Abelian ideals and cohomology of symplectic type
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- by Li Luo
- Proc. Amer. Math. Soc. 137 (2009), 479-485
- DOI: https://doi.org/10.1090/S0002-9939-08-09685-8
- Published electronically: September 29, 2008
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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 $C_n$.References
- Raoul Bott, Homogeneous vector bundles, Ann. of Math. (2) 66 (1957), 203–248. MR 89473, DOI 10.2307/1969996
- Paola Cellini and Paolo Papi, Abelian ideals of Borel subalgebras and affine Weyl groups, Adv. Math. 187 (2004), no. 2, 320–361. MR 2078340, DOI 10.1016/j.aim.2003.08.011
- James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
- Bertram Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329–387. MR 142696, DOI 10.2307/1970237
- Bertram Kostant, The set of abelian ideals of a Borel subalgebra, Cartan decompositions, and discrete series representations, Internat. Math. Res. Notices 5 (1998), 225–252. MR 1616913, DOI 10.1155/S107379289800018X
- Bertram Kostant, Powers of the Euler product and commutative subalgebras of a complex simple Lie algebra, Invent. Math. 158 (2004), no. 1, 181–226. MR 2090363, DOI 10.1007/s00222-004-0370-7
- Dmitri I. Panyushev, Abelian ideals of a Borel subalgebra and long positive roots, Int. Math. Res. Not. 35 (2003), 1889–1913. MR 1995141, DOI 10.1155/S1073792803211053
- Dmitri Panyushev and Gerhard Röhrle, Spherical orbits and abelian ideals, Adv. Math. 159 (2001), no. 2, 229–246. MR 1825058, DOI 10.1006/aima.2000.1959
- I. Schur, Zur Theorie der vertauschbaren Matrizen, J. Reine Angew. Math. 130 (1905), 66–76.
- Ruedi Suter, Abelian ideals in a Borel subalgebra of a complex simple Lie algebra, Invent. Math. 156 (2004), no. 1, 175–221. MR 2047661, DOI 10.1007/s00222-003-0337-0
Bibliographic Information
- Li Luo
- Affiliation: Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
- Email: luoli@amss.ac.cn
- Received by editor(s): January 24, 2008
- Published electronically: September 29, 2008
- Communicated by: Gail R. Letzter
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 479-485
- MSC (2000): Primary 17B05, 17B56; Secondary 17B20, 17B30
- DOI: https://doi.org/10.1090/S0002-9939-08-09685-8
- MathSciNet review: 2448567