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Positive Systems of Kostant Roots

Abstract

Let \(\mathfrak {g}\) be a simple complex Lie algebra and let \(\mathfrak {t} \subset \mathfrak {g}\) be a toral subalgebra of \(\mathfrak {g}\). As a \(\mathfrak {t}\)-module \(\mathfrak {g}\) decomposes as

$$\mathfrak{g} = \mathfrak{s} \oplus \left( \oplus_{\nu \in \mathcal{R}}~ \mathfrak{g}^{\nu}\right)$$

where \(\mathfrak {s} \subset \mathfrak {g}\) is the reductive part of a parabolic subalgebra of \(\mathfrak {g}\) and \(\mathcal {R}\) is the Kostant root system associated to \(\mathfrak {t}\). When \(\mathfrak {t}\) is a Cartan subalgebra of \(\mathfrak {g}\) the decomposition above is nothing but the root decomposition of \(\mathfrak {g}\) with respect to \(\mathfrak {t}\); in general the properties of \(\mathcal {R}\) resemble the properties of usual root systems. In this note we study the following problem: “Given a subset \(\mathcal {S} \subset \mathcal {R}\), is there a parabolic subalgebra \(\mathfrak {p}\) of \(\mathfrak {g}\) containing \(\mathcal {M} = \oplus _{\nu \in \mathcal {S}} \mathfrak {g}^{\nu }\) and whose reductive part equals \(\mathfrak {s}\)?”. Our main results is that, for a classical simple Lie algebra \(\mathfrak {g}\) and a saturated \(\mathcal {S} \subset \mathcal {R}\), the condition \((\text {Sym}^{\cdot }(\mathcal {M}))^{\mathfrak {s}} = \mathbb {C}\) is necessary and sufficient for the existence of such a \(\mathfrak {p}\). In contrast, we show that this statement is no longer true for the exceptional Lie algebras F4,E6,E7, and E8. Finally, we discuss the problem in the case when \(\mathcal {S}\) is not saturated.

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References

  1. Bourbaki, N.: Éléments de mathématique. Groupes et algèbres de Lie, Ch. IV – VI, Herman, Paris, 288 pp (1968)

  2. Dimitrov, I., Futorny, V., Grantcharov, D.: Parabolic sets of roots. Contemp. Math. 499, 61–74 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  3. Dimitrov, I., Penkov, I.: Weight modules of direct limit Lie algebras. IMRN 5, 223–249 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  4. Dimitrov, I., Roth, M.: Cup products of line bundles on homogeneous varieties and generalized PRV components of multiplicity one, to appear in Algebra & Number Theory

  5. Kostant, B.: Root Systems for Levi Factors and Borel–de Siebenthal Theory, Symmetry and Spaces. Progr. Math., vol. 278, pp 129–152. Birkhäuser Boston, Inc., Boston (2010)

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Correspondence to Ivan Dimitrov.

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Presented by Peter Littelmann.

Research of I. Dimitrov and M. Roth was partially supported by NSERC grants

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Dimitrov, I., Roth, M. Positive Systems of Kostant Roots. Algebr Represent Theor 20, 1365–1378 (2017). https://doi.org/10.1007/s10468-017-9691-2

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  • DOI: https://doi.org/10.1007/s10468-017-9691-2

Keywords

  • Parabolic subalgebras
  • Kostant root systems
  • Positive roots

Mathematics Subject Classification (2010)

  • Primary 17B22; Secondary 17B20
  • 17B25