Abelian Ideals of a Borel Subalgebra and Root Systems, II

  • Dmitri I. PanyushevEmail author


Let 𝔤 be a simple Lie algebra with a Borel subalgebra 𝔟 and 𝔄𝔟 the set of abelian ideals of 𝔟. Let Δ+ be the corresponding set of positive roots. We continue our study of combinatorial properties of the partition of 𝔄𝔟 parameterised by the long positive roots. In particular, the union of an arbitrary set of maximal abelian ideals is described, if 𝔤 ≠ 𝔰𝔩n. We also characterise the greatest lower bound of two positive roots, when it exists, and point out interesting subposets of Δ+ that are modular lattices.


Root system Borel subalgebra Abelian ideal Modular lattice 

Mathematic Subject Classification (2010)

17B20 17B22 06A07 20F55 


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Part of this work was done during my stay at the Max-Planck-Institut für Mathematik (Bonn). I would like to thank the Institute for its warm hospitality and excellent working conditions.


  1. 1.
    Bourbaki, N.: Groupes et algèbres de Lie” Chapitres 4,5 et, vol. 6. Hermann, Paris (1975)Google Scholar
  2. 2.
    Cellini, P., Papi, P.: ad-nilpotent ideals of a Borel subalgebra. J. Algebra 225, 130–141 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cellini, P., Papi, P.: Abelian ideals of Borel subalgebras and affine Weyl groups. Adv. Math. 187, 320–361 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gorbatsevich, V.V., Onishchik, A.L., Vinberg, E.B.: “Lie Groups and Lie Algebras” III (Encyclopaedia Math Sci.), vol. 41. Springer, Berlin (1994)Google Scholar
  5. 5.
    Joseph, A.: Orbital varieties of the minimal orbit. Ann. Scient. Éc. Norm. Sup. (4) 31, 17–45 (1998)CrossRefzbMATHGoogle Scholar
  6. 6.
    Kostant, B.: The set of abelian ideals of a Borel subalgebra, Cartan decompositions, and discrete series representations. Intern. Math. Res. Notices 5, 225–252 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kostant, B.: Root Systems for Levi Factors and Borel-de Siebenthal Theory. “Symmetry and spaces”, 129–152, Progr. Math., vol. 278. Birkhäuser Boston, Inc, Boston (2010)Google Scholar
  8. 8.
    Panyushev, D.: Abelian ideals of a Borel subalgebra and long positive roots. Intern. Math. Res. Notices 35, 1889–1913 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Panyushev, D.: The poset of positive roots and its relatives. J. Algebraic Combin. 23, 79–101 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Panyushev, D.: Two covering polynomials of a finite poset, with applications to root systems and ad-nilpotent ideals. J. Comb. 3, 63–89 (2012)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Panyushev, D.: Abelian ideals of a Borel subalgebra and root systems. J. Eur. Math. Soc. 16(12), 2693–2708 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Panyushev, D.: Minimal inversion complete sets and maximal abelian ideals. J. Algebra 445, 163–180 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Panyushev, D.: Normalisers of abelian ideals of Borel subalgebras and Z-gradings of a simple Lie algebra. J. Lie Theory 26, 659–672 (2016)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Panyushev, D., Röhrle, G.: Spherical orbits and abelian ideals. Adv. Math. 159, 229–246 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Stanley, R.P: “Enumerative Combinatorics”, vol. 1 (Cambridge Stud. Adv. Math. vol. 49). Cambridge Univ Press (1997)Google Scholar
  16. 16.
    Suter, R.: Abelian ideals in a Borel subalgebra of a complex simple Lie algebra. Invent Math. 156, 175–221 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Institute for Information Transmission Problems of the R.A.S.MoscowRussia

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