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.
Theorem 1 (Birkhoff-von Neumann).
(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 and , 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 .
Given a reflexive non-degenerate sesquilinear form we define
This is called the isometry group of in .
is a subgroup of .
Given , then
Hence . Clearly as well. Finally,
So and is a subgroup of . ∎
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:
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.
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 -matrix over some field , and row vectors we have a reflexive bilinear form defined by
Whence is non-degenerate if and only if .
The most common example is the identity matrix . For then
The isometry group of is nothing more than the invertible matrices where
Thus it is common for to denote the orthogonal group over and be given by
For symplectic groups the form is the typical matrix found in the definition of symplectic matrices. Hence the isometry condition for an alternating form
show that . Thus it is common to define
Thus the symplectic matrices form a group of isometries.
3 Special subgroups
4 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 such that for some fixed , 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 .)
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.
- 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.
|Date of creation||2013-03-22 15:50:13|
|Last modified on||2013-03-22 15:50:13|
|Last modified by||Algeboy (12884)|
|Synonym||linear algebraic groups|