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!

 

[paranormal]

A left-invariant metric on a Lie group is a metric that is preserved under left translations, meaning that the metric at the identity element is the same as the metric at any other element obtained by left multiplication. Similarly, a right-invariant metric is preserved under right translations. In the case of SU(2), a special unitary group of 2x2 complex matrices with determinant 1, the most common choice for a left-invariant metric is the Frobenius inner product, defined as follows:

For two elements A and B in SU(2), the Frobenius inner product is given by <A,B> = Tr(A^*B), where A^* is the conjugate transpose of A and Tr() denotes the matrix trace. This inner product is left-invariant because for any element g in SU(2), we have <gA, gB> = Tr((gA)^*(gB)) = Tr(A^*B) = <A,B>.

Similarly, a right-invariant metric on SU(2) can be defined as <A,B> = Tr(A^*B) = <A,B>, since for any element g in SU(2), we have <Ag, Bg> = Tr((Ag)^*(Bg)) = Tr(A^*B) = <A,B>.

Both of these metrics are natural choices for SU(2) since they are invariant under the group operations and are also positive definite, making them suitable for defining distances and angles between elements in SU(2).