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Borel subalgebra

From Wikipedia, the free encyclopedia

In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra is a maximal solvable subalgebra.[1] The notion is named after Armand Borel.

If the Lie algebra is the Lie algebra of a complex Lie group, then a Borel subalgebra is the Lie algebra of a Borel subgroup.

Borel subalgebra associated to a flag

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Let be the Lie algebra of the endomorphisms of a finite-dimensional vector space V over the complex numbers. Then to specify a Borel subalgebra of amounts to specify a flag of V; given a flag , the subspace is a Borel subalgebra,[2] and conversely, each Borel subalgebra is of that form by Lie's theorem. Hence, the Borel subalgebras are classified by the flag variety of V.

Borel subalgebra relative to a base of a root system

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Let be a complex semisimple Lie algebra, a Cartan subalgebra and R the root system associated to them. Choosing a base of R gives the notion of positive roots. Then has the decomposition where . Then is the Borel subalgebra relative to the above setup.[3] (It is solvable since the derived algebra is nilpotent. It is maximal solvable by a theorem of Borel–Morozov on the conjugacy of solvable subalgebras.[4])

Given a -module V, a primitive element of V is a (nonzero) vector that (1) is a weight vector for and that (2) is annihilated by . It is the same thing as a -weight vector (Proof: if and with and if is a line, then .)

See also

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References

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  1. ^ Humphreys, Ch XVI, § 3.
  2. ^ Serre 2000, Ch I, § 6.
  3. ^ Serre 2000, Ch VI, § 3.
  4. ^ Serre 2000, Ch. VI, § 3. Theorem 5.
  • Chriss, Neil; Ginzburg, Victor (2009) [1997], Representation Theory and Complex Geometry, Springer, ISBN 978-0-8176-4938-8.
  • Humphreys, James E. (1972), Introduction to Lie Algebras and Representation Theory, Springer-Verlag, ISBN 978-0-387-90053-7.
  • Serre, Jean-Pierre (2000), Algèbres de Lie semi-simples complexes [Complex Semisimple Lie Algebras], translated by Jones, G. A., Springer, ISBN 978-3-540-67827-4.