Glorious pairs of roots and Abelian ideals of a Borel subalgebra

Abstract

Let \({{\mathfrak {g}}}\) be a simple Lie algebra with a Borel subalgebra \({{\mathfrak {b}}}\). Let \(\Delta ^+\) be the corresponding (po)set of positive roots and \(\theta \) the highest root. A pair \(\{\eta ,\eta '\}\subset \Delta ^+\) is said to be glorious, if \(\eta ,\eta '\) are incomparable and \(\eta +\eta '=\theta \). Using the theory of abelian ideals of \({{\mathfrak {b}}}\), we (1) establish a relationship of \(\eta ,\eta '\) to certain abelian ideals associated with long simple roots, (2) provide a natural bijection between the glorious pairs and the pairs of adjacent long simple roots (i.e., some edges of the Dynkin diagram), and (3) point out a simple transform connecting two glorious pairs corresponding to the incident edges in the Dynkin diagram. In types \({{\mathbf {\mathsf{{{DE}}}}}}_{}\), we prove that if \(\{\eta ,\eta '\}\) corresponds to the edge through the branching node of the Dynkin diagram, then the meet \(\eta \wedge \eta '\) is the unique maximal non-commutative root. There is also an analogue of this property for all other types except type \({{\mathbf {\mathsf{{{A}}}}}}_{}\). As an application, we describe the minimal non-abelian ideals of \({{\mathfrak {b}}}\).

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Correspondence to Dmitri I. Panyushev.

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This research is partially supported by the R.F.B.R. Grant No. 16-01-00818.

Appendix A. Data for the exceptional Lie algebras

Appendix A. Data for the exceptional Lie algebras

A.1 For \({{\mathbf {\mathsf{{{E}}}}}}_{6}\), \({{\mathbf {\mathsf{{{E}}}}}}_{7}\), and \({{\mathbf {\mathsf{{{E}}}}}}_{8}\), the numbering of the simple roots is

For \({{\mathbf {\mathsf{{{E}}}}}}_{6}\), \({{\mathbf {\mathsf{{{E}}}}}}_{7}\), and \({{\mathbf {\mathsf{{{E}}}}}}_{8}\), the numbering of the simple roots is

figured

respectively.

The glorious pairs \((\eta ,\eta ')\) associated with adjacent simple roots \((\alpha ,\alpha ')\) are given below. Recall that here \(\eta \in \min (I(\alpha )_{\textsf {min}})\) and \({\mathsf {cl}}(\eta )=\alpha '\).

A.2 For \({{\mathbf {\mathsf{{{D}}}}}}_{n}\) and \({{\mathbf {\mathsf{{{E}}}}}}_{n}\), the explicit correspondence between tails and odd roots is presented below. The numbering of tails follows Fig. 1. That is, according to the chosen numbering of \(\Pi \), see Example 3.8 and figures in Sect. 1, \({{\mathcal {T}}}_1\) contains \(\alpha _1\), \({{\mathcal {T}}}_2\) contains \(\alpha _{n-1}\), and \({{\mathcal {T}}}_3\) contains \(\alpha _n\). In particular, we always have \(\#{{\mathcal {T}}}_3=1\). Then the odd root \(\beta _i\) corresponding to \({{\mathcal {T}}}_i\) is:

A.3 Here we provide the transition roots for the pairs of incident edges in types \({{\mathbf {\mathsf{{{E}}}}}}_{6}\), \({{\mathbf {\mathsf{{{E}}}}}}_{7}\), and \({{\mathbf {\mathsf{{{E}}}}}}_{8}\), and thereby explicitly present the bijection of Remark 3.13.

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Panyushev, D.I. Glorious pairs of roots and Abelian ideals of a Borel subalgebra. J Algebr Comb 52, 505–525 (2020). https://doi.org/10.1007/s10801-019-00911-9

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Keywords

  • Root system
  • Borel subalgebra
  • Abelian ideal
  • Adjacent simple roots

Mathematics Subject Classification

  • 17B20
  • 17B22
  • 06A07
  • 20F55