1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Isomorphism between so(3) and su(2)

  1. Jul 21, 2016 #1
    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. jcsd
  3. Jul 21, 2016 #2

    fresh_42

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Isomorphism between so(3) and su(2)
  1. SO(3),SU(2) connection (Replies: 4)

  2. Prove: SO(3)/SO(2)=S^2 (Replies: 5)

Loading...