Nilpotent Lie algebras of small dimension

The finite-dimensional nilpotent simply connected Lie groups or nilpotent Lie algebras are all obtained by successive central extensions (in a non unique way in general) from $1$ or the null algebra. This results in a oriented graph listing the nilpotent Lie algebras, and on which the arrows designate these extensions. We have drawn this graph for real Lie algebras of dimension at most 6, or 7 in the case of filiform Lie algebras. The names are those of De Graaf up to dimension 6 and of Magnin in dimension 7 (who followed older conventions taken by Carles, Dixmier and Vergne). Technically, de Graaf lists the algebras over any field, while Magnin lists the Lie algebras over $\mathbf C$. If the ground field is taken to be $\mathbf R$, the discriminant $\delta$ which appears in some algebra can be taken to be $-1,0$ or $1$; these are real forms of the same complex Lie algebra, and if $\delta \neq 0$ their associated Lie groups have infinitely many distinct commensurability classes of lattices. The notation $\mathfrak{g} \stackrel{r}{\longrightarrow}\mathfrak{h}$ means that $\mathfrak{h}$ is a $r$-central extension of $\mathfrak{g}$ by $\mathbf R$, that is, there is a short exact sequence \[ 0 \to \mathbf R \to \mathfrak g \to \mathfrak h \to 0, \] where the image of $\mathbf R$ lies in $C^i \mathfrak g \cap Z(\mathfrak g)$. $C^i \mathfrak g$ is the descending central series of $\mathfrak g$, that is, $C^1 \mathfrak g = \mathfrak g$ and $C^{i+1} \mathfrak g = [\mathfrak g, C^i \mathfrak g]$ for $i \geqslant 1$. The notation $\mathfrak{g} \stackrel{\mathrm{gr}}{\dashrightarrow} \mathfrak{h}$ means that $\mathfrak{h}=\operatorname{gr}(\mathfrak{g})$, where \[ \operatorname{gr}(\mathfrak{g}) = \bigoplus_{i=1}^{+\infty} C^i \mathfrak g / C^{i+1} \mathfrak g. \] This is an extended version of the picture in Cone-equivalent nilpotent groups with different Dehn functions with Llosa Isenrich and Tessera. \[ \xymatrix@C=1.2cm@R=2cm{ & & & & & & & & & \mathcal{G}_{7,0.2} \ar@{-->}@/^0.5pc/[rr] & & \fbox{$\mathcal{G}_{7,2.3}$} & & & & \\ & & & & & \fbox{$\mathscr L_{6,19}(\delta)$} & & & \mathcal{G}_{7,1.1(i_\lambda)} \ar@{-->}@/^1pc/[rrru] & \mathcal{G}_{7,1.1(iv)} \ar@{-->}@/^0.5pc/[rru] & \mathcal{G}_{7,1.1(ii)} \ar@{-->}[ru] & & \mathcal{G}_{7,1.6} \ar@{-->}[lu] & \mathcal{G}_{7,1.4} \ar@{-->}@/_0.5pc/[llu] & \mathcal{G}_{7,0.1} \ar@{-->}@/_1pc/[lllu] & \mathcal{G}_{7,0.3} \ar@{-->}@/_1.5pc/[llllu] & & & \\ \fbox{$\mathscr L_{6,4}$} & \fbox{$\mathscr L_{6,2}$} & \fbox{$\mathscr L_{6,3}$} & \fbox{$\mathscr L_{6,8}$} & \fbox{$\mathscr L_{6,25}$} & \mathscr{L}_{6,6} \ar@{-->}[r] & \fbox{$\mathscr{L}_{6,7}$} & \mathscr{L}_{6,13} \ar@{-->}[l] & \mathscr{L}_{6,9} & \fbox{$\mathscr{L}_{6,21}(\delta)$} & \fbox{$\mathscr{L}_{6,16}$} & \fbox{$\mathscr{L}_{6,18}$} \ar[uu]_6 \ar[lu]_6 \ar[ru]_6 & & & & \\ \fbox{$\mathbf R^6$} & \fbox{$\mathscr{L}_{6,22}(\delta)$} & \mathscr{L}_{6,10} \ar@{-->}[u] & \mathscr{L}_{6,5} \ar@{-->}[lu] & \mathscr{L}_{6,23} \ar@/_2pc/@{-->}[u] & \fbox{$\mathscr{L}_{6,20}$} & \mathscr{L}_{6,12} \ar@{-->}[u] & \mathscr{L}_{6,11} \ar@{-->}[ul] & \mathscr{L}_{6,24}(\delta) \ar@{-->}[u] & \mathscr{L}_{6,14} \ar@{-->}[ru] & \mathscr{L}_{6,15} \ar@{-->}[ru] \ar[luu]_6 \ar[luuu]^6 \ar@/^2pc/[lluu]^6 & \mathscr{L}_{6,17} \ar@{-->}[u] \ar[rruu]^6 \ar[rrruu]^6 \ar@/_1pc/[rrrruu]_6 & & & & \\ & \fbox{$\mathbf R^5$} \ar[lu]^1 & \fbox{$\mathscr{L}_{5,4}$} \ar[lu]^2 \ar@/^2pc/[lluu]^1 & \fbox{$\mathscr{L}_{5,2}$} \ar[lu]^{3} \ar[u]^3 \ar[uul]_3 \ar@/^2pc/[lluu]^1 & \fbox{$\mathscr{L}_{5,8}$} \ar[u]^3 \ar[ru]^3 \ar@/^1pc/[luu]_1 \ar@/^2pc/[uu]^3 \ar[ruuu]_3 & &\fbox{$\mathscr{L}_{5,3}$} \ar@/_2pc/[uu] \ar[u]_4 \ar[ru] %\ar@/_6pc/[lllluu]^1 & \mathscr{L}_{5,5} \ar@/_1pc/[ru] \ar@{-->}[l] \ar@/_2pc/[uu]_4 %\ar@/_6pc/[llllu]_1 & \fbox{$\mathscr L_{5,9}$} \ar@/_1pc/[ruu]_4 \ar@/_2pc/[uu]_1 & \mathscr{L}_{5,6} \ar@{-->}[r]^{\mathrm{gr}} \ar[u]_5 \ar[ru] & \fbox{$\mathscr{L}_{5,7}$} \ar@/_1pc/[ru]_5 \ar@/_1pc/[ruu]^5 \ar@/_1.5pc/[uu]_5 & & & & \\ & &\fbox{$\mathbf R^4$} \ar[ur]^2 \ar[ul]^1 \ar[u]^2 & & \fbox{$\mathscr{L}_{4,2}$} \ar[urr]^3 \ar@/_2pc/[urrr]_3 \ar[ul]^1 \ar[u]^2 & & & & \fbox{$\mathscr{L}_{4.3}$} \ar[u]^3 \ar[ur]^4 \ar[rru]_4 & & & & & & \\ & & & \fbox{$\mathbf R^3$} \ar[ul]^1 \ar[ur]^2 & & & \fbox{$\mathscr{L}_{3,2}$} \ar[ull]^1 \ar[urr]_3 & & & & & & & \\ & & & & \fbox{$\mathbf R^2$} \ar[ul]^1 \ar[urr]_2 & & & & \\ & & & & \fbox{$\mathbf R$} \ar[u]^1 & & & } \] This picture was displayed here with the help of the XYpic extension for MathJax.