified on 2013-03-22 15:50:13
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 23
Author Algeboy (12884)
Entry type Definition
Classification msc 11E57
Synonym linear algebraic groups
Related topic SemilinearTransformation
Related topic PolaritiesAndForms
Related topic SesquilinearFormsOverGeneralFields
Defines classical group
Defines isometry