Matrix Lie Group

Lie Algebra of a Matrix Lie Group

Let X be nxn real or complex matrix. The exponential of X is defined by the usual power series e^tX = ∑ X^m / m!  for m = 1 ... ∞. e^tX is a smooth curve in M_n(C).

d/dt (e^tX) = Xe^tX = e^tXX.  In particular, d/dt (e^tX) |_t=0= X.

A function A: R --> GL(n;C) is called a one-parameter subgroup of GL(n;C) if 1) A is continuous; 2) A(0) = I; 3) A(t+s) = A(t)A(s) for all t, s ∈ R. If A is a one-parameter subgroup of GL(n;C), then there exists a unique nxn complex matrix X such that A(t) = e^tX.

Recall that a matrix Lie group is any subgroup G of GL(n;C) with the following properties: If X_m is any sequence of matrices in G, and X_m converges to some matrix X then either X ∈ G, or X is not invertible. Every matrix Lie group is a smooth embedded submanifold of M_n(C) and is thus a Lie group.  But not every Lie group is isomorphic to a matrix Lie group. 

Let G be a matrix Lie group. The Lie algebra of G, denoted g, is the set of all matrices X such that e^tX is in G for all real numbers t. X is in g if and only if the one-parameter subgroup generated by X lies in G. It only require that e^tX be in G for all real numbers t. If e^X ∈ G but e^tX ∉ G for some real values of t. Then X is not in the Lie algebra of G.  

Every matrix Lie group is an embedded submanifold of GL(n;C). g is the tangent space to G at the identity. So a matrix X is in g if and only if there exists a smooth curve Υ in M_n(C) such that 1) Υ(t) lies in G for all t;  2) Υ(0) = I;  3) dΥ/dt |_t=0 = X. This means that g can alternatively be defined as the set of all derivatives of smooth curves through the identity in G. 

Let X and Y be nxn complex matrices, some further properties of the matrix exponential.

1) The Lie Product Formula,  e^X+Y = lim_m-->∞ (e^X/m e^Y/m)^m .

2) For any X ∈ M_n(C), we have det(e^tX) = e^trace(X).

General Linear Groups gl(n;C), gl(n;R)

The Lie algebra of GL(n;C) is the space of all nxn complex matrices, denoted gl(n;C). 

The Lie algebra of GL(n;R) is the space of all nxn real matrices, denoted gl(n;R). If G is a subgroup of GL(n:R), then the Lie algebra of G must consist entirely of real matrices. 

(If e^tX is real for all real numbers t, then X = d/dt e^tX |_t=0 will also be real. )

Special Linear Groups sl(n;C), sl(n;R)

The Lie algebra of SL(n;C) is the space of all nxn complex matrices with trace zero, denoted by sl(n;C).

(If X is any nxn matrix such that det(e^tX )=1 for all t, then e^{t trace(X)} =1 for all t. Therefore, t trace(X) is an integer multiple of 2πi for all t, which is only possible if trace(X) = 0.)

The Lie algebra of SL(n;R) is the space of all nxn real matrices with trace zero, denoted by sl(n;R).

Unitary Groups u(n), su(n)

The Lie algebra of U(n) is the space of all nxn complex matrices X such that X^* = -X, denoted u(n).

(Recall that a matrix U is unitary if and only if U^* = -U^-1. Thus, e^tX is unitary if and only if (e^tX )^* = (e^tX )^-1 = e^-tX . Since (e^tX )^* = e^tX* . Therefore e^tX* = e^-tX.)

The Lie algebra of SU(n) is the space of all nxn complex matrices X such that X^* = -X and trace(X) = 0, denoted u(n).

Orthogonal Groups so(n), so(n;C)

The identity component of O(n) is just SO(n). Since the exponential of a matrix in the Lie algebra is automatically in the identity component, the Lie algebra of O(n) is the same as the Lie algebra of SO(n). The Lie algebra of O(n) and SO(n) is the space of all nxn real matrices X with X^tr = -X, denoted so(n). Note that the condition X^tr = -X forces the diagonal entries of X to be zero, so necessarily the trace of X is zero. 

(An nxn real matrix R is orthogonal if and only if R^tr = -R^-1. So, given an nxn real matrix X, e^tX is orthogonal if and only if (e^tX )^tr = e^tX )^-1 or e^{t tr(X)} = e^-tX. If it holds for all t, then by differentiating at t = 0, we would have  X^tr = -X.)

The Lie algebra of SO(n;C) is the space of nxn complex matrices satisfying X^tr = -X, denoted by so(n;C).

Symplectic Groups sp(n;R), sp(n;C), sp(n)

Let J be the matrix in the definition of the symplectic groups. sp(n;R) is the space of 2n x 2n real matrices X such that JX^trJ = J. 

sp(n;C) is the space of 2n x 2n complex matrices satisfying the same condition. The elements of sp(n;C) are precisely 2n x 2n matrices of the form (A,C; B, -A^tr), where A is an arbitrary nxn matrix and B and C are arbitrary symmetric matrices. 

sp(n) = sp(n;C) ∩ u(2n).

Generalized Orthogonal Groups so(n;k)

The Lie algebra of O(n;k), which is the same as the Lie algebra of SO(n;k), consists of all (n+k) x (n+k) real matrices X with gX^trg = -X, is denoted by so(n;k).

(a matrix A is in O(n;k) if and only if A^trgA = g, where g is the (n+k) x (n+k) diagonal matrix with the first n diagonal entries equal to one and the last k diagonal entries equal to minus one. This condition is equivalent to the condition g^-1A^trg = A^-1. Since g^-1 = g, gA^trg = A^-1. Now, if X is an (n+k) x (n+k) real matrix, then e^tX is in O(n;k) if and only if ge^{t tr(X)}g  = e^{t gtr(X)g} = e^-tX. This condition holds for all real t if and only if gtr(X)g = -X.)

Heisenberg Group

The Lie algebra of the Heisenberg group is the space of all 3 x 3 real matrices that are strictly upper triangular. 

(Recall that H is the group of all 3 x 3 real matrices A = (1,0,0; a,1,0; b,c,1), the exponential of the matrix of the form (0,0,0; α,0,0; β,γ,0) was in H.)

Euclidean and Poincare Groups

The Lie algebra of E(n) is the space of all (n+1) x (n+1) real matrices of the form (Y⁽¹⁾,0 : Y⁽²⁾,0 : ... : y₁, ... , y_n,0) with Y satisfying Y^tr = -Y.

The Lie algebra of P(n;1) is the space of all (n+2) x (n+2) real matrices of the form (Y⁽¹⁾,0 : Y⁽²⁾,0 : ... : y₁, ... , y_n+1,0) with Y ∈ so(n;1).