On the index of the quotient of a Borel subalgebra by an ad-nilpotent ideal

TL;DR

This paper investigates bounds for the index of quotients of Borel subalgebras by ad-nilpotent ideals in simple Lie algebras, generalizing known formulas and proposing conjectures for exactness.

Contribution

It provides new upper bounds for these indices and extends Panov's formula from type A to more general cases, including conjecturing exactness.

Findings

Upper bounds for the index of Borel quotient are established.

The bound for the nilpotent radical quotient generalizes Panov's type A formula.

The bound for the Borel quotient is exact in type A and conjectured to be in general.

Abstract

In this paper, we give upper bounds for the index of the quotient of the Borel subalgebra of a simple Lie algebra or its nilpotent radical by an ad-nilpotent ideal. For the nilpotent radical quotient, our bound is a generalization of the formula for the index given by Panov in the type A case. In general, this bound is not exact. Using results from Panov, we show that the upper bound for the Borel quotient is exact in the type $A$ case, and we conjecture that it is exact in general.