"""Complete projective-plane case division, Theorem 8.9. CC0-1.0."""
from pathlib import Path
from html import escape
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<title id="title">Every complex projective-plane Lie algebra</title>
<desc id="desc">Irreducible matrix actions are the full sl3 algebra or the three-dimensional conic stabilizer. Reducible nonsolvable actions give six affine or block models. Every solvable action preserves a complete flag and lies in its five-dimensional stabilizer; projection onto the affine x-line and all vertical kernels give the complete covering list.</desc>
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text(40,48,'The complete projective-plane reduction','title')
text(40,85,'h ⊂ pgl₃(ℂ) ≅ sl₃(ℂ), acting on ℙ(ℂ³); ordinary projective conjugacy first.','math')
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text(640,147,'Does the matrix action preserve a proper nonzero subspace of ℂ³?','head','middle')
text(640,180,'A matrix line is a projective point; a matrix plane is a projective line.','small','middle')
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text(65,273,'No: irreducible','head')
text(65,312,'Every solvable ideal acts by scalars, hence is zero.','small')
text(65,350,'A maximal toral algebra has dimension 1 or 2.','small')
text(65,391,'rank 1: C₃, the nonsingular conic stabilizer','math')
text(65,431,'rank 2: full sl₃, dimension 8','math')
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text(665,273,'Yes: reducible','head')
text(665,312,'A point or a line is invariant.','small')
text(665,350,'Point and line cases are exchanged by duality.','small')
text(665,391,'Separate nonsolvable and solvable actions.','head')
text(665,431,'Ordinary projective types remain distinct.','small')
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text(65,531,'Reducible and nonsolvable: six types','head')
text(65,572,'A preserved line gives affine matrices (L,b) ∈ gl₂ ⋉ ℂ².','math')
text(65,612,'The image is sl₂ or gl₂; its translation kernel is 0 or ℂ².','small')
text(65,652,'Full kernel: A₅, A₆ and their duals.   Zero kernel: L₃, L₄ (self-dual).','math')
text(65,686,'The zero-kernel graph is removed by the explicit equation β([L,K]) = Lβ(K) − Kβ(L).','small')
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text(65,778,'Solvable: an incident point / line flag is invariant','head')
text(65,819,'Choose the line at infinity and its vertical point:  b = ⟨p,q,R,A,B⟩, R = xq.','math')
text(65,861,'Projection onto the affine x-line:','head')
text(65,903,'image 0','math');text(420,903,'image ⟨p⟩ or ⟨A⟩','math');text(900,903,'image ⟨p,A⟩','math')
text(65,947,'The vertical kernel lies in k = ⟨q,R,B⟩. Its complete list is:','small')
text(65,988,'0;   ⟨q⟩, ⟨R⟩, ⟨B⟩;   ⟨q,R⟩, ⟨q,B⟩, ⟨R,B⟩;   k.','math')
text(65,1030,'A kernel with a nonconstant translation coefficient forces its root to be fixed by the image.','small')
text(65,1068,'Lemma 8.8 solves every lift and every residual bracket, including resonant coefficients.','small')
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text(65,1162,'Changing the flag gives the final projective parameter relations','head')
text(65,1203,'F₄(a,c), K₂(a,c): [a:c] exact.    T₃(a,c): [a:c] ~ [c:a].','math')
text(65,1244,'W₃(a,c): [a:c] ~ [a:a−c].    One-dimensional distinct eigenvalues: six values.','math')
text(65,1285,'Nilpotent ranks, common kernels/images and Jordan parts separate all remaining types.','small')
text(65,1320,'No seven-dimensional algebra occurs; zero and the full algebra are included.','small')
text(40,1387,'Proof: Lemmas 8.6–8.8 and Theorem 8.9. Exact dual pairings: Corollary 8.10.','small')
text(40,1430,'The schematic concerns complex connected local projective actions and their matrix algebras.','small')
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