Fine Gradings of Low-Rank Complex Lie Algebras and of Their Real Forms
Milena Svobodova

TL;DR
This review explores the classification of fine gradings of low-rank complex Lie algebras and their real forms, clarifying the relationship between gradings and group gradings, and developing methods for their systematic identification.
Contribution
The paper revises the correspondence between fine gradings and MAD-groups for simple Lie algebras and introduces new methods for finding fine gradings, especially for real forms.
Findings
Revised the relation between gradings and group gradings for simple Lie algebras.
Developed methods for identifying fine (group) gradings of complex and real Lie algebras.
Applied these methods to classify gradings of specific low-rank Lie algebras.
Abstract
In this review paper, we treat the topic of fine gradings of Lie algebras. This concept is important not only for investigating the structural properties of the algebras, but, on top of that, the fine gradings are often used as the starting point for studying graded contractions or deformations of the algebras. One basic question tackled in the work is the relation between the terms 'grading' and 'group grading'. Although these terms have originally been claimed to coincide for simple Lie algebras, it was revealed later that the proof of this assertion was incorrect. Therefore, the crucial statements about one-to-one correspondence between fine gradings and MAD-groups had to be revised and re-formulated for fine group gradings instead. However, there is still a hypothesis that the terms 'grading' and 'group grading' coincide for simple complex Lie algebras. We use the MAD-groups as the…
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