Ad-nilpotent ideals of a Borel subalgebra II

By Paola Cellini, Paolo Papi. 2001

120 citations recorded by the source corpus.

We provide an explicit bijection between the ad-nilpotent ideals of a Borel subalgebra of a simple Lie algebra g and the orbits of Q/(h+1)Q under the Weyl group (Q being the coroot lattice and h the Coxeter number of g). From this result we deduce in a uniform way a counting formula for the ad-nilpotent ideals.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/689c8f518d1780a98c692e1bb8167b3522ddbec0

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Citations in the corpus

Spherical Orbits and Abelian Ideals (2001, background): “Following [4], we say that an ideal of b is ad nilpotent whenever it consists of nilpotent elements.”
Generalized Catalan Numbers, Weyl Groups and Arrangements of Hyperplanes (2004, methodology): “A case-free proof was given recently by Cellini and Papi [8], via a bijection with the W -orbits of Q̌/(h + 1)Q̌, thus answering a question raised by Krattenthaler et al.”
On a refinement of the generalized Catalan numbers for Weyl groups (2004, methodology): “A bijection between ideals in Φ and W -orbits of T was later described by Cellini and Papi [5] and is based on the construction of a map which assigns an element of the affine Weyl group Wa to each ideal in Φ [4].”
Cluster algebras: Notes for the CDM-03 conference (2003, background): “These numbers are known to count [6, 12, 14, 29, 34, 35, 45] a variety of combinatorial objects related to the root system Φ.”
ad-nilpotent ideals of a Borel subalgebra: generators and duality (2003, methodology): “There is a simple procedure for obtaining the affine-linear action on V from the linear action on V̂ , which is explained in [7], but we do not need this.”
Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups (2006, methodology): “On the other hand, Cellini and Papi [45] gave a bijective proof that the positive chambers in the Catalan arrangement are counted by (1.”
B-Stable Ideals in the Nilradical of a Borel Subalgebra (2003, methodology): “(1) [12], [3] There exists w ∈Wa such that k(α,w) = αI,+ for all α ∈ Φ.”
Powers of the Euler product and commutative subalgebras of a complex simple Lie algebra (2003, background): “= X ξ∈Ξ Vξ (0.8) Furthermore Cand, a fortiori, Ck, are multiplicity-free g-modules. It is a beautiful later result of Dale Peterson that cardΞ = 2ℓ. A simpler proof of Peterson’s theorem was given in [C-P]. See Theorem 2.9 in that reference. It is clear that there are 2ℓalcoves in 2A1. They are para”