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Highest-Weight Representation Theory

In: Representations of Lie Algebras and Partial Differential Equations

Author

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  • 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 study finite-dimensional irreducible representations of a finite-dimensional semisimple Lie algebra over $$\mathbb {C}$$ C . First we introduce the universal enveloping algebra of a Lie algebra and prove the Poincaré-Birkhoff-Witt (PBW) Theorem. Then we use the universal enveloping algebra o to construct Verma modules. Moreover, we prove that any finite-dimensional-module is the quotient of a Verma module modulo its maximal proper submodule, whose generators are explicitly given. Furthermore, the Weyl’s character formula is derived and the dimensional formula is determined.

Suggested Citation

  • Xiaoping Xu, 2017. "Highest-Weight Representation Theory," Springer Books, in: Representations of Lie Algebras and Partial Differential Equations, chapter 0, pages 125-151, Springer.
    • Handle: RePEc:spr:sprchp:978-981-10-6391-6_5
    DOI: 10.1007/978-981-10-6391-6_5
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