Lie algebra of $S O (3)$

We can reach it by taking the rotations in three directions, e.g. a rotation around the z axis:

R_{z} (θ) = ⎣ ⎢ ⎡ c o s (θ) s i n (θ) 0 - s i n (θ) c o s (θ) 0 001 ⎦ ⎥ ⎤

then we derive and evaluate at 0:

L_{z} = \frac{d R _{z} ( θ )}{d θ} ∣ ∣ ∣ ∣ ∣_{0} = ⎣ ⎢ ⎡ - s i n (θ) c o s (θ) 0 - c o s (θ) - s i n (θ) 0 001 ⎦ ⎥ ⎤ ∣ ∣ ∣ ∣ ∣_{0} = ⎣ ⎢ ⎡ 010 - 1 00 000 ⎦ ⎥ ⎤

L_{z}

therefore represents the infinitesimal rotation.

Note that the exponential map reverses this and gives a finite rotation around the Z axis back from the infinitesimal generator

L_{z}

e^{θ L_{z}} = R_{z} (θ)

Repeating the same process for the other directions gives:

L_{x} = T O D O L_{y} = T O D O

We have now found 3 linearly independent elements of the Lie algebra, and since

S O (3)

has dimension 3, we are done.

 Ancestors