Abstract
Let \({\mathfrak{g}=\mathfrak{g}^{\bar 0}\oplus \mathfrak{g}^{\bar 1}}\) be a \({\mathbb{Z}_2}\)-graded Lie algebra. We study the posets of abelian subalgebras of \({\mathfrak{g}^{\bar 1}}\) which are stable w.r.t. a Borel subalgebra of \({\mathfrak{g}^{\bar 0}}\). In particular, we find a natural parametrization of maximal elements and dimension formulas for them. We recover as special cases several results of Kostant, Panyushev, and Suter.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cellini P., Papi P.: ad-nilpotent ideals of a Borel subalgebra. J. Algebra 225, 130–141 (2000)
Cellini P., Papi P.: Abelian ideals of Borel subalgebras and affine Weyl groups. Adv. Math. 187, 320–361 (2004)
Cellini P., Möseneder Frajria P., Papi P.: Abelian subalgebras in \({\mathbb{Z}_2}\)-graded Lie algebras and affine Weyl groups. Int. Math. Res. Notices 43, 2281–2304 (2004)
Cellini P., Kac V.G., Möseneder Frajria P., Papi P.: Decomposition rules for conformal pairs associated to symmetric spaces and abelian subalgebras of \({\mathbb{Z}_2}\)-graded Lie algebras. Adv. Math. 207, 156–204 (2006)
Deodhar V.: A note on subgroups generated by reflections in Coxeter groups. Arch. Math. (Basel) 53(6), 543–546 (1989)
Iwahori N., Matsumoto H.: On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups. Inst. Hautes tudes Sci. Publ. Math. No. 25, 5–48 (1965)
Jacobson N.: Schur’s theorems on commutative matrices. Bull. Am. Math. Soc. 50, 431–436 (1944)
Kac V.G., Möseneder Frajria P., Papi P.: On the Kernel of the affine Dirac operator. Moscow Math. J. 8(4), 759–788 (2008)
Kostant B.: Eigenvalues of the Laplacian and commutative Lie subalgebras. Topology 3(Suppl 2), 147–159 (1965)
Kac V.G.: Infinite-dimensional Lie algebras. 3rd edn. Cambridge University Press, Cambridge (1990)
Kostant B.: The set of abelian ideals of a borel subalgebra, cartan decompositions, and discrete series representations. Int. Math. Res. Notices 5, 225–252 (1998)
Kostant B.: Powers of the Euler product and commutative subalgebras of a complex simple Lie algebra. Invent. Math. 158, 181–226 (2004)
Malcev, A.: Commutative subalgebras of semi-simple Lie algebras. Bull. Acad. Sci. URSS Sér. Math. 9, 291–300 (1945) [Izvestia Akad. Nauk SSSR] A. I. Malcev, Commutative subalgebras of semi-simple Lie algebras Am. Math. Soc. Transl. 40 (1951)
Möseneder Frajria P., Papi P.: Casimir operators, abelian subalgebras and \({\mathfrak u}\)-cohomology. Rendiconti di Matematica 27, 265–276 (2007)
Panyushev D.: Isotropy representations, eigenvalues of a Casimir element, and commutative Lie subalgebras. J. Lond. Math. Soc. 64(1), 61–80 (2001)
Panyushev D., Röhrle G.: Spherical orbits and abelian ideals. Adv. Math. 159, 229–246 (2001)
Panyushev D.: Abelian ideals of a Borel subalgebra and long positive roots. Int. Math. Res. Notices 35, 1889–1913 (2003)
Schur I.: Zur Theorie der vertauschbaren Matrizen. J. Reine Angew. Math. 130, 66–76 (1905)
Suter R.: Abelian ideals in a Borel subalgebra of a complex simple Lie algebra. Invent. Math. 156(1), 175–221 (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cellini, P., Frajria, P.M., Papi, P. et al. On the structure of Borel stable abelian subalgebras in infinitesimal symmetric spaces. Sel. Math. New Ser. 19, 399–437 (2013). https://doi.org/10.1007/s00029-012-0097-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00029-012-0097-z