Homework Help: Left and right invariant metric on SU(2)

1. Dec 19, 2012

popbatman

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

I nedd some help to write a left-invariant and right invariant metric on SU(2)

2. Relevant equations

3. The attempt at a solution

2. Dec 19, 2012

dextercioby

What parametrization are you using ? And write the definitions you're working with. As per the guidelines of this particular forum, you're asked to post your work first and ask for advice/help later.

And this looks like pure mathematics subject, why did you place it here ?

3. Dec 20, 2012

popbatman

I'm using the parametrization:

\begin{array}{cc}
x_{0} - ix_{3} & x_{1}+ix_{2} \\
-x_{1}+iy_{2} & y_{0}+iy_{3} \end{array}

Now I know that left invariant vector fields are obtained starting from vector tanget to the identity of the group (pauli matrices).
by duality i can find also a basis for the left invariant forms.
a) Is this right?

Once I've found the left invariant forms $θ^{i}$

I can define the metric g=$g_{\mu\nu}$$θ^{\mu}\otimesθ^{\nu}$

b)Is this the left invariant metric i'm looking for? Are the metric coefficient totally arbitrary, a part the constraints to make the metric non degenerate and strictly positive (if i want a riemannian metric)?

I'm sorry if this is not the the right place for my post! Thank you for helping!

4. Dec 20, 2012

popbatman

I'm sorry, obvyously in the matrix is x everywhere!