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group_theory:notions:forms_of_a_lie_algebra

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group_theory:notions:forms_of_a_lie_algebra [2016/12/22 12:38]
jakobadmin [Weyl's Unitary Trick]
group_theory:notions:forms_of_a_lie_algebra [2017/12/06 09:33] (current)
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 $SO(4)$ is the real **compact** form of $SO(4)_\mathbb{C}=SL(2)_\mathbb{C}\times SL(2)_\mathbb{C}/​\mathbb{Z}_2$ and $Lor({1,​3})$ is a real **non-compact** form. $SO(4)$ is the real **compact** form of $SO(4)_\mathbb{C}=SL(2)_\mathbb{C}\times SL(2)_\mathbb{C}/​\mathbb{Z}_2$ and $Lor({1,​3})$ is a real **non-compact** form.
  
-Moreover, the simply connected group $SU(2) \times SU(2)$ is the double cover of $SO(4)$ and therefore their Lie algebras are isomorphic. In addition $\mathfrak{su}(2)$ is a compact real form of $\mathfrak{sl}(2)_\mathbb{C}$. This means that we can study the representations of $\mathfrak{lor}_{3,​1}$,​ by studying the representations of $\mathfrak{sl}(2)_\mathbb{C}$.+Moreover, the simply connected group $SU(2) \times SU(2)$ is the double cover of $SO(4)$ and therefore their Lie algebras are isomorphic. In addition$\mathfrak{su}(2)$ is a compact real form of $\mathfrak{sl}(2)_\mathbb{C}$. This means that we can study the representations of $\mathfrak{lor}_{3,​1}$,​ by studying the representations of $\mathfrak{sl}(2)_\mathbb{C}$.
  
 <WRAP tip> Weyl's unitary trick allows us to derive the representations of the Lorentz algebra $\mathfrak{lor}_{3,​1}$,​ by computing the representations of the universal cover of $\mathfrak{so}(4)$,​ which is $\mathfrak{su}(2) \times \mathfrak{su}(2)$. This is incredibly useful, because all irreducible representations of $\mathfrak{su}(2)$ are known and can be classified easily.</​WRAP>​ <WRAP tip> Weyl's unitary trick allows us to derive the representations of the Lorentz algebra $\mathfrak{lor}_{3,​1}$,​ by computing the representations of the universal cover of $\mathfrak{so}(4)$,​ which is $\mathfrak{su}(2) \times \mathfrak{su}(2)$. This is incredibly useful, because all irreducible representations of $\mathfrak{su}(2)$ are known and can be classified easily.</​WRAP>​
group_theory/notions/forms_of_a_lie_algebra.txt ยท Last modified: 2017/12/06 09:33 (external edit)