Proving A_μ' in Lie(SU(N)) for U in SU(N)

[latentcorpse]

How would you go about proving that if $A_\mu \in \text{Lie}(SU(N))$ then $A_\mu' \in \text{Lie}(SU(N)) \forall U \in SU(N)$

where $A_\mu'=U A_\mu U^{-1}-\frac{1}{g} ( \partial_\mu U) U^{-1}$?

Presumably we need to check some defining property of being in the Lie Group?

Thanks.

 

[praharmitra]

Could tell me what $$g$$ is?

 

[latentcorpse]

Could tell me what $$g$$ is?

Thanks for your reply.
Well this is to do with the physics of non abelian gauge theory so I think we just take it as the dimensionless coupling constant?
In short, I think we just treat it as a constant.

 

[praharmitra]

Ah! well, here is what I have done

Property for Lie(SU(N)) : $$A^\dagger = A$$

Property for SU(N) : $$A^\dagger = A^{-1}$$

What I am able to show is that

$A'_\mu^\dagger = U A_\mu ^\dagger U^{-1}+\frac{1}{g} ( \partial_\mu U) U^{-1}$

There is a plus sign where there should be a minus. I am trying to figure that out.EDIT: Ahh! Got it. I forgot that $$\partial_\mu^\dagger = -\partial_\mu$$ that fixes it.

 

[latentcorpse]

Ah! well, here is what I have done

Property for Lie(SU(N)) : $$A^\dagger = A$$

Property for SU(N) : $$A^\dagger = A^{-1}$$

What I am able to show is that

$A'_\mu^\dagger = U A_\mu ^\dagger U^{-1}+\frac{1}{g} ( \partial_\mu U) U^{-1}$

There is a plus sign where there should be a minus. I am trying to figure that out.


EDIT: Ahh! Got it. I forgot that $$\partial_\mu^\dagger = -\partial_\mu$$ that fixes it.

Shouldn't the Lie group, $\text{Lie}(SU(N))$ be anti hermtian matrices i.e. $A^\dagger=-A$?

 

[praharmitra]

Well,
$$U^\dagger = U^{-1} \Rightarrow exp(iA)^\dagger = exp(iA)^{-1}$$

$$\Rightarrow exp(-iA^\dagger) = exp(-iA) \Rightarrow A^\dagger = A$$

 

[latentcorpse]

Well,
$$U^\dagger = U^{-1} \Rightarrow exp(iA)^\dagger = exp(iA)^{-1}$$

$$\Rightarrow exp(-iA^\dagger) = exp(-iA) \Rightarrow A^\dagger = A$$

I think in my notes, we have the i absorbed into the matrix A since we define the lie algebra to be the vector space of traceless, anti-hermitian matrices...