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Lie Groups, Physics, and Geometry by Robert Gilmore

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11

Real forms

 

 

Root space diagrams classify all the simple Lie algebras and summarize their commutation relations. The Lie algebras so classified exist over the field of complex numbers. Each simple Lie algebra over images of complex dimension n has a number of inequivalent real subalgebras over images of real dimension n. These are obtained by putting reality restrictions on the coordinates in the complex Lie algebra. The different real forms of a complex simple Lie algebra are obtained systematically by a simple eigenvalue decomposition. For the classical ...

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