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Semisimple Lie Algebras

In: Representations of Lie Algebras and Partial Differential Equations

Author

Listed:
  • Xiaoping Xu

    (Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Institute of Mathematics
    University of Chinese Academy of Sciences, School of Mathematics)

Abstract

We use the Killing form to derive the decomposition of a finite-dimensional semisimple Lie algebra over $$\mathbb {C}$$ C into a direct sum of simple ideals. Moreover, we prove the Weyl’s theorem of complete reducibility, and the equivalence of the complete reducibility of real and complex modules is also given. Cartan’s root-space decomposition of a finite-dimensional semisimple Lie algebra over $$\mathbb {C}$$ C is derived.

Suggested Citation

  • Xiaoping Xu, 2017. "Semisimple Lie Algebras," Springer Books, in: Representations of Lie Algebras and Partial Differential Equations, chapter 0, pages 33-59, Springer.
    • Handle: RePEc:spr:sprchp:978-981-10-6391-6_2
    DOI: 10.1007/978-981-10-6391-6_2
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