Parabolic Lie algebra

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In algebra, a parabolic Lie algebra is a subalgebra of a semisimple Lie algebra satisfying one of the following two conditions:

  • contains a maximal solvable subalgebra (a Borel subalgebra) of ;
  • the orthogonal complement with respect to the Killing form of in is the nilradical of .

These conditions are equivalent over an algebraically closed field of characteristic zero, such as the complex numbers. If the field is not algebraically closed, then the first condition is replaced by the assumption that

  • contains a Borel subalgebra of

where is the algebraic closure of .

Examples

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For the general linear Lie algebra  , a parabolic subalgebra is the stabilizer of a partial flag of  , i.e. a sequence of nested linear subspaces. For a complete flag, the stabilizer gives a Borel subalgebra. For a single linear subspace  , one gets a maximal parabolic subalgebra  , and the space of possible choices is the Grassmannian  .

In general, for a complex simple Lie algebra  , parabolic subalgebras are in bijection with subsets of simple roots, i.e. subsets of the nodes of the Dynkin diagram.

See also

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Bibliography

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  • Baston, Robert J.; Eastwood, Michael G. (2016) [1989], The Penrose Transform: its Interaction with Representation Theory, Dover, ISBN 9780486816623
  • Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
  • Grothendieck, Alexander (1957), "Sur la classification des fibrés holomorphes sur la sphère de Riemann", Amer. J. Math., 79 (1): 121–138, doi:10.2307/2372388, JSTOR 2372388.
  • Humphreys, J. (1972), Linear Algebraic Groups, Springer, ISBN 978-0-387-90108-4