Fine gradings of complex simple Lie algebras and Finite Root Systems

TL;DR

This paper introduces finite root systems associated with complex simple Lie algebras, classifies maximal abelian subgroups inducing quasi-good gradings, and constructs bases for these Lie algebras using these systems.

Contribution

It defines finite root systems analogous to classical root systems, classifies maximal abelian subgroups inducing quasi-good gradings, and links these to the structure of complex simple Lie algebras.

Findings

Finite root systems satisfy specific axioms similar to classical root systems.

Five series of maximal abelian subgroups induce quasi-good gradings in classical Lie algebras.

The set of roots of these subgroups forms a finite root system.

Abstract

A $G$-grading on a complex semisimple Lie algebra $L$, where $G$ is a finite abelian group, is called quasi-good if each homogeneous component is 1-dimensional and 0 is not in the support of the grading. Analogous to classical root systems, we define a finite root system $R$ to be some subset of a finite symplectic abelian group satisfying certain axioms. There always corresponds to $R$ a semisimple Lie algebra $L(R)$ together with a quasi-good grading on it. Thus one can construct nice basis of $L(R)$ by means of finite root systems. We classify finite maximal abelian subgroups $T$ in $\Aut(L)$ for complex simple Lie algebras $L$ such that the grading induced by the action of $T$ on $L$ is quasi-good, and show that the set of roots of $T$ in $L$ is always a finite root system. There are five series of such finite maximal abelian subgroups, which occur only if $L$ is a classical simple Lie algebra.