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Abelian ideals and cohomology of symplectic type

Author: Li Luo
Journal: Proc. Amer. Math. Soc. 137 (2009), 479-485
MSC (2000): Primary 17B05, 17B56; Secondary 17B20, 17B30
Published electronically: September 29, 2008
MathSciNet review: 2448567
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Abstract: Let $ \mathfrak{b}$ be a Borel subalgebra of the symplectic Lie algebra $ \mathfrak{sp}(2n,\mathbb{C})$ and let $ \mathfrak{n}$ be the corresponding maximal nilpotent subalgebra. We find a connection between the abelian ideals of $ \mathfrak{b}$ and the cohomology of $ \mathfrak{n}$ with trivial coefficients. Using this connection, we are able to enumerate the number of abelian ideals of $ \mathfrak{b}$ with given dimension via the Poincaré polynomials of Weyl groups of types $ A_{n-1}$ and $ C_n$.

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Additional Information

Li Luo
Affiliation: Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

Keywords: Abelian ideal, cohomology, symplectic Lie algebra, Weyl group, Poincar\'e polynomial.
Received by editor(s): January 24, 2008
Published electronically: September 29, 2008
Communicated by: Gail R. Letzter
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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