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Appendix B Complete classification in low dimension

Subsection B.1 Overview

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
  • rank of the grading
  • 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

\begin{equation*} V_{(1,0)} = \langle Y_1\rangle,\quad V_{(0,1)} = \langle Y_2\rangle,\quad V_{(1,1)} = \langle Y_3\rangle,\quad \end{equation*}

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

\begin{equation*} V_{1} = \langle Y_1,Y_2\rangle,\quad V_{2} = \langle Y_3\rangle \end{equation*}

is a rank 1 grading with layers of dimensions 1 and 2 and is classified as 1.11b to distinguish it from the grading

\begin{equation*} V_{0} = \langle Y_1\rangle,\quad V_{1} = \langle Y_2,Y_3\rangle\text{.} \end{equation*}

Gradings that admit a positive realization are highlighted in blue and the stratification is highlighted in red. Any parent class is similarly highlighted in blue or red, so for instance any Lie algebra which admits a stratification has a red highlight, and any grading type that admits a positive grading has a blue highlight.