Parabolic Lie algebra: Difference between revisions
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A '''parabolic Lie algebra''' '''p''' is a subalgebra of a [[semisimple Lie algebra]] '''g''' satisfying one of the following two equivalent conditions: |
A '''parabolic Lie algebra''' '''p''' is a subalgebra of a [[semisimple Lie algebra]] '''g''' satisfying one of the following two equivalent conditions: |
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* '''p''' contains a maximal [[Borel subalgebra]] of '''g'''; |
* '''p''' contains a maximal [[solvable Lie algebra|solvable]] subalgebra (a [[Borel subalgebra]]) of '''g'''; |
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* the [[nilradical]] of '''p''' is its [[Killing form|Killing perp]] in '''g'''. |
* the [[nilradical]] of '''p''' is its [[Killing form|Killing perp]] in '''g'''. |
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Revision as of 19:14, 26 September 2008
A parabolic Lie algebra p is a subalgebra of a semisimple Lie algebra g satisfying one of the following two equivalent conditions:
- p contains a maximal solvable subalgebra (a Borel subalgebra) of g;
- the nilradical of p is its Killing perp in g.
References
- Robert J. Baston and Michael G. Eastwood, The Penrose Transform: its Interaction with Representation Theory, Oxford University Press, 1989.
- William Fulton and Joe Harris (1991), Representation theory. A first course, Readings in Mathematics 129, Springer-Verlag.
- J. Humphreys (1972). Linear Algebraic Groups. New York: Springer. ISBN 0-387-90108-6.