# Isomorphism between so(3) and su(2)

1. Jul 21, 2016

### MrRobot

1. The problem statement, all variables and given/known data

How do I use the commutation relations of su(2) and so(3) to construct a Lie-algebra isomorphism between these two algebras?
2. Relevant equations
The commutation relations are [ta, tb] = i epsilonabc tc, the ts being the basis elements of the algebras. They basically have the same commutation relation, only ta are two by two by the su(2) while 3X3 by so(3).

3. The attempt at a solution

2. Jul 21, 2016

### Staff: Mentor

How they are represented by matrices isn't important. If you have a linear, bijective mapping
$$φ : \mathfrak{su}(2) \longrightarrow \mathfrak{so}(3)$$
e.g. if you map all basis vectors $t^α \longmapsto {s}^α$ then you have to check whether $φ([t^α,t^β]) = [φ(t^α),φ(t^β)] = [{s}^α,s^β].$ If this is the case for all pairs $(α,β)$ then it is a Lie-algebra isomorphism.