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parabolic subgroup (Definition)

Let $G$ be a complex semi-simple Lie group. Then any subgroup $P$ of $G$ containg a Borel subgroup $B$ is called parabolic. Parabolics are classified in the following manner. Let $\fr g$ be the Lie algebra of $G$ , $\fr h$ the unique Cartan subalgebra contained in $\fr b$ , the algebra of $B$ , $R$ the set of roots corresponding to this choice of Cartan, and $R^+$ the set of positive roots whose root spaces are contained in $\fr b$ and let $\fr p$ be the Lie algebra of $P$ . Then there exists a unique subset $\Pi_P$ of $\Pi$ , the base of simple roots associated to this choice of positive roots, such that $\{\fr b,\fr g_{-\alpha}\}_{\alpha\in\Pi_P}$ generates $\fr p$ . In other words, parabolics containing a single Borel subgroup are classified by subsets of the Dynkin diagram, with the empty set corresponding to the Borel, and the whole graph corresponding to the group $G$ .




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Cross-references: group, graph, empty set, Dynkin diagram, generates, simple roots, base, subset, root spaces, positive roots, roots, algebra, contained, Cartan subalgebra, Lie algebra, Borel subgroup, subgroup, Lie group, semi-simple, complex
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This is version 1 of parabolic subgroup, born on 2003-02-13.
Object id is 4036, canonical name is ParabolicSubgroup.
Accessed 3881 times total.

Classification:
AMS MSC17B20 (Nonassociative rings and algebras :: Lie algebras and Lie superalgebras :: Simple, semisimple, reductive )

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