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According to my notes on SUSY 'as everyone knows, every continuous group defines a manifold', via

[tex]\Lambda : G \to \mathcal{M}_{G}[/tex]

[tex]\{ g = e^{i\alpha_{a}T^{a}} \} \to \{ \alpha_{a} \}[/tex]

It gives the examples of U(1) having the manifold [tex]\mathcal{M}_{U(1)} = S^{1}[/tex] and SU(2) has [tex]\mathcal{M}_{SU(2)} = S^{3}[/tex].

It then gives the example of SL(2,C) which I don't follow :

[tex]G = SL(2,\mathbb{C})[/tex] and [tex]g=HV[/tex] where [tex]V = SU(2)[/tex] and [tex]H=H^{\dagger}[/tex], positive and detH = 1.

[tex]H = x^{\mu}\sigma_{\mu} = \left( \begin{array}{cc} x^{0}+x^{3} & x^{1}+ix^{2} \\ x^{1}-ix^{2} & x^{0}-x^{3} \end{array} \right)[/tex]

Therefore [tex](x^{0})^{2} - \sum_{i}^{3} |x^{i}|^{2} = 1[/tex] so the manifold is [tex]\mathbb{R}^{3}[/tex], thus giving the full manifold [tex]\mathcal{M}_{SL(2,\mathbb{C})} = \mathbb{R}^{3} \times S^{3}[/tex]

I don't see how it's obvious to consider g as the product of two separate group elements. Is this just a case of knowing the answer and skipping part of the derivation or is there something obvious about SL(2,C) which tells you it's manifold is the direct product of two simpler manifolds?

Also, though I might be about to look very stupid, why does [tex](x^{0})^{2} - \sum_{i}^{3} |x^{i}|^{2} = 1[/tex] imply H has [tex]\mathcal{M}_{H} = \mathbb{R}^{3}[/tex], it would seem to me the defining equation isn't pointing at [tex]\mathbb{R}^{3}[/tex] but something slightly different.

Thanks in advance.

[tex]\Lambda : G \to \mathcal{M}_{G}[/tex]

[tex]\{ g = e^{i\alpha_{a}T^{a}} \} \to \{ \alpha_{a} \}[/tex]

It gives the examples of U(1) having the manifold [tex]\mathcal{M}_{U(1)} = S^{1}[/tex] and SU(2) has [tex]\mathcal{M}_{SU(2)} = S^{3}[/tex].

It then gives the example of SL(2,C) which I don't follow :

[tex]G = SL(2,\mathbb{C})[/tex] and [tex]g=HV[/tex] where [tex]V = SU(2)[/tex] and [tex]H=H^{\dagger}[/tex], positive and detH = 1.

[tex]H = x^{\mu}\sigma_{\mu} = \left( \begin{array}{cc} x^{0}+x^{3} & x^{1}+ix^{2} \\ x^{1}-ix^{2} & x^{0}-x^{3} \end{array} \right)[/tex]

Therefore [tex](x^{0})^{2} - \sum_{i}^{3} |x^{i}|^{2} = 1[/tex] so the manifold is [tex]\mathbb{R}^{3}[/tex], thus giving the full manifold [tex]\mathcal{M}_{SL(2,\mathbb{C})} = \mathbb{R}^{3} \times S^{3}[/tex]

I don't see how it's obvious to consider g as the product of two separate group elements. Is this just a case of knowing the answer and skipping part of the derivation or is there something obvious about SL(2,C) which tells you it's manifold is the direct product of two simpler manifolds?

Also, though I might be about to look very stupid, why does [tex](x^{0})^{2} - \sum_{i}^{3} |x^{i}|^{2} = 1[/tex] imply H has [tex]\mathcal{M}_{H} = \mathbb{R}^{3}[/tex], it would seem to me the defining equation isn't pointing at [tex]\mathbb{R}^{3}[/tex] but something slightly different.

Thanks in advance.

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