Case 4*a**3 +27 =0$ generic derivation : delta:= mat((xi(1,1),0,0,0,0,0),(xi(2,1),xi(2,2),0,0,0,0),(xi(3,1),xi(3,2),2*xi(1,1),0,0 ,0),(xi(4,1),xi(4,2), - xi(2,1),xi(2,2) + xi(1,1),0,0),(xi(5,1),xi(5,2),xi(5,3), xi(4,2) - xi(3,1),xi(2,2) + 2*xi(1,1),xi(5,6)),(xi(6,1),xi(6,2),xi(6,3),0,0,xi(6 ,6)))$ [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] adx1 := [ ] [0 1 0 0 0 0] [ ] [0 0 0 1 0 0] [ ] [0 0 0 0 0 0] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] adx2 := [ ] [-1 0 0 0 0 0] [ ] [0 0 1 0 0 0] [ ] [0 0 0 0 0 0] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] adx3 := [ ] [0 0 0 0 0 0] [ ] [0 -1 0 0 0 0] [ ] [0 0 0 0 0 0] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] adx4 := [ ] [0 0 0 0 0 0] [ ] [-1 0 0 0 0 0] [ ] [0 0 0 0 0 0] The generic nilpotent derivation : the eigenvalues are 0 xi(1,1):=0 xi(2,2):=0 xi(6,6):=0 by subtracting adjoints one then may suppose: xi(4,1):=0,xi(4,2):=0,xi(5,1):=0,xi(5,2):=0 delta:= [ 0 0 0 0 0 0 ] [ ] [xi(2,1) 0 0 0 0 0 ] [ ] [xi(3,1) xi(3,2) 0 0 0 0 ] [ ] [ 0 0 - xi(2,1) 0 0 0 ] [ ] [ 0 0 xi(5,3) - xi(3,1) 0 xi(5,6)] [ ] [xi(6,1) xi(6,2) xi(6,3) 0 0 0 ] We denote this delta by the shortform shortformdelta:={xi(2,1), ss, xi(3,1), xi(3,2), ss, xi(5,3), xi(5,6), ss, xi(6,1), xi(6,2), xi(6,3)} paramindexeslist:={{2,1},{3,1},{3,2},{5,3},{5,6},{6,1},{6,2},{6,3}} a neq conditionssura$ a:=a$ delta:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(0,1,0,0,0,0),(0,0,0,0,0,0),(0,0,a,0,0,1),(1,0,0 ,0,0,0))$ shortformdelta:={0, ss, 0, 1, ss, a, 1, ss, 1, 0, 0}$ on resout l'equation {{0,1},0} qui est maintenant AA:= - d(0,6)$ Unknown: d(0,6) Unknown: d(0,6) bonne inconnue W:=d(0,6)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,1},1} qui est maintenant AA:= - d(1,6)$ Unknown: d(1,6) Unknown: d(1,6) bonne inconnue W:=d(1,6)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,1},2} qui est maintenant AA:= - d(2,6)$ Unknown: d(2,6) Unknown: d(2,6) bonne inconnue W:=d(2,6)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,1},3} qui est maintenant AA:= - d(3,6) + d(2,1)$ Unknowns: {d(3,6),d(2,1)} Unknowns: {d(3,6),d(2,1)} bonne inconnue W:=d(3,6)$ sa valeur doit etre WW:=d(2,1)$ on resout l'equation {{0,1},4} qui est maintenant AA:= - (d(4,6) + d(2,0))$ Unknowns: {d(4,6),d(2,0)} Unknowns: {d(4,6),d(2,0)} bonne inconnue W:=d(4,6)$ sa valeur doit etre WW:= - d(2,0)$ on resout l'equation {{0,1},5} qui est maintenant AA:=d(6,1) - d(5,6) - d(4,0) + d(3,1)*a$ Unknowns: {d(6,1),d(5,6),d(4,0),d(3,1),a} Unknowns: {d(6,1),d(5,6),d(4,0),d(3,1),a} bonne inconnue W:=d(6,1)$ sa valeur doit etre WW:=d(5,6) + d(4,0) - d(3,1)*a$ on resout l'equation {{0,1},6} qui est maintenant AA:= - d(6,6) + d(1,1) + d(0, 0)$ Unknowns: {d(6,6),d(1,1),d(0,0)} Unknowns: {d(6,6),d(1,1),d(0,0)} bonne inconnue W:=d(6,6)$ sa valeur doit etre WW:=d(1,1) + d(0,0)$ on resout l'equation {{0,2},0} qui est maintenant AA:= - d(0,3)$ Unknown: d(0,3) Unknown: d(0,3) bonne inconnue W:=d(0,3)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,2},1} qui est maintenant AA:= - d(1,3)$ Unknown: d(1,3) Unknown: d(1,3) bonne inconnue W:=d(1,3)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,2},2} qui est maintenant AA:= - d(2,3)$ Unknown: d(2,3) Unknown: d(2,3) bonne inconnue W:=d(2,3)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,2},3} qui est maintenant AA:= - d(3,3) + d(2,2) + d(0, 0)$ Unknowns: {d(3,3),d(2,2),d(0,0)} Unknowns: {d(3,3),d(2,2),d(0,0)} bonne inconnue W:=d(3,3)$ sa valeur doit etre WW:=d(2,2) + d(0,0)$ on resout l'equation {{0,2},4} qui est maintenant AA:= - d(4,3) + d(1,0)$ Unknowns: {d(4,3),d(1,0)} Unknowns: {d(4,3),d(1,0)} bonne inconnue W:=d(4,3)$ sa valeur doit etre WW:=d(1,0)$ on resout l'equation {{0,2},5} qui est maintenant AA:=d(6,2) - d(5,3) + d(3,2)* a - d(3,0)$ Unknowns: {d(6,2),d(5,3),d(3,2),d(3,0),a} Unknowns: {d(6,2),d(5,3),d(3,2),d(3,0),a} bonne inconnue W:=d(6,2)$ sa valeur doit etre WW:=d(5,3) - d(3,2)*a + d(3,0)$ on resout l'equation {{0,2},6} qui est maintenant AA:= - d(6,3) + d(1,2)$ Unknowns: {d(6,3),d(1,2)} Unknowns: {d(6,3),d(1,2)} bonne inconnue W:=d(6,3)$ sa valeur doit etre WW:=d(1,2)$ on resout l'equation {{0,3},0} qui est maintenant AA:= - d(0,5)*a$ Unknowns: {d(0,5),a} Unknowns: {d(0,5),a} pas de selection possible de variable a coefficient numerique dans - d(0,5)*a on resout l'equation {{0,3},1} qui est maintenant AA:= - d(1,5)*a$ Unknowns: {d(1,5),a} Unknowns: {d(1,5),a} pas de selection possible de variable a coefficient numerique dans - d(1,5)*a on resout l'equation {{0,3},2} qui est maintenant AA:= - d(2,5)*a$ Unknowns: {d(2,5),a} Unknowns: {d(2,5),a} pas de selection possible de variable a coefficient numerique dans - d(2,5)*a on resout l'equation {{0,3},3} qui est maintenant AA:= - d(3,5)*a$ Unknowns: {d(3,5),a} Unknowns: {d(3,5),a} pas de selection possible de variable a coefficient numerique dans - d(3,5)*a on resout l'equation {{0,3},4} qui est maintenant AA:= - d(4,5)*a$ Unknowns: {d(4,5),a} Unknowns: {d(4,5),a} pas de selection possible de variable a coefficient numerique dans - d(4,5)*a on resout l'equation {{0,3},5} qui est maintenant AA:= - d(5,5)*a + d(2,2)*a + d(2,0) + d(1,2) + 2*d(0,0)*a$ Unknowns: {d(5,5),d(2,2),d(2,0),d(1,2),d(0,0),a} Unknowns: {d(5,5),d(2,2),d(2,0),d(1,2),d(0,0),a} bonne inconnue W:=d(1,2)$ sa valeur doit etre WW:=d(5,5)*a - d(2,2)*a - d(2,0) - 2*d(0,0)*a$ on resout l'equation {{0,3},6} qui est maintenant AA:= - d(6,5)*a$ Unknowns: {d(6,5),a} Unknowns: {d(6,5),a} pas de selection possible de variable a coefficient numerique dans - d(6,5)*a on resout l'equation {{0,4},3} qui est maintenant AA:=d(2,4)$ Unknown: d(2,4) Unknown: d(2,4) bonne inconnue W:=d(2,4)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,4},5} qui est maintenant AA:=d(6,4) + d(3,4)*a + d(1,0 )$ Unknowns: {d(6,4),d(3,4),d(1,0),a} Unknowns: {d(6,4),d(3,4),d(1,0),a} bonne inconnue W:=d(6,4)$ sa valeur doit etre WW:= - (d(3,4)*a + d(1,0))$ on resout l'equation {{0,4},6} qui est maintenant AA:=d(1,4)$ Unknown: d(1,4) Unknown: d(1,4) bonne inconnue W:=d(1,4)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,5},3} qui est maintenant AA:=d(2,5)$ Unknown: d(2,5) Unknown: d(2,5) bonne inconnue W:=d(2,5)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,5},5} qui est maintenant AA:=d(6,5) + d(3,5)*a$ Unknowns: {d(6,5),d(3,5),a} Unknowns: {d(6,5),d(3,5),a} bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:= - d(3,5)*a$ on resout l'equation {{0,5},6} qui est maintenant AA:=d(1,5)$ Unknown: d(1,5) Unknown: d(1,5) bonne inconnue W:=d(1,5)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,6},0} qui est maintenant AA:= - d(0,5)$ Unknown: d(0,5) Unknown: d(0,5) bonne inconnue W:=d(0,5)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,6},3} qui est maintenant AA:= - d(3,5)$ Unknown: d(3,5) Unknown: d(3,5) bonne inconnue W:=d(3,5)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,6},4} qui est maintenant AA:= - d(4,5)$ Unknown: d(4,5) Unknown: d(4,5) bonne inconnue W:=d(4,5)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,6},5} qui est maintenant AA:= - d(5,5) + d(2,1)*a + d( 1,1) + 2*d(0,0)$ Unknowns: {d(5,5),d(2,1),d(1,1),d(0,0),a} Unknowns: {d(5,5),d(2,1),d(1,1),d(0,0),a} bonne inconnue W:=d(5,5)$ sa valeur doit etre WW:=d(2,1)*a + d(1,1) + 2*d(0,0)$ on resout l'equation {{1,2},0} qui est maintenant AA:= - d(0,4)$ Unknown: d(0,4) Unknown: d(0,4) bonne inconnue W:=d(0,4)$ sa valeur doit etre WW:=0$ on resout l'equation {{1,2},3} qui est maintenant AA:= - d(3,4) + d(0,1)$ Unknowns: {d(3,4),d(0,1)} Unknowns: {d(3,4),d(0,1)} bonne inconnue W:=d(3,4)$ sa valeur doit etre WW:=d(0,1)$ on resout l'equation {{1,2},4} qui est maintenant AA:= - d(4,4) + d(2,2) + d(1, 1)$ Unknowns: {d(4,4),d(2,2),d(1,1)} Unknowns: {d(4,4),d(2,2),d(1,1)} bonne inconnue W:=d(4,4)$ sa valeur doit etre WW:=d(2,2) + d(1,1)$ on resout l'equation {{1,2},5} qui est maintenant AA:= - d(5,4) + d(4,2) - d(3, 1)$ Unknowns: {d(5,4),d(4,2),d(3,1)} Unknowns: {d(5,4),d(4,2),d(3,1)} bonne inconnue W:=d(5,4)$ sa valeur doit etre WW:=d(4,2) - d(3,1)$ on resout l'equation {{1,2},6} qui est maintenant AA:=d(1,0) - d(0,2) + d(0,1)* a$ Unknowns: {d(1,0),d(0,2),d(0,1),a} Unknowns: {d(1,0),d(0,2),d(0,1),a} bonne inconnue W:=d(1,0)$ sa valeur doit etre WW:=d(0,2) - d(0,1)*a$ on resout l'equation {{1,3},5} qui est maintenant AA:=d(2,1) + d(0,2)$ Unknowns: {d(2,1),d(0,2)} Unknowns: {d(2,1),d(0,2)} bonne inconnue W:=d(2,1)$ sa valeur doit etre WW:= - d(0,2)$ on resout l'equation {{1,4},5} qui est maintenant AA:=d(2,2) + d(1,1) + d(0,2)* a - 2*d(0,0)$ Unknowns: {d(2,2),d(1,1),d(0,2),d(0,0),a} Unknowns: {d(2,2),d(1,1),d(0,2),d(0,0),a} bonne inconnue W:=d(2,2)$ sa valeur doit etre WW:= - d(1,1) - d(0,2)*a + 2*d(0,0)$ on resout l'equation {{1,6},5} qui est maintenant AA:= - d(2,0) + d(0,1)$ Unknowns: {d(2,0),d(0,1)} Unknowns: {d(2,0),d(0,1)} bonne inconnue W:=d(0,1)$ sa valeur doit etre WW:=d(2,0)$ on resout l'equation {{2,3},5} qui est maintenant AA:=3*( - d(1,1) + d(0,0))$ Unknowns: {d(1,1),d(0,0)} Unknowns: {d(1,1),d(0,0)} bonne inconnue W:=d(1,1)$ sa valeur doit etre WW:=d(0,0)$ Derivation equations to cancel (Reduce output) : \\{{{{0,1},0},0}, {{{0,1},1},0}, {{{0,1},2},0}, {{{0,1},3},0}, {{{0,1},4},0}, {{{0,1},5},0}, {{{0,1},6},0}, {{{0,2},0},0}, {{{0,2},1},0}, {{{0,2},2},0}, {{{0,2},3},0}, {{{0,2},4},0}, {{{0,2},5},0}, {{{0,2},6},0}, {{{0,3},0},0}, {{{0,3},1},0}, {{{0,3},2},0}, {{{0,3},3},0}, {{{0,3},4},0}, {{{0,3},5},0}, {{{0,3},6},0}, {{{0,4},3},0}, {{{0,4},5},0}, {{{0,4},6},0}, {{{0,5},3},0}, {{{0,5},5},0}, {{{0,5},6},0}, {{{0,6},0},0}, {{{0,6},1},0}, {{{0,6},2},0}, {{{0,6},3},0}, {{{0,6},4},0}, {{{0,6},5},0}, {{{0,6},6},0}, {{{1,2},0},0}, {{{1,2},1},0}, {{{1,2},2},0}, {{{1,2},3},0}, {{{1,2},4},0}, {{{1,2},5},0}, {{{1,2},6},0}, {{{1,3},4},0}, {{{1,3},5},0}, {{{1,3},6},0}, {{{1,4},0},0}, {{{1,4},1},0}, {{{1,4},2},0}, {{{1,4},3},0}, {{{1,4},4},0}, {{{1,4},5},0}, {{{1,4},6},0}, {{{1,5},4},0}, {{{1,5},5},0}, {{{1,5},6},0}, {{{1,6},4},0}, {{{1,6},5},0}, {{{1,6},6},0}, {{{2,3},0},0}, {{{2,3},1},0}, {{{2,3},2},0}, {{{2,3},3},0}, {{{2,3},4},0}, {{{2,3},5},0}, {{{2,3},6},0}, {{{2,4},3},0}, {{{2,4},4},0}, {{{2,4},5},0}, {{{2,5},3},0}, {{{2,5},4},0}, {{{2,5},5},0}, {{{2,6},3},0}, {{{2,6},4},0}, {{{2,6},5},0}, {{{3,4},5},0}, {{{3,5},5},0}, {{{3,6},5},0}, {{{4,5},5},0}, {{{4,6},5},0}, {{{5,6},5},0}}$ Il n'y a pas de phase 2$ derivation generique de gtildedelta:$ MATD:= mat((d(0,0),d(2,0),d(0,2),0,0,0,0),( - d(2,0)*a + d(0,2),d(0,0), - d(2,0),0,0,0, 0),(d(2,0), - d(0,2), - d(0,2)*a + d(0,0),0,0,0,0),(d(3,0),d(3,1),d(3,2), - d(0, 2)*a + 2*d(0,0),d(2,0),0, - d(0,2)),(d(4,0),d(4,1),d(4,2), - d(2,0)*a + d(0,2), - d(0,2)*a + 2*d(0,0),0, - d(2,0)),(d(5,0),d(5,1),d(5,2),d(5,3),d(4,2) - d(3,1), - d(0,2)*a + 3*d(0,0),d(5,6)),(d(6,0),d(5,6) + d(4,0) - d(3,1)*a,d(5,3) - d(3,2 )*a + d(3,0), - d(2,0), - d(0,2),0,2*d(0,0)))$ pour delta:= [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 1 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 a 0 0 1] [ ] [1 0 0 0 0 0] pour shortformdelta:={0, ss, 0, 1, ss, a, 1, ss, 1, 0, 0} Unknowns: {d(6,0), d(5,6), d(5,3), d(5,2), d(5,1), d(5,0), d(4,2), d(4,1), d(4,0), d(3,2), d(3,1), d(3,0), d(2,0), d(0,2), d(0,0), a} Unknowns: {d(6,0), d(5,6), d(5,3), d(5,2), d(5,1), d(5,0), d(4,2), d(4,1), d(4,0), d(3,2), d(3,1), d(3,0), d(2,0), d(0,2), d(0,0), a} listeparametresMATD{d(6,0), d(5,6), d(5,3), d(5,2), d(5,1), d(5,0), d(4,2), d(4,1), d(4,0), d(3,2), d(3,1), d(3,0), d(2,0), d(0,2), d(0,0)}$ dim Der(gtildedelta):=15$ t1:=D(0,0):= [1 0 0 0 0 0 0] [ ] [0 1 0 0 0 0 0] [ ] [0 0 1 0 0 0 0] [ ] [0 0 0 2 0 0 0] [ ] [0 0 0 0 2 0 0] [ ] [0 0 0 0 0 3 0] [ ] [0 0 0 0 0 0 2] Unknowns: {d(2,0),d(0,2),d(0,0),a} Unknowns: {d(2,0),d(0,2),d(0,0),a} commutant de t1 dans der(gtildedelta): mat((d(0,0),d(2,0),d(0,2),0,0,0,0), ( - d(2,0)*a + d(0,2),d(0,0), - d(2,0),0,0,0,0), (d(2,0), - d(0,2), - d(0,2)*a + d(0,0),0,0,0,0), (0,0,0, - d(0,2)*a + 2*d(0,0),d(2,0),0, - d(0,2)), (0,0,0, - d(2,0)*a + d(0,2), - d(0,2)*a + 2*d(0,0),0, - d(2,0)), (0,0,0,0,0, - d(0,2)*a + 3*d(0,0),0), (0,0,0, - d(2,0), - d(0,2),0,2*d(0,0))) Unknowns: {d(2,0),d(0,2),d(0,0),a} Unknowns: {d(2,0),d(0,2),d(0,0),a} t2:=D(0,2):= [0 0 1 0 0 0 0 ] [ ] [1 0 0 0 0 0 0 ] [ ] [0 -1 - a 0 0 0 0 ] [ ] [0 0 0 - a 0 0 -1] [ ] [0 0 0 1 - a 0 0 ] [ ] [0 0 0 0 0 - a 0 ] [ ] [0 0 0 0 -1 0 0 ] Unknowns: {d(2,0),d(0,2),d(0,0),a} Unknowns: {d(2,0),d(0,2),d(0,0),a} t3:=D(2,0):= [ 0 1 0 0 0 0 0 ] [ ] [ - a 0 -1 0 0 0 0 ] [ ] [ 1 0 0 0 0 0 0 ] [ ] [ 0 0 0 0 1 0 0 ] [ ] [ 0 0 0 - a 0 0 -1] [ ] [ 0 0 0 0 0 0 0 ] [ ] [ 0 0 0 -1 0 0 0 ] Neither t2 nor t3 is semisimple (double eigenvalue with 1-dimensional eigenspa\ ec). We consider s:= t2 + beta*t3. s:= [ 0 beta 1 0 0 0 0 ] [ ] [ - (a*beta - 1) 0 - beta 0 0 0 0 ] [ ] [ beta -1 - a 0 0 0 0 ] [ ] [ 0 0 0 - a beta 0 -1 ] [ ] [ 0 0 0 - (a*beta - 1) - a 0 - beta] [ ] [ 0 0 0 0 0 - a 0 ] [ ] [ 0 0 0 - beta -1 0 0 ] Hand computations prove that s is semisimple if and only if beta = -3/a. Hence the rank is 2. s:= [ - 3 ] [ 0 ------ 1 0 0 0 0 ] [ a ] [ ] [ 3 ] [ 4 0 --- 0 0 0 0 ] [ a ] [ ] [ - 3 ] [------ -1 - a 0 0 0 0 ] [ a ] [ ] [ - 3 ] [ 0 0 0 - a ------ 0 -1 ] [ a ] [ ] [ 3 ] [ 0 0 0 4 - a 0 ---] [ a ] [ ] [ 0 0 0 0 0 - a 0 ] [ ] [ 3 ] [ 0 0 0 --- -1 0 0 ] [ a ] In what follows, we shift the notations : t2 or t2prime if confusion could arise will denote the foregoing semisimple s. t2:= [ - 3 ] [ 0 ------ 1 0 0 0 0 ] [ a ] [ ] [ 3 ] [ 4 0 --- 0 0 0 0 ] [ a ] [ ] [ - 3 ] [------ -1 - a 0 0 0 0 ] [ a ] [ ] [ - 3 ] [ 0 0 0 - a ------ 0 -1 ] [ a ] [ ] [ 3 ] [ 0 0 0 4 - a 0 ---] [ a ] [ ] [ 0 0 0 0 0 - a 0 ] [ ] [ 3 ] [ 0 0 0 --- -1 0 0 ] [ a ] rank 2 with maximal torus t1,t2 2 matrice de passage de la base X(0)=delta, X(1),..., X(6) a une base diagonali\ sant le tore maximal: on peut prendre - 27 a**3:= ------- 4 P:= [ 2 ] [ - 4*a ] [ 1 0 --------- 0 0 0 0 ] [ 9 ] [ ] [ - 4*a ] [ 0 1 -------- 0 0 0 0 ] [ 3 ] [ ] [ 9 3 ] [---- --- 1 0 0 0 0 ] [ 2 a ] [ a ] [ ] [ - a ] [ 0 0 0 1 ------ 0 0 ] [ 3 ] [ ] [ 3 9 ] [ 0 0 0 --- 1 0 ------] [ a 2 ] [ 4*a ] [ ] [ 0 0 0 0 0 1 0 ] [ ] [ 2 ] [ - a ] [ 0 0 0 0 ------- 0 1 ] [ 9 ] P**(-1)*t1*P:= [1 0 0 0 0 0 0] [ ] [0 1 0 0 0 0 0] [ ] [0 0 1 0 0 0 0] [ ] [0 0 0 2 0 0 0] [ ] [0 0 0 0 2 0 0] [ ] [0 0 0 0 0 3 0] [ ] [0 0 0 0 0 0 2] P**(-1)*t2*P:= [ 9 ] [---- 0 0 0 0 0 0 ] [ 2 ] [ a ] [ ] [ 9 ] [ 0 ---- 0 0 0 0 0 ] [ 2 ] [ a ] [ ] [ - 45 ] [ 0 0 ------- 0 0 0 0 ] [ 2 ] [ 4*a ] [ ] [ - 9 ] [ 0 0 0 ------ 0 0 0 ] [ 2 ] [ 4*a ] [ ] [ 18 ] [ 0 0 0 0 ---- 0 0 ] [ 2 ] [ a ] [ ] [ 0 0 0 0 0 - a 0 ] [ ] [ - 9 ] [ 0 0 0 0 0 0 ------] [ 2 ] [ 4*a ] matrice des derivations dans cette base diagonalisante Y(1),...,Y(7): PP:= [ 2 ] [ - 4*a ] [ 1 0 --------- 0 0 0 0 ] [ 9 ] [ ] [ - 4*a ] [ 0 1 -------- 0 0 0 0 ] [ 3 ] [ ] [ 9 3 ] [---- --- 1 0 0 0 0 ] [ 2 a ] [ a ] [ ] [ - a ] [ 0 0 0 1 ------ 0 0 ] [ 3 ] [ ] [ 3 9 ] [ 0 0 0 --- 1 0 ------] [ a 2 ] [ 4*a ] [ ] [ 0 0 0 0 0 1 0 ] [ ] [ 2 ] [ - a ] [ 0 0 0 0 ------- 0 1 ] [ 9 ] avec PP:=P*Q:= [ 2 ] [ - 4*a ] [ 1 0 --------- 0 0 0 0 ] [ 9 ] [ ] [ - 4*a ] [ 0 1 -------- 0 0 0 0 ] [ 3 ] [ ] [ 9 3 ] [---- --- 1 0 0 0 0 ] [ 2 a ] [ a ] [ ] [ - a ] [ 0 0 0 1 ------ 0 0 ] [ 3 ] [ ] [ 3 9 ] [ 0 0 0 --- 1 0 ------] [ a 2 ] [ 4*a ] [ ] [ 0 0 0 0 0 1 0 ] [ ] [ 2 ] [ - a ] [ 0 0 0 0 ------- 0 1 ] [ 9 ] on voit apparaitre les poids sur la diagonale 2 9*d(0,2) + d(0,0)*a r(1) := ---------------------- 2 a - (3*d(2,0) - d(0,0)*a) r(2) := -------------------------- a 2 - (9*d(0,2) - 4*d(0,0)*a - 12*d(2,0)*a) r(3) := ------------------------------------------- 2 4*a 2 27*d(0,2) + 8*d(0,0)*a + 12*d(2,0)*a r(4) := --------------------------------------- 2 4*a 2 9*d(0,2) + 2*d(0,0)*a - 3*d(2,0)*a r(5) := ------------------------------------- 2 a r(6) := - (d(0,2)*a - 3*d(0,0)) 2 - (9*d(0,2) - 8*d(0,0)*a ) r(7) := ----------------------------- 2 4*a 3*(d(2,0)*a + 3*d(0,2)) r(4)-(r(2)+r(3)):= ------------------------- 2 a - 3*(8*d(2,0)*a - 3*d(0,2)) r(5)-(r(1)+r(3)):= ------------------------------ 2 4*a r(5)-(r(1)+r(2)):= 0 r(6)-(r(2)+r(4)):= 0 r(6)-(r(3)+r(5)):= 0 The torus consists of t1=d(0,0) and tprime2=d(0,2) -(3/a)*d(2,0) Hence the dual basis of the torus consists of the d(0,0)* and d(0,2)* (the stars for the projection here), and one has on the torus d(2,0)* = -(3/a) d(0,2)*. Hence in writing down the weights on the torus, (we simply stops writing the stars for the dual basis), one has to identify d(2,0) and -(3/a)*d(0,2) - 3*d(0,2) d(2,0):= ------------- a r(1) := gamma2 r(2) := gamma2 r(3) := gamma1 r(4) := gamma1 + gamma2 r(5) := 2*gamma2 r(6) := gamma1 + 2*gamma2 r(7) := gamma1 + gamma2 Le systeme de poids est le systeme 2.2 calcul de relations de commutation de la base diaY(j) diagonalisant le tore listcommutateursdesx := {{{0,1},x(6)}, {{0,2},x(3)}, {{0,3},a*x(5)}, {{0,4},0}, {{0,5},0}, {{0,6},x(5)}, {{1,2},x(4)}, {{1,3},0}, {{1,4},x(5)}, {{1,5},0}, {{1,6},0}, {{2,3},x(5)}, {{2,4},0}, {{2,5},0}, {{2,6},0}, {{3,4},0}, {{3,5},0}, {{3,6},0}, {{4,5},0}, {{4,6},0}, {{5,6},0}} 2 9*x(2) + x(0)*a diaY(1):=------------------ 2 a 3*x(2) + x(1)*a diaY(2):=----------------- a 2 9*x(2) - 12*x(1)*a - 4*x(0)*a diaY(3):=-------------------------------- 9 3*x(4) + x(3)*a diaY(4):=----------------- a 2 - x(6)*a + 9*x(4) - 3*x(3)*a diaY(5):=-------------------------------- 9 diaY(6):=x(5) 2 4*x(6)*a + 9*x(4) diaY(7):=-------------------- 2 4*a liste des commutateurs des diaY(i) :$ listcommutateurdesdiaY:={{{1,2},( - 9*diay(5))/a**2}, {{1,3},( - 4*diay(7)*a + 15*diay(4))/3}, {{1,4},( - diay(6)*a)/3}, {{1,5},0}, {{1,6},0}, {{1,7},diay(6)}, {{2,3},(4*a*(diay(7)*a + 3*diay(4)))/9}, {{2,4},(6*diay(6))/a}, {{2,5},0}, {{2,6},0}, {{2,7},( - diay(6)*a)/3}, {{3,4},0}, {{3,5}, - 3*diay(6)*a}, {{3,6},0}, {{3,7},0}, {{4,5},0}, {{4,6},0}, {{4,7},0}, {{5,6},0}, {{5,7},0}, {{6,7},0}}$ Now we make explicit the isomorphism with an algebra of the book:$ Namely g_{7,2.2}$ i.e. we go from the basis diaY(i) to the new basis ZZ(i) defined by the matrix:$ on pose :$ avec comme matrice de changement de base :$ This isom computed by calculisom6_53xCVIII2.red$ mat((0,( - 2*a)/3,0,0,0,0,0),(0,1,1,0,0,0,0),(1,0,0,0,0,0,0),(0,0,0,0,2*a,( - 4* a)/3,0),(0,0,0,6/a,0,0,0),(0,0,0,0,0,0,-9),(0,0,0,0,( - 4*a**2)/3,( - 4*a**2)/9, 0))$ det(isom):= 648$ ZZ(1):=diay(3)$ ZZ(2):=(3*diay(2) - 2*diay(1)*a)/3$ ZZ(3):=diay(2)$ ZZ(4):=(6*diay(5))/a$ ZZ(5):=( - 2*(2*diay(7)*a - 3*diay(4))*a)/3$ ZZ(6):=( - 4*(diay(7)*a + 3*diay(4))*a)/9$ ZZ(7):= - 9*diay(6)$ listcommutateursdesZZ:=$ {{1,2},zz(5)}$ {{1,3},zz(6)}$ {{1,4},2*zz(7)}$ {{1,5},0}$ {{1,6},0}$ {{1,7},0}$ {{2,3},zz(4)}$ {{2,4},0}$ {{2,5},0}$ {{2,6},zz(7)}$ {{2,7},0}$ {{3,4},0}$ {{3,5}, - zz(7)}$ {{3,6},zz(7)}$ {{3,7},0}$ {{4,5},0}$ {{4,6},0}$ {{4,7},0}$ {{5,6},0}$ {{5,7},0}$ {{6,7},0}$ We get the commutation relations of$ g_{7,2.2}$ Et cela pour a**3:=( - 27)/4$ shortformdelta:={0, ss, 0, 1, ss, a, 1, ss, 1, 0, 0}$ delta:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(0,1,0,0,0,0),(0,0,0,0,0,0),(0,0,a,0,0,1),(1,0,0 ,0,0,0))$