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Journal of Lie Theory 33 (2023), No. 2, 497--526
Copyright Heldermann Verlag 2023

Combinatorial and Geometric Constructions Associated with the Kostant Cascade

Dmitri I. Panyushev
Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia


\newcommand{\g}{{\mathfrak g}} \newcommand{\be}{{\mathfrak b}} \newcommand{\te}{{\mathfrak t}} \newcommand{\ut}{{\mathfrak u}} \newcommand{\gK}{{\mathcal K}} Let $\g$ be a complex simple Lie algebra and $\be=\te\oplus\ut^+$ a fixed Borel subalgebra. Let $\Delta^+$ be the set of positive roots associated with $\ut^+$ and $\gK\subset\Delta^+$ the Kostant cascade. We elaborate on some constructions related to $\gK$ and applications of $\gK$. This includes the cascade element $x_\gK$ in the Cartan subalgebra $\te$ and properties of certain objects naturally associated with $\gK$: an abelian ideal of $\be$, a nilpotent $G$-orbit in $\g$, and an involution of $\g$.

Keywords: Root system, cascade element, abelian ideal, Frobenius algebra, nilpotent orbit.

MSC: 17B22, 17B20, 17B08, 14L30.

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