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- Classification of finite-dimensional representations of semisimple Lie algebras)be completely reducible (or semisimple) if it is isomorphic to a direct sum of irreducible representations (cf. semisimple module). If V is finite-dimensional...28 KB (4,308 words) - 17:20, 8 November 2023
- In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any...41 KB (5,731 words) - 17:17, 10 April 2024
- Representations of finite groups of Lie type)Borel–Weil–Bott construction of representations of algebraic groups using coherent sheaf cohomology is also similar. For real semisimple groups there is an analogue...28 KB (4,066 words) - 03:03, 24 March 2024
- Semisimple group)symplectic group Sp(2n). Simple algebraic groups and (more generally) semisimple algebraic groups are reductive. Claude Chevalley showed that the classification...55 KB (7,845 words) - 18:28, 24 April 2024
- Semisimple)ask whether the category of finite-dimensional representations of a group or a Lie algebra is semisimple, that is, whether every finite-dimensional representation...13 KB (1,867 words) - 10:13, 18 February 2024
- Discrete series (of representations))the discrete series representations of connected semisimple groups G. In particular, such a group has discrete series representations if and only if it...10 KB (1,366 words) - 21:06, 12 August 2023
- field of characteristic 0 {\displaystyle 0} . In a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero...14 KB (2,026 words) - 03:13, 15 March 2024
- Lie group (section Representations)classification of semisimple Lie algebras, Cartan's theory of symmetric spaces, and Hermann Weyl's description of representations of compact and semisimple Lie groups...64 KB (9,440 words) - 19:52, 2 March 2024
- irreducible representation of a semisimple Lie group or Lie algebra can be constructed from the fundamental representations by a procedure due to Élie Cartan...3 KB (473 words) - 08:23, 28 August 2022
- groups. In particular, this implies there are no non-trivial semisimple representations. Of course, the group of matrices U n {\displaystyle \mathbb {U}...11 KB (1,815 words) - 08:42, 20 January 2023
- theory of Lie groups, the study of representations of SU(2) is fundamental to the study of representations of semisimple Lie groups. It is the first case...19 KB (3,397 words) - 22:24, 18 July 2023
- Representations of the Poincaré group)representation theory of a Lie group that is neither a compact group nor a semisimple group. It is fundamental in theoretical physics. In a physical theory...5 KB (584 words) - 21:42, 18 August 2023
- Springer representations)the Springer representations are certain representations of the Weyl group W associated to unipotent conjugacy classes of a semisimple algebraic group...7 KB (823 words) - 09:22, 8 December 2023
- states that A is semisimple when the characteristic of the base field is zero. The tensor product of two irreducible representations V 1 , V 2 {\displaystyle...16 KB (2,895 words) - 23:55, 6 May 2024