Left and right invariant metric on SU(2)

[popbatman]

Homework Statement

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

Homework Equations

The Attempt at a Solution

 

[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 ?

 

[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!

 

[popbatman]

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