Classical Groups

Group

A group is a set G of which the product operation is a map of G x G into G (closure), denoted by g*h, has the following properties:

Associativity

For all g, h, k ∈ G,  g*(h*k) = (g*h)*k

Identity

There exists an element e in G such that for all g ∈ G, g*e = e*g = g

Inverse

For all g ∈ G, there exist h ∈ G with g*h = h*g = e.

If  g*h = h*g for all g, h ∈ G, then the group is said to be commutative (abelian).


The special and general linear group, the orthogonal and unitary groups, and the symplectic groups make up the classical groups.

General Linear Groups, GL(n;R) and GL(n;C)

The general linear group over the real numbers, GL(n;R), is the group of all nxn invertible matrices with real entries. The general linear group over the complex numbers, GL(n;C), is the group of all nxn invertible matrices with complex entries. 

A matrix Lie group is any subgroup G of GL(n;C) with the following property: If A_m is any sequence of matrices in G, and A_m converges to some matrix A then either A ∈ G, or A is not invertible, i.e., a matrix Lie group is a closed subgroup of GL(n;C).

Special Linear Groups, SL(n;R) and SL(n;C)

The special linear group is the group of nxn invertible matrices, with real or complex entries, having determinant one. Both of these are subgroups of GL(n;C). Furthermore, if A_m is a sequence of matrices with determinant one and A_m converges to A, then A also has determinant one, because determinant is a continuous function. Thus, SL(n;R) and SL(n;C) are matrix Lie groups.

Orthogonal Groups and Special Orthogonal Groups, O(n) and SO(n)

The set of all nxn real orthogonal matrices is the orthogonal group O(n). It is a subgroup of GL(n;C). An nxn real matrix A is said to be orthogonal if the column vectors that make up A are orthonormal, that is, if

∑_l A_ljA_lk = δ_jk ,  1 ≤ j, k ≤ n,  l=1 to n. Kronecker  delta, δ_jk =1 for j = k, δ_jk = 0 for j ≠ k.

Equivalently, A is orthogonal if it preserves the inner product, namely <x, y> = <Ax, Ay> = ∑_k x_ky_k for all vectors x, y in Rⁿ. Still another equivalent definition is that A is orthogonal if A^TA = I, i.e., A^T = A^-1. The limit of a sequence of orthogonal matrices is orthogonal, because the A^TA = I is preserved under limits. Thus, O(n) is a matrix Lie group. 

The set of nxn orthogonal matrices with determinant one is the special orthogonal group SO(n). Both orthogonality and the property of having determinant one are preserved under limits, so SO(n) is a matrix Lie group. Since elements of O(n) already have determinant ±1, SO(n) is "half" of O(n). 

Geometrically, elements of O(n) are either rotations or combinations of rotations and reflections. The elements of SO(n) are just the rotations.

Unitary and Special Unitary Groups, U(n) and SU(n)

The set of all nxn unitary matrices is the unitary group U(n), and it is a subgroup of GL(n; C). An nxn real matrix A is said to be unitary if the column vectors that make up A are orthonormal, that is, if ∑_l A^*_jlA_lk = δ_jk.

Equivalently, A is unitary if it preserves the inner product, namely <x, y> = <Ax, Ay> = ∑_k conj(x)_ky_k for all vectors x, y in Cⁿ. Still another equivalent definition is that A is unitary if A^*A = I, i.e., A^* = A^-1, here A^* is the adjoint of A. The limit of unitary matrices is unitary, so U(n) is a matrix Lie group. The set of unitary matrices with determinant one is the special unitary group SU(n). Note that a unitary matrix can have determinant e^iθ for any θ, and so SU(n) is a smaller subset of U(n) than SO(n) is of O(n). 

Complex Orthogonal Groups, O(n;C) and SO(n;C)

Consider the bilinear form (• , •) on Cⁿ defined by (x, y) = ∑_k x_ky_k. This form is not an inner product because it is symmetric rather than conjugate-symmetric. The set of all nxn complex matrices A which preserve the form, (Ax, Ay) = (x, y),  is the complex orthogonal group O(n;C), it is a subgroup of GL(n;C). The group SO(n;C) is defined to be the set of all A in O(n;C) with det A = 1 and it is also a matrix Lie group. 

Symplectic Groups, Sp(n;R), Sp(n;C), and Sp(n)

The set of all 2nx2n matrices A which preserves B (i.e., B[Ax, Ay] = B[x, y] for all x, y ∈ R²ⁿ) is the real symplectic group Sp(n;R), and it is a subgroup of GL(2n;R), here B is the skew-symmetric bilinear form on R²ⁿ defined as:

B[x, y] = ∑_k x_ky_n+k - x_n+ky_k. 

This group arises naturally in the study of classical mechanics. If J is the 2nx2n matrix  J = ( 0, -I : I, 0), then B[x, y] = (x, Jy), and a 2nx2n real matrix A is in Sp(n;R) if and only if A^TJA = J. This relation shows that det A = ±1, for all A ∈ Sp(n;R). In fact, det A = 1 for all A ∈ Sp(n;R).

We can define a bilinear form on C²ⁿ by the same formula. The set of 2nx2n complex matrices which preserve this form is the complex symplectic group Sp(n;C).  A 2nx2n complex matrix A is in Sp(n;C) if and only if A^TJA = J. It can show that det A = ±1, for all A ∈ Sp(n;R). This relation shows that det A = ±1, for all A ∈ Sp(n;C). In fact, det A = 1 for all A ∈ Sp(n;C).

Finally, the compact symplectic group Sp(n) is defined as

Sp(n) = Sp(n;C) ∩ U(2n)