AppendixAComplete classification in low dimension

SubsectionA.1Overview

The following pages contain the classification of gradings of all nilpotent Lie algebras up to dimension 6 and a representative sample of nilpotent Lie algebras of dimension 7. The gradings are organized by

• dimension
• central series (only in dimension 7)
• Lie algebra
• dimensions of the layers

Gradings are named as $$r.d_1d_2\ldots d_nl\text{,}$$ where $$r$$ is the rank of the grading, $$d_1,\ldots,d_n$$ are the dimensions of the layers (in increasing order) and $$l$$ is a letter $$a,b,c,\ldots$$ used to distinguish non-equivalent gradings with the same rank and type. For example consider the Heisenberg Lie algebra $$L_{3,2}$$ in the basis $$Y_1,Y_2,Y_3$$ with the Lie bracket defined by $$[Y_1,Y_2]=Y_3\text{.}$$ The maximal grading is a rank 2 grading

with 3 layers of dimension 1 and is classified as grading 2.3a of $$L_{3,2}\text{.}$$ The stratification