# 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 $b:V\times V\rightarrow k$, 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 $2$.

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 $b:V\times V\rightarrow k$ 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 $GL(V)$.

###### Proposition 3.

$Isom(b)$

###### 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,

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

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:

###### 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

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

Whence $b$ is non-degenerate if and only if $\det B\neq 0$.

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

 $AIA^{t}=I;\qquad AA^{t}=I.$

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

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

For symplectic groups the form is the typical $J=\begin{bmatrix}0&I\\ -I&0\end{bmatrix}$ matrix found in the definition of symplectic matrices. Hence the isometry condition for an alternating form

 $b(vA,wA)=vAJA^{t}w^{t}=vJw^{t}=b(v,w)$

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

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

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

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

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

## 4 Projective groups

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 $k^{\times}$.)

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

• $PSp(V)=Sp(V)/(Z\cap Sp(V))$

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

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

## References

 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