Fine gradings and gradings by root systems on simple Lie algebras (Q2340466)

The author continues in this paper a series of previous studies on the classifications of gradings by abelian groups. Indeed, these gradings are already classified for the classical simple Lie algebras and for the exceptional simple algebras. On the other hand, gradings by root systems were introduced by Berman and Moody to study some classes of infinite-dimensional Lie algebras. In this paper, the author relates both types of gradings, showing that any fine grading with infinite universal grading group on a simple finite-dimensional Lie algebra over an algebraically closed field of characteristic \(0\) induces a grading by a (possibly non-reduced) root system. Concretely, he shows that any fine grading is determined by a grading by a root system and a special grading on the coordinate algebra of the root grading. Finally, he also obtains certain consequences for the classification of the fine gradings on the simple exceptional simple Lie algebras.