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Glorious pairs of roots and Abelian ideals of a Borel subalgebra

  • Dmitri I. PanyushevEmail author
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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 \({{\mathfrak {b}}}\).

Keywords

Root system Borel subalgebra Abelian ideal Adjacent simple roots 

Mathematics Subject Classification

17B20 17B22 06A07 20F55 

Notes

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute for Information Transmission Problems of the R.A.SMoscowRussia

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