Special Weights and Roots for Finite Lie Algebras

TL;DR

This paper introduces special roots and weights in finite Lie algebras, providing explicit constructions and showing how they offer an alternative perspective on Weyl group actions.

Contribution

It presents new special weights and roots, along with explicit methods to construct them, enhancing understanding of Weyl group actions in finite Lie algebras.

Findings

Existence of special weights with unique properties

Explicit construction methods for special weights and roots

Alternative expression of Weyl group actions

Abstract

With the introduction of special roots, we show the existence of some special weights with quite interesting properties for finite Lie algebras. We propose and discuss two statements which lead us to an explicit construction of these special weights and roots. It is seen that special weights provides us an alternative way of expressing the whole action of Weyl groups.