Parabolic subalgebras, parabolic buildings and parabolic projection

TL;DR

This paper introduces an elementary approach to the geometry of parabolic subalgebras in reductive Lie algebras over characteristic zero fields, deriving structure theory from root systems and establishing a parabolic projection process.

Contribution

It provides a new, elementary method to understand parabolic subalgebras and their associated buildings without relying on the traditional structure theory.

Findings

Constructed the Tits building of a reductive Lie algebra.

Established a parabolic projection process to subalgebras of Levi quotients.

Outlined potential applications to geometric configurations and moduli spaces.

Abstract

Reductive (or semisimple) algebraic groups, Lie groups and Lie algebras have a rich geometry determined by their parabolic subgroups and subalgebras, which carry the structure of a building in the sense of J. Tits. We present herein an elementary approach to the geometry of parabolic subalgebras, over an arbitrary field of characteristic zero, which does not rely upon the structure theory of semisimple Lie algebras. Indeed we derive such structure theory, from root systems to the Bruhat decomposition, from the properties of parabolic subalgebras. As well as constructing the Tits building of a reductive Lie algebra, we establish a "parabolic projection" process which sends parabolic subalgebras of a reductive Lie algebra to parabolic subalgebras of a Levi subquotient. We indicate how these ideas may be used to study geometric configurations and their moduli.