# Questions tagged [group-theory]

Group theory is a branch of abstract algebra. A group is a set of objects, together with a binary operation, that satisfies four axioms. The set must be closed under the operation and contain an identity object. Every object in the set must have an inverse, and the operation must be associative. Groups are used in physics to describe symmetry operations of physical systems.

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### Decompose $SU(4)$ into $SU(3) \times U(1)$

I'm solving these problems concerning the $SU(4)$ group and I've reached the point where I have determined the Cartan matrix of $SU(4)$, its inverse and the weight schemes for $(1 0 0)$ and $(0 1 0)$ ...
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### How to formulate with a tensor multiplication method $2\otimes 2\otimes 2$ in $SU(2)$ group?

In $SU(2)$ we can write $2\otimes 2=3\oplus 1$ or \begin{equation} q_iq^j=\left(q_iq^j-\frac{1}{2}\delta^i_jq^kq_k\right)+\frac{1}{2}\delta^i_jq^kq_k, \end{equation} where $q_i$ is a $SU(2)$ doublet, ...
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### $SU(2)$ vs $SO(3)$ in Quantum Mechancs

When we're talking about spatial rotations is quantum mechanics, why do we need to resort to $SU(2)$? Why isn't $SO(3)$ enough? I've read that $SO(3)$ isn't simply connected, and I've read about ...
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### Why are covering groups more 'fundamental'?

So I understand that the Lie algebra of $SO(1,3)$ is isomorphic to the Lie algebra of $SU(2)\oplus SU(2)$, and the Lie algebra of $SO(3)$ is isomorphic to one copy of $SU(2)$ (at the group level we ...
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### Lorentz Invariance of Weyl Lagrangian

I have been reading 'Quantum Field Theory and the Standard Model' by Schwartz and have gotten stuck on a line of reasoning in Section 10.2.2. I understand that we can construct a (right-handed) four-...