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classical groups

(Definition)

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 over any field , the set of all linear transformations on is denoted and called the general linear group. We now define several significant related groups of .

Theorem 1 (Birkhoff-von Neumann) Given a reflexive non-degenerate sesquilinear form $b:V\times V\rightarrow k$ , then up to a constant is one of the following:

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

(Refer to [3, Theorem 7.1] and [1, Chapter V].) We prefer the definition

over

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 act on the vector space on the right hand side. This means given $v\in V$ and $f\in GL(V)$ , corresponds the vector in which sends to. If one thinks of as a matrix this requires to be a row vector. It is also common to consider as a function and use the notation .

Definition 2 Given a reflexive non-degenerate sesquilinear form $b:V\times V\rightarrow k$ we define

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

This is called the isometry group of in .

Proposition 3 is a subgroup of .

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

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

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

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

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

is a subgroup of

. $\qedsymbol$

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 if is alternating.
Unitary group if is hermitian.
Orthogonal group if is symmetric.

A vector space equipped with a reflexive non-degenerated sesquilinear form 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 instead of . 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.

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 over some field , and row vectors $v,w\in k^n$ we have a reflexive bilinear form defined by

$\displaystyle b(v,w)=vBw^t.$

Whence

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

The most common example is the identity matrix . For then

$\displaystyle b(v,w)=vw^t$

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 $\mathbb{R}$ .

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

$\displaystyle AIA^t=I;\qquad AA^t=I.$

Thus it is common for

to denote the orthogonal group over $\mathbb{R}$ and be given by

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

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

$\displaystyle b(vA,wA)=vAJA^t w^t=vJw^t=b(v,w)$

show that

. Thus it is common to define

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

Thus the symplectic matrices form a group of isometries.

Special subgroups

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

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

These get the names

$% latex2html id marker 1000 $\displaystyle 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.

Projective groups

The projective geometry of a vector space , denoted 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 -dimensional subspaces to other -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 - they fix . Therefore when we consider the action of on we factor out the kernel of the action - that is the scalar transforms (matrices.) We denote this group by . Because scalar matrices commute with all other, and not other matrices do, we notice this is the same as factoring by the center .

Immediately this gives rise the projective versions of each of the classical groups: Let - the set of scalar transformations (a group isomorphic to $k^\times$ .)

, $PSL(V)=SL(V)/(Z\intersect SL(V))$
$PSp(V)=Sp(V)/(Z\intersect Sp(V))$
$PU(V)=U(V,b)/(Z\intersect U(V,b))$ , $PSU(V,b)=SU(V,b)/(Z\intersect SU(V,b))$
$PO(V)=O(V,b)/(Z\intersect O(V,b))$ , $PSO(V,b)=SO(V,b)/(Z\intersect SO(V,b))$

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.

Bibliography

1: Gruenberg, K. W. and Weir, A.J. Linear Geometry 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.

"classical groups" is owned by Algeboy.

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Other names:

linear algebraic groups

Also defines:

classical group, isometry

Attachments:: orders and structure of classical groups (Topic) by Algeboy

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Cross-references: simple groups, center, Transforms, kernel, factor, action, fixed, transformations, scalar, lines, points, invertible linear maps, subspaces, lattice, projective geometry, determinant, invertible linear transformations, special linear group, commutator, group of isometries, symplectic matrices, invertible, reals, rationals, ordered fields, axiom, positive definite, dot product, identity matrix, non-degenerate, bilinear forms, necessary, isomorphism, isomorphic, isometric, dimension, hyperbolic basis, orthogonal, unitary, sesquilinear form, Reflexive, orthogonal group, unitary group, types, subgroup, isometry group, function, row vector, vector, right hand side, act on, equivalent, properties, characteristic, symmetric, order, automorphism, Hermitian, alternating, reflexive non-degenerate sesquilinear, groups, general linear group, linear transformations, field, vector space, finite dimensional, basis, theory, matrices
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This is version 20 of classical groups, born on 2006-04-08, modified 2007-06-23.
Object id is 7812, canonical name is ClassicalGroups.
Accessed 4061 times total.

Classification:

AMS MSC:

11E57 (Number theory :: Forms and linear algebraic groups :: Classical groups)

Pending Errata and Addenda

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