Special Groups

Generalized Orthogonal and Lorentz Groups, O(n;k) and O(3;1)

Let n and k be positive integers, and consider R^n+k . Define a symmetric bilinear form: [• , •]_n,k on R^n+k by [x, y]_n,k = x₁y₁ + ... + x_ny_n - x_n+1y_n+1 - ... - x_n+ky_n+k

The set of (n+k) x (n+k) real matrices A which preserve this form, i.e., [Ax ,Ay]_n,k = [x, y]_n,k for all x, y ∈ R^n+k,  is the generalized orthogonal group O(n;k). It is a subgroup of GL(n+k;R) and a matrix Lie group.

If A is an (n+k) x (n+k) real matrix, let A⁽ⁱ⁾ denote the i-th column vector of A, that is A⁽ⁱ⁾ = (A_1,i ... A_n+k,i). Then, A is in O(n;k) iff the following conditions are satisfied:

[A^(l), A^(j)]_n,k = 0  , l ≠ j,

[A^(l), A^(l)]_n,k = 1  , 1 ≤ l ≤ n,

[A^(l), A^(l)]_n,k = -1  , n+1 ≤ l ≤ n+k.

For any A in O(n;k), det A = ±1. Of particular interest in physics is the Lorentz group O(3;1). 

Heisenberg Group H

The set of all 3 x 3 real matrices A of the form 

A = (1,0,0 : a,1,0 : b,c,1), 

where a, b, and c are arbitrary real numbers, is the Heisenberg group. The Lie algebra of H gives a realization of the Heisenberg commutation relations of quantum mechanics. 

The product of two matrices of this form is again of the same form. The inverse of  A is A^-1 = (1,0,0 : -a,1,0 : ac-b,-c,1)

H is a subgroup of GL(3:R). The limit of matrices of this form is again of the same form, and so H is a matrix Lie group.

Euclidean Group E(n)

By definition, it is the group of all one-to-one, onto, distance-preserving maps of Rⁿ to itself, i.e., f : Rⁿ --> Rⁿ such that d(f(x), f(y)) = d(x, y) for all x, y  ∈ Rⁿ. Here, d is the usual distance on Rⁿ : d(x, y) = |x - y|.  f need not be linear. The orthogonal group O(n) is a subgroup of E(n) and is the group of all linear distance-preserving maps of Rⁿ to itself. For x ∈ Rⁿ , define the translation by x, denoted T_x by,

T_x(y) = x + y.

The set of translations is also a subgroup of E(n).

Every element T of E(n) can be written uniquely as an orthogonal linear transformation followed by a translation in the form T = T_xR, with x ∈ Rⁿ and R  ∈ O(n).

We can write an element of T of E(n) as a pair of {x, R}. For y ∈ Rⁿ, {x, R}y = Ry + x. The product operation for E(n) is :

{x₁, R₁} {x₂, R₂}  =  {x₁+ R₁x₂, R₁R₂ } 

The inverse of an element of E(n) is given by {x, R}^-1 = {-R^-1x, R^-1}.

E(n) is not a subgroup of GL(n;R), since translations are not linear maps. However, E(n) is isomorphic to a subgroup of GL(n+1;R), via the map which associates to {x,R} ∈ E(n) the following matrix:

(R⁽¹⁾,0 : R⁽²⁾,0 : ... : x₁, ... , x_n,1)  with R  ∈ O(n). Because the limit of things of this form is in the same form. So Euclidean group can be expressed as a matrix Lie group.

Poincare Group P(n:1)

Similarly, we can define the Poincare group P(n;1) to be the group of all transformations of Rⁿ⁺¹ of the form T = T_xA, with x ∈ Rⁿ⁺¹ and A  ∈ O(n;1). This is the group of affine transformations of Rⁿ⁺¹ which preserve the Lorentz "distance" d_L(x, y) = (x₁ - y₁)² + ... + (x_n - y_n)² - (x_n+1 - y_n+1)² . An affine transformation is one of the form x --> Ax + b, where A is a linear transformation and b is a constant. The Poincare group P(n;1) is isomorphic to the group of (n+2) x (n+2) matrices of the form

(A⁽¹⁾,0 : A⁽²⁾,0 : ... : x₁, ... , x_n+1,1)  with A  ∈ O(n;1). The set of matrices of this form is a matrix Lie group.