Post-Lie algebra structures for perfect Lie algebras

Figures & data

Table

$\mathfrak{g}$	$\mathrm{\text{dim}}\mathfrak{g}$	$\mathrm{\text{dim}}Z(\mathfrak{g})$	Turkowski	Alev et al.
$\mathfrak{s}{\mathfrak{l}}_2(\mathbb{C})⋉V(2)$	5	0	$L_{5,1}$	L5
$\mathfrak{s}{\mathfrak{l}}_2(\mathbb{C})⋉V(3)$	6	0	$L_{6,4}\cong L_{6,1}$	$L_{6,1}$
$\mathfrak{s}{\mathfrak{l}}_2(\mathbb{C})⋉{\mathfrak{n}}_3(\mathbb{C})$	6	1	$L_{6,2}$	$L_{6,3}$
$\mathfrak{s}{\mathfrak{l}}_2(\mathbb{C})⋉V(4)$	7	0	$L_{7,6}$	$L_{7,1}$
$\mathfrak{s}{\mathfrak{l}}_2(\mathbb{C})⋉(V(2)\oplus V(2))$	7	0	$L_{7,7}$	$L_{7,2}$
$\mathfrak{s}{\mathfrak{l}}_2(\mathbb{C})\oplus (\mathfrak{s}{\mathfrak{l}}_2(\mathbb{C})⋉V(2))$	8	0	$\mathfrak{s}{\mathfrak{l}}_2\oplus L_{5,1}$	$\mathfrak{s}{\mathfrak{l}}_2\oplus L_5$
$\mathfrak{s}{\mathfrak{l}}_2(\mathbb{C})⋉V(5)$	8	0	$L_{8,21}$	$L_{8,1}$
$\mathfrak{s}{\mathfrak{l}}_2(\mathbb{C})⋉(V(2)\oplus V(3))$	8	0	$L_{8,22}$	$L_{8,2}$
$\mathfrak{s}{\mathfrak{l}}_2(\mathbb{C})⋉(V(2)\oplus {\mathfrak{n}}_3(\mathbb{C}))$	8	1	$L_{8,13}^{\epsilon =0}$	$L_{8,13}$
$\mathfrak{s}{\mathfrak{l}}_2(\mathbb{C})⋉{\mathfrak{f}}_{2,3}(\mathbb{C})$	8	0	$L_{8,15}$	$L_{8,18}$
$\mathfrak{s}{\mathfrak{l}}_2(\mathbb{C}){⋉}_{\phi }{\mathfrak{n}}_5(\mathbb{C})$	8	1	$L_{8,13}^1\cong L_{8,13}^{-1}$	$L_{8,15}$
$\mathfrak{s}{\mathfrak{l}}_2(\mathbb{C}){⋉}_{\psi }{\mathfrak{n}}_5(\mathbb{C})$	8	1	$L_{8,19}$	$L_{8,16}$

Table

$\mathfrak{g}$	$\mathrm{\text{dim}}\mathfrak{g}$	$\mathrm{\text{dim}}Z(\mathfrak{g})$	Turkowski
$\mathfrak{s}{\mathfrak{l}}_2(\mathbb{C})\oplus (\mathfrak{s}{\mathfrak{l}}_2(\mathbb{C})⋉V(3))$	9	0	$\mathfrak{s}{\mathfrak{l}}_2\oplus L_{6,1}$
$\mathfrak{s}{\mathfrak{l}}_2(\mathbb{C})\oplus (\mathfrak{s}{\mathfrak{l}}_2(\mathbb{C})⋉{\mathfrak{n}}_3(\mathbb{C}))$	9	1	$\mathfrak{s}{\mathfrak{l}}_2\oplus L_{6,2}$
$\mathfrak{s}{\mathfrak{l}}_2(\mathbb{C})⋉V(6)$	9	0	$L_{9,59}$
$\mathfrak{s}{\mathfrak{l}}_2(\mathbb{C})⋉(V(2)\oplus V(4))$	9	0	$L_{9,60}$
$\mathfrak{s}{\mathfrak{l}}_2(\mathbb{C})⋉(V(3)\oplus V(3))$	9	0	$L_{9,61}$
$\mathfrak{s}{\mathfrak{l}}_2(\mathbb{C})⋉(V(2)\oplus V(2)\oplus V(2))$	9	0	$L_{9,63}$
$\mathfrak{s}{\mathfrak{l}}_2(\mathbb{C})⋉(V(3)\oplus {\mathfrak{n}}_3(\mathbb{C}))$	9	1	$L_{9,58}$
$\mathfrak{s}{\mathfrak{l}}_2(\mathbb{C})⋉({\mathfrak{n}}_3(\mathbb{C})\oplus {\mathfrak{n}}_3(\mathbb{C}))$	9	2	$L_{9,37}\cong L_{9,42}$
$\mathfrak{s}{\mathfrak{l}}_2(\mathbb{C})⋉{\mathfrak{f}}_{3,2}(\mathbb{C})$	9	0	$L_{9,62}$
$\mathfrak{s}{\mathfrak{l}}_2(\mathbb{C})⋉{\mathcal{A}}_{6,4}(\mathbb{C})$	9	2	$L_{9,41}$

Table

$(\mathfrak{g},\mathfrak{n})$	$\mathfrak{n}$ abe	$\mathfrak{n}$ nil	$\mathfrak{n}$ sol	$\mathfrak{n}$ sim	$\mathfrak{n}$ sem	$\mathfrak{n}$ red	$\mathfrak{n}$ com	$\mathfrak{n}$ per
$\mathfrak{g}$ abelian	✓	✓	✓	–	–	–	✓	–
$\mathfrak{g}$ nilpotent	✓	✓	✓	–	–	–	✓	–
$\mathfrak{g}$ solvable	✓	✓	✓	✓	✓	✓	✓	✓
$\mathfrak{g}$ simple	–	–	–	✓	–	–	–	–
$\mathfrak{g}$ semisimple	–	–	–	–	✓	–	–	–
$\mathfrak{g}$ reductive	✓	✓	✓	–	$?$	✓	✓	✓
$\mathfrak{g}$ complete	✓	✓	✓	✓	✓	✓	✓	✓
$\mathfrak{g}$ perfect	–	$?$	–	$?$	$?$	✓	–	✓