Real forms of a Lie algebra

introduction

• a complex Lie algbera L can be regarded as a real Lie algebra \(L^{R}\)
• if \(L^{R}=L_0\oplus i L_0\) for some real subalgebra \(L_0\)
• \(L_0\) is called a real from of \(L\)
• split real forms
• compact real forms

compact real forms

• of all the real forms of a given simple complex Lie algebra, there is precisely one which is the real Lie algebra of a compact Lie group
• a real Lie algebra which is the Lie algebra of some compact group is called compact

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• ID : Q7301156