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Ideals of Heisenberg type and minimax elements of affine Weyl groups. (English) Zbl 1086.17005

Vinberg, Ernest (ed.), Lie groups and invariant theory. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3733-8/hbk). Translations. Series 2. American Mathematical Society 213. Advances in the Mathematical Sciences 56, 191-213 (2005).
Let \({\mathfrak g}\) be a complex simple Lie algebra, \({\mathfrak b}\) a fixed Borel subalgebra of \({\mathfrak g}\) and \(\Delta^+\) the set of positive roots of \({\mathfrak g}\) respect to a Cartan subalgebra contained in \({\mathfrak b}\). Ad-nilpotent ideals (that is, ideals of \({\mathfrak b}\) contained in \([{\mathfrak b},{\mathfrak b}]\)) are determined by subsets of \(\Delta^+\); the subsets without simple roots correspond to the so called strictly positive ad-nilpotent ideals.
Denoting by \(Ud({\mathfrak g})\) and \({Ud({\mathfrak g})}_0\) the sets of all ad-nilpotent and strictly positive ad-nilpotent) ideals of \({\mathfrak g}\) respectively, in [P. Cellini and P. Papi, J. Algebra 258, No. 1, 112–121 (2002; Zbl 1033.17008)], a bijection is established between \(Ud({\mathfrak g})\) and certain elements (admissible elements for Cellini-Papi, minimal elements in this paper) of \(\widehat W\), the affine Weyl group. More recently, N. Sommers [Can. Math. Bull. 48, No. 3, 460–472 (2005; Zbl 1139.17303)] described a bijection between \({Ud({\mathfrak g})}_0\) and the set maximal elements of \(\widehat W\). Each bijection lead to a formula which compute the cardinality of the sets \(Ud({\mathfrak g})\) and \({Ud({\mathfrak g})}_0\).
After introduction, in this paper Cellini-Papi and Sommers previous results are surveyed, the so called ideals of Heisenberg type are studied and minimax elements in \(\widehat W\) (elements which are simultaneously minimal and maximal) are characterized. In the last part, the author counts the number of minimax elements (minimax ideals) for each classical \({\mathfrak g}\) and shows that these numbers are in connection with some famous integer sequences as \(n\)th the Motzkin number in case \({\mathfrak g}= \text{sl}_{n+1}\).
For the entire collection see [Zbl 1063.22001].

MSC:

17B20 Simple, semisimple, reductive (super)algebras
20F55 Reflection and Coxeter groups (group-theoretic aspects)
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