Abelian Ideals and Cohomology of Symplectic Type

TL;DR

This paper explores the connection between abelian ideals in Borel subalgebras of symplectic Lie algebras and the cohomology of their nilpotent parts, providing enumeration methods linked to Weyl group polynomials.

Contribution

It establishes a novel relationship between abelian ideals and cohomology in symplectic Lie algebras, enabling enumeration via Weyl group Poincare polynomials.

Findings

Enumerates abelian ideals using Weyl group polynomials

Establishes a relationship between abelian ideals and cohomology

Provides formulas for counting ideals of specific dimensions

Abstract

For symplectic Lie algebras $\mathfrak{sp}(2n,\mathbb{C})$, denote by $\mathfrak{b}$ and $\mathfrak{n}$ its Borel subalgebra and maximal nilpotent subalgebra, respectively. We construct a relationship between the abelian ideals of $\mathfrak{b}$ and the cohomology of $\mathfrak{n}$ with trivial coefficients. By this relationship, we can enumerate the number of abelian ideals of $\mathfrak{b}$ with certain dimension via the Poincare polynomials of Weyl groups of type $A_{n-1}$ and $C_n$.