Normalisers of abelian ideals of a Borel subalgebra and $\mathbb{Z}$-gradings of a simple Lie algebra

Abstract

Let $\mathfrak{g}$ be a simple Lie algebra and $\mathfrak{Ab}$ the poset of all abelian ideals of a fixed Borel subalgebra of $\mathfrak{g}$. If $\mathfrak{a}\in \mathfrak{Ab}$, then the normaliser of $\mathfrak{a}$ is a standard parabolic subalgebra of $\mathfrak{g}$. We give an explicit description of the normaliser for a class of abelian ideals that includes all maximal abelian ideals. We also elaborate on a relationship between abelian ideals and $\mathbb{Z}$-gradings of $\mathfrak{g}$ associated with their normalisers.

13 pp

References in corpus (3)

• Abelian ideals in a Borel subalgebra of a complex simple Lie algebra
• Powers of the Euler product and commutative subalgebras of a complex simple Lie algebra
• Minimal inversion complete sets and maximal abelian ideals

Cited by in corpus (1)

• Abelian ideals of a Borel subalgebra and root systems, II