classical groups

1 Classical Groups

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

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

Theorem 1 (Birkhoff-von Neumann).

Given a reflexive non-degenerate sesquilinear form $b\mathrm{:}V\mathrm{\times }V\mathrm{\to }k$, then up to a constant $b$ is one of the following:

• Alternating: that is $b(v,v)=0$.

• Hermitian: so $b(v,w)=b{(w,v)}^{\sigma }$ where $\sigma$ is a field automorphism of $k$ of order $2$.

• Symmetric: so $b(v,w)=b(w,v)$.

(Refer to [3, Theorem 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 characteristic 2. In all other characteristics these two properties are equivalent.

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 $v\in V$ and $f\in GL(V)$, $vf$ 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 vector. It is also common to consider $f$ as a function and use the notation $f(v)$.

Definition 2.

Given a reflexive non-degenerate sesquilinear form $b\mathrm{:}V\mathrm{\times }V\mathrm{\to }k$ we define

$$Isom(b)=\{f\in GL(V):b(vf,wf)=b(v,w),v,w\in V\}.$$

This is called the isometry group of $b$ in $G\mathit{}L\mathit{}\mathrm{(}V\mathrm{)}$.

Proposition 3.

$Isom(b)$ is a subgroup of $G\mathit{}L\mathit{}\mathrm{(}V\mathrm{)}$.

Proof.

Given $f,g\in Isom(b)$, $v,w\in V$ then

$$b(v(fg),w(fg))=b((vf)g,(wf)g)=b(vf,wf)=b(v,w).$$

Hence $fg\in Isom(b)$. Clearly $1\in Isom(b)$ as well. Finally,

$$b(v{f}^{-1},w{f}^{-1})=b((v{f}^{-1})f,(w{f}^{-1})f)=b(v,w).$$

So $f^{-1}\in Isom(b)$ 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:

• Symplectic group $Sp(V,b)$ if $b$ is alternating.

• Unitary group $U(V,b)$ if $b$ is hermitian.

• Orthogonal group $O(V,b)$ if $b$ is symmetric.

A vector space $V$ equipped with a reflexive non-degenerated sesquilinear form $b$ is also given the designation symplectic, unitary, and orthogonal 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 dimension is isometric, meaning isomorphic as vector spaces but with an isomorphism 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 forms with matrices. Given any $n\times n$-matrix $B$ over some field $k$, and row vectors $v,w\in k^n$ we have a reflexive bilinear form defined by

$$b(v,w)=vB{w}^t.$$

Whence $b$ is non-degenerate if and only if $detB\ne 0$.

The most common example is the identity matrix $B=I_n$. For then

$$b(v,w)=v{w}^t$$

is the usual dot product, only perhaps without the positive definite axiom which makes sense only for ordered fields like the rationals $\mathbb{Q}$ and reals $ℝ$.

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

$$AI{A}^t=I;A{A}^t=I.$$

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

$$O(n)=\{A\in GL(n):A{A}^t=I\}.$$

For symplectic groups the form is the typical matrix found in the definition of symplectic matrices. Hence the isometry condition for an alternating form

$$b(vA,wA)=vAJ{A}^t{w}^t=vJ{w}^t=b(v,w)$$

show that $AJ{A}^t=J$. Thus it is common to define

$$Sp(2m)=\{A\in GL(2m):AJ{A}^t=J\}.$$

Thus the symplectic matrices form a group of isometries.

3 Special subgroups

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

$$SIsom(V,b)=Isom(V,b)\cap SL(V).$$

These get the names

$$Sp(V),SU(V,b),\text{ and }SO(V,b).$$

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

4 Projective groups

The projective geometry of a vector space $V$, denoted $PG(V)$ is its lattice of subspaces. Clearly invertible linear maps 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 transformations, i.e.: those $f\in GL(V)$ such that $vf=v\lambda$ for some fixed $\lambda \in k$, 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 $k^{\times }$.)

• $PGL(V)=GL(V)/Z$, $PSL(V)=SL(V)/(Z\cap SL(V))$

• $PSp(V)=Sp(V)/(Z\cap Sp(V))$

• $PU(V)=U(V,b)/(Z\cap U(V,b))$, $PSU(V,b)=SU(V,b)/(Z\cap SU(V,b))$

• $PO(V)=O(V,b)/(Z\cap O(V,b))$, $PSO(V,b)=SO(V,b)/(Z\cap SO(V,b))$

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

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