# Monopoles in non-abelian semi-simple gauge groups

Relative to the following

Indeed, the modern point of view is that the operator of electric charge is the generator of a U(1) group. The charge quantization condition arises in models of unification if the electromagnetic subgroup is embedded into a semi-simple non-Abelian gauge group of higher rank. In this case, the electric charge generator forms nontrivial commutation relations with all other generators of the gauge group.

I have a few questions:

Could someone explain me why it is important that the gauge group $U(1)$ embedded in the a large gauge group should have non-trivial commutation relations to guarantee charge quantization. Isn't it enough that it should be compactly embedded?

Why does the group has to be semi-simple? Aren't the main Grand Unified Theories that are now considered simple group, like $SO(5)$ or $SO(10)$ ?

And finally: the group has to be non-abelian to embed the groups $SU(2)$ and $SU(3)$ describing the other forces?

• Looking up the definitions of these terms and thinking about them for a few minutes would give the answers. Have you checked wikipedia? Simple Lie Algebra, Non-Abelian Group – Flint72 May 23 '14 at 18:38

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Firstly, if the Lie brackek relations were trivial then you would not have any charge.

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For your second point, any simple algebra is semi-simple. Look up the definition of semi-simple. An algebra is semi-simple if it has non non-trivial ideals, and thus can be decomposed into a direct sum of simple components. Hence any simple algebra $\mathfrak{g}$ is trivially semi-simple, since it is a direct of itself and the 'zero algebra'

$$\mathfrak{g} = \mathfrak{g} \oplus \mathfrak{0}$$

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As for your third point, in order for a group $G$ to have a non-Abelian subgroup $H$ we must have that $G$ is non-Abelian.

I suggest you study group theory, Lie groups, Lie algebras and Representation theory. Fulton & Harris is a good place to start.

• @user41736: In addition to Flint's Fulton and Harris resource recommendation, you can quickly brush up on some group theory, etc. in Peskin and Schroeder's chapter before non-abelian gauge theory. – JamalS May 24 '14 at 9:11