Quantum Mechanics for Mathematicians

5.2 Lie algebras of the orthogonal and unitary groups

The groups we are most interested in are the groups of linear transformations
preserving an inner product: the orthogonal and unitary groups. We have seen
that these are subgroups ofGL(n,R) orGL(n,C), consisting of those elements
Ω satisfying the condition
ΩΩ†= 1

In order to see what this condition becomes on the Lie algebra, write Ω =etX,
for some parametert, andXa matrix in the Lie algebra. Since the transpose of
a product of matrices is the product (order-reversed) of the transposed matrices,
i.e.,
(XY)T=YTXT

and the complex conjugate of a product of matrices is the product of the complex
conjugates of the matrices, one has

(etX)†=etX

†

The condition
ΩΩ†= 1

thus becomes
etX(etX)†=etXetX

†

= 1

Taking the derivative of this equation gives

etXX†etX

†

+XetXetX

†

= 0

Evaluating this att= 0 gives

X+X†= 0

so the matrices we want to exponentiate must be skew-adjoint (it can be shown
that this is also a sufficient condition), satisfying

X†=−X

Note that physicists often choose to define the Lie algebra in these cases
as self-adjoint matrices, then multiplying byibefore exponentiating to get a
group element. We will not use this definition, with one reason that we want to
think of the Lie algebra as a real vector space, so want to avoid an unnecessary
introduction of complex numbers at this point.