Characterization of complex filiform Lie algebras of dimension 0 according to whether they are or not derived from others

Re . Academia de Ciencias. Za agoza. 48 (1993) CHARACTERIZATION OF COMPLEX FILIFORM LIE ALGEBRAS OF DIMENSION 8 ACCORDING TO WHETHER THEY ARE OR NOT DERIVED FROM OTHERS By F. J. ECHARTE REULA 1,. ,R. GOMEZ MARTII y J. NU n Z VALDEs'" . Uni e sidad de Se illa. Facul ad de Ma emá ica s. Dp o de Alg eb a, Compu ación, Geome ía y Topología. C/ Ta i a s/n. 41012 Se illa. España. 2 Uni e sidad de Se illa. Facul ad de In o má ica y Es adí s i ca. Dp o de Ma emá ica Aplicada. C/ Ta ia s/n . 41012 Se illa , E spa ñ a. AMS (1980) Subj ec Cla ssi ic a ion: 17 B 30 Aos ac .- In h is pape , W9 cha ac e ize hose Comp Lex FiLilo m Lie AL6eo as 01 dimens ion 8 which a e de i ed I o m o he SoL aoLe Lie AL6eo as 01 hi6he dimensiono This esuL and h e p e ious one 6i en in ({8]) aLLow us o lind a compLe e Lis 01 Cha ac e is icaLLy NiLpo en F iL ilo m Lie AL6eo as 01 di mens io n 8 . 1.- In oduc ion and No a ions. The e does no e xis , a p esen , a ny cla s s i ic a i on o compl ex Nilpo en L ie Alg eb as (NLA) o dim ens ion g e a e h an 7 . Goze and Ancochea, by he in oduc ion o a new in a ian wh ich he y c all he "cha ac e is ic sequence", which co e sponds o h e max i mal d imen sions o Jo dan blocks o a ce a in n ilpo en ma ix , ob a ined he classi ica io n o complex Fi l i o m Li e Alg eb a s (FLA) o dim ension 8 ([1). These FLA, as i is known, a e a sub s e o NLA. The a u ho s a e su pp o e d by he p ojec PAICYT o he Ju n a de And alucía. E spaña (1990). 35 [[ A,B],C) + [[B,C],A] + [[C,A],B) =O and we deno e by ma FLA o dim ensio n ~ ([8), ha is, a complex NLA ad mi ing a b asi s o o ep esen he Jacobi O 36 F om now on, we w i e (A,B,C) ( 1) s uch ha X. ii! [m ,m i. ( 2 ) id en i y A co mplex FLA may be de i ed om an SLA o highe dim ensiono In ([8) we p o ed he ollowing: (1 .1) Theo em: A complex FLA o dimens ion ~ is ei h e de i ed om a SLA o dimension n+1 o no d e i ed om any LA. The ma in pu p o se o his pap e is, h e e o e, s a ing om he classi ic a ion o FLA o dimension 8 by Goze and Ancochea, o cha ac e iz e wo g oups o hese LA, acco d ing o whe he o no hey a e de i ed om o he Sol able Lie Algeb as (SLA) o highe dim ens iono This cha ac e i za ion we p opose could be in e es ing in he sense o k no wing which o hese FLA a e al so Cha ac e is ically Ni l po e n Li e Algeb as (CNLA) since, as we p e iously p o ed in an ea li e pa p e ([8), i a FLA is no de i ed om any LA hen i i s a CNLA. Th is he o em will allow us o w i e he lis o Cha ac e is ic al ly Nilp o en Fili o m Lie Algeb as (CNFLA) o dimens i on 8. Th e e o e, al hough he classi ica ion o Lie Algeb as (LA) o d ime ns ion g e a e h an 7 h as no be en ob ained ye , excep o FLA, he main p oblem which is now consid e ed is he sea ch o new esul s, a he han he cla ssi ica ion i s el , such as he a ainmen o new in a ian s o he desc ip ion o he i educible componen s o a ie ies o LA. Fo his pu pose, Vale i a s p o ed ecen ly ha h e a i e y o NLA o dimen sion 8 has 8 i educibl e componen s and ha only h e wo i s o h em mee h e open s e o FLA ([Q). lIow e e , he impo ance o hese au ho s' wo k is no only o ha e ob ai n ed hi s cl a s si ica ion, bu , abo e all, o ha e de ised echn i ques wh ich c an be appli ed in any dim ension, which, apa om h e ac o ha ing p e m i ed Gómez Ma ín o clasSi y FLA o dimens ion 9 ([5]) , o a lly so l ed he p obl em o h is cl assi ica ion o FLA o g ea e d imens ion , al hough, o cou se, his equi es ha d and comp li ca e d calcula ions which a e imp ossible wi hou he use o a compu e . Mo eo e , ~ will deno e a complex SLA o dim en sion n+1 be aXz Sn Sn 3 S X. J-' IX .x.: , J (1 Sj'S n-1) [X ',X.' I=O (1 <jS • J sui able way e i ying 1<j A necessa y and su icien condi ion o a FLA ID o be de i ed om he SLA ~ is ha a "" O. u A complex FLA ID is a CNLA i and only i ID i s no de i ed om any LA. (3) (1. 3) Theo em As we men ioned aboye, in ([8]) he ollowing wo h eo em s a e p o ed: (1 . 2) Theo em So, o sepa a e FLA o dimension 8 in o wo g oups acco din g o whe he o no hey a e de i ed om ano he SLA (Theo em (1.1) we will use a me hod p e iou sly indica ed by us o he ca se o NLA o dimension 7 ([4]). Thi s p oceedu e, which we call ed he "m e hod o de e mina ion o he anishing o he co e ici en a,," , c an be applie d o FLA. I consis s o a se o i e a i e calc ula ion s ba s ed on all (6 ) 37 wi h (Xj, ~ , U) = O, 1 S Lh S n , The FLA ID is said o be de i ed om he SLA ~ i [~,~I =ID, whe e [~,~] ep esen s any linea combina ion o all b acke s among ields o he basis (1). Cha ac e is ically Nilpo en Lie Algeb as a e hos e LA i n which all hei de i a ions a e nilpo en . The i s examples o CNLA gi en in he li e a u e we e o LA ha a e no de i ed om a ny LA, un il Luks ga e an example o a CNLA which is d e i ed om a LA o dimension 18 ([7]) . Howe e , i is e y easy o gi e ex ampl es o Nilpo en Lie Algeb as (NLA) which a e de i ed and no CNLA. X· =X jj hen i esul s ha Mo eo e , s i n ce [ X.,X jl O 1 <j<nand [X.,X "J ma E C), i we con sid e h e change o ba s i s gi e n by X' = X + a X " " , n) . S o, a ba s i s (1) c an always chosen in 'a (4) such ha (5 ) (X" •• , X" ' U} is a ba s í.s , whe e (X"X z, . .. , X,, } is he basis (1} men ion ed aboye, and U a de i a i on o ID such ha . and These i e a i e s eps a e he ollowing: 38 wi h >3 hei ~ ,. hs a,. wi h 'J , 4.- Repea ing he o me h ee s eps wi h U, X,. and h,. >2 in he i s place. Secondly, wi h U, Xs and l), wi h s In he case ha he FLA m is de i ed om ano he SLA o dimension one g ea e han he dimension o m, we can ob ain he simples SLA ~ by aking a.. =1 and he es o he coe icien s aLj O (i, j .. 1) . Mo eo e , we can check ha in FLA o dimension S 9, ei he di ec ly o by sui able changes o base, i is always possible o ge (7 ) [l), , U1=ah ~ wi h h= 1, 2, ... , n (al hough his is no p o ed in he gene al case o dimension mnI . 5.- Pinally, obse ing i in he new exp essions (6) ob ained in his way he coe icien a.. appea s explici y o no ; his will indica e, acco ding o heo em (1.2) whe he he PLA m which we a e s udying is de i ed om ano he SLA ~ o dimension one mo e han he dimension o m. and so on, un il e mina ing his p oceeding wi h U, Xn_. and Xn' 2.- Di ision o complex FLA o dimension ~ in o wo g oups depending on whe he hey ~ o no de i ed om o he LA. 3.- Replacing in (6) he coe icien s co esponding alues ob ained p e iously in s ep 2. 2.- Using each equa ion ob ained in his way o exp ess in e ms o he o he s he coe icien ~ , whose pai o subindices is 'J he g ea e (in lexicog aphic o de ) . (Po ins ance, om he equa ion aaaa=Owe will ha e a a a a. 7) • ,.,. •• 44 '7 44 ,.,. •• 1.- Using he Jacobi iden i y (X.' l), , U) =O wi h h. > 1, i de i ing om such iden i y he se o equa ions ob ained by se ing o ze o he coe icien o each ield X.. J pos sible J a cobi iden i ie s o he kind (~, ~, U) =o wi h 1 S i,j S 8 , om which we can d e e mine whe he he co e i cien a•• anishes o no o This in u n de e mines, by heo em (1.2), whe he he PLA m is de i ed om he SLA ~ w i h {X.' .. ,X.,U} a s a ba sis. By appliying he me hod men ioned abo e o each o hem we ob ain he ollowing e sul s: By he me hod m en ioned abo e, we i nd ha 6o he 13 FLA and 2 o he 7 amilies a e de i ed om SLA o d imens io n8w he eas he o he 7FLA and he 5 amilies a e no o • Z 8 law s ¡Jo' ¡Jo' , ¡JB ' a e no de i ed The FLA o d imens ion 8 wi h la w ¡J: i s no de i ed o m an y LA, sinc e: 39 a X+a X + a X+a X+ a X+ a X u z .0 o ..... i"" .,," .7 7 O aX oz z (a." + a. 7)Xz+ (a oz + a.?) Xo a"z Xz+ 3/5 a.? Xa+ (a oz + a.?) X.. (3/5 a"z + a ... + 3/5 a." + 6/25 ai ,, )Xz+ + (a"z + a.,,) X o+ 1/5 a.? X.. + ( ao z+ a .?) X" a?Z Xz+ (3/5 a"z + a." + 6/25 a.,, ) + + (a"z + a." + 2/5 a.,,) X.. + 1/5 aoX" + (aoz + a. ? ) X" aoz Xz +(a?Z- a ... )X o+ (3/ 5 a" z - a ... + 2/ 5a . ,,+ 6/25 a. ,, ) X.. + + (a"z + 1/5 a . ,, ) X"- a ." X" + a. zX? The FLA o dimension 8 wi h law ¡.¡~ is no de i ed om a ny LA, since : [X o' Uj [X. ,U] [Xz ,U] [X o' U] [X.. ,U] [X",U] [X",U] ( 2. 2) (2.1) So we ob ain he ollowing P oo : We sepa a e FLA o dim ens io n 8 i n o wo g ou p s d ependin g on whe h e hey a e de i ed om an o he SLA, s a ing om he clas si ica ion o hes e FLA by G oz e an d Anc o chea ([1 ]). This cla s si ic a ion (which i s indica ed in an A pp end i x a he end o his pap e ) con ains a lis o 13 i sola ed FLA and 7 on e-p a ame e a milies o FLA. Theo em 1.- The complex PLA o dimension 8 wi h ... c:S,O 8 P,Ol :l0,0I :ll S.8,o . u:J ,a and 17 ¡Jo, ¡Jo ' ¡JB' ¡Jo ,¡Jo ,¡Jo' ¡Jo ,¡Jo "« om any o he LA. The FLA o di o ension 8 wi h law ~o is no de i ed o o any • X7 X7 X7 a oz +a oz X7 ) Xcs X+a CS 02 +a 17 aX .2 " (a + oz (a + oz X - a " 1CS X+aX es 17? X+aX 8 82" +o ) a17 X¿ + 40 a X+a X +a X +a X+a X iZ 2 18 81.'" '" 115 i 1.es d O aX az z ex a X + a X 1 7 2 82 8 (a 1 CS +a 1 7) X2+ (1 + ex) a 17 acsz Xz+ (a 1 CS + a 1 7) X. + (2 a1 CS X2+ (a 02 +a1 7) X. a"2 Xz+ ( a. 2+ a 17) X¿ (a 1 CS+ a 1¿) X2+ (a"2 + a 1" + a17) Xo+ ( a 0 2 + a72 X2+ ( a" 2 + a 1" + a17) X¿ + (a. 2+ a1 7) . Xcs a. 2·X2+ a 7 2 X. - (a 1¿ + a1 d) X¿ + a"2 X" - a 1 CS Xcs LA, s ince : a X + a X+a X + a X+a X +a X 12 2 18 9 14" la 5 les es 17 7 O aX .2 2 The FLA o di o ension 8 wi h l aw ~¿ is no de i ed o o any o T he FLA o di o ens ion 8 wi h law ~=.ex i s no de i ed o o any LA, s i n ce : LA, s inc e: a X+ a X +a X+a X +a X+a X 12 Zsa 8 14" 115 es id eS 1.7 7 O aX .z z a1 CS Xz+ (a. z+ a 1 7) X. a"z Xz+ (a 02 + a 1 7) X¿ (a 17+ a .. ) X2+ (a"2 + a 1,,) X.+ (a 0 2+ a1 7) X" a72 X2+ a 17 X. + (a"2 + a 1,,) X¿ + (a 0 2 +a17 a. 2X2+ (a 72 -a1 CS) X. - a 1¿ X¿ + a"2 X" - a 1 CS aX .z z a1 CS Xz+ (a oz + a 1 7) X. a" z Xz+ (a oz + a 17) X¿ (a 17+a1 CS + a 1 ¿) Xz+ (a"z + a 1 7+ a 1,,) Xo+ a7Z Xz+ a17 Xo+ (a"z + a 1" + a 1 7) X¿ + aoz X Z+ (a 7 Z- a 1 CS ) X. - (a 1CS-a 1¿) X¿ + a"z a X+ a X+a X+ a X + a 12 2 1 8 a 1. '" llS a id O ( 2.5) [Xi , U ) [XZ ,U] [X. ,U) [ X¿,U) [ X",U) [XCS,U] [X 7, U) [XO'U) [X i' U) [XZ ,U] [X. , U) [X¿ , U) [X ",U) [ XCS , U] [Xi ,U) [XZ ,U) [X O' U] [X¿,U ) [X",U) [XCS,U) [X 7,U] [X. ,U) (2 . 4) [Xi ,U] [X z ,U) [X. , U] [X¿ , U] [ X",U) [XCS,U) [X7 ,U] [XO ,U] (2.3) (2 .8) Th eFLA o di mension 8 wi b l a w ¡.¡ ~O,OI is n o de i ed om any LA, s inc e: [X" Uj a X + a X +a X + a X + aX + a X '2 2 ,a a, ¿ ¿ '" ",., a ,? 7 [X2 ,U] O [X a' U] aX 82 2 [X¿' U] a X + a X ¿2 2 82 8 [ X",U] a X + a X + aX "2 2¿ 2 8 82 ¿ [ X."U] aX + a X +a X + aX .,2 2 "2 8¿ 2 ¿82 " [ X?' Uj a X + a X + a X+a X + aX ?2 2 "2 8"2 ¿¿ 2 "82 es [X 8,Uj aX + (a72 - ex a - a -a , ¿) X + (a - a- a..,, ) X+ 82 2 '7 ,., 8" 2 ' 7 ¿ +( a"2 - a , ., ) X+(a¿2 -a)X + aX " ,? ., 82 ? 41 Tbe FLA o dimension 8 wi h law ¡.¡: i S' no d c i ed om any LA, si nce : a X+ a X +a X+a X+a X +a X i 2 2 sa 8 1'" '" 115 !S 1 d es 17 7 O any X+ ¿ aX+aX 1 7!5 92 d ( 2+01) a, ,,- al CS )X¿ aX 82 2 -aX+ a X , ? 2 82 a aX+aX ,., 2 82 ¿ a X + (a + a,? )X+a X + a X .,2 2,., 8..? ¿8 2 e a X + (a"2 -a, ,, - a, .,) X + aX+a X + aX ?2 28,? ¿, ? "82 ., aX+ (a?2 + a , ¿ ) X.+ (a"2 - a,,, - 2 a , ., ) X .,.. aX + aX 82 28¿,., "82 7 The FLA o dimen sion 8w i h law ¡.¡p, CJ. is no d e i ed om 8 LA, s ince : a X. +aX+ a X + a X+a X + a X '2 2 ..a 8 ,¿ ¿'" ",., ., ,? 7 O a ? 2 X2+ (a" 2- a,,,) Xo+a, ? X¿ + ( 2+01) a02 X2+ (a?2- 01 a , ¿ - a, ,, ) X8+( a.,2 - - (2 +01) a, ., X" + aa 2 X7 aX 82 2 a X + a X ,? 282 a aX + a X + a X "2 2 ..? 882 ¿ a X + (a"2 + a.. ?) X+ a X+a X .,2 28 ,? ¿82 " aX+ (a"2 - a, ., ) X + (a"2 +a ,? )X + aX + a X ?2 28¿ ,? "82 a aX+ (a?2 - a, ¿ - 01 a,.,) X + (a - a-2a ex a'7 ) 82 28 "2 '" ,., + (a -a.. .,) X + a X "2 "82 7 [X" U] [X2 ,U] [Xa ,U] [X ¿,U] [X ",U] [ X."U] [X? ,U] [XO ,U] [X" U] [X2'U] [Xa'U] [X ¿' U] [ X" ,U] [ X."U] [X? ,U] [XO'U] ( 2. 6) (2 .7) NOTE: In h i s alg eb a i is e i ied ha a= 3/2 (o. -1)(o. - 2/ 3) 42 ( 2.11 ) The FLA o di oension 8 wi h law ¡.I~" ' o. is no de i ed o o any LA, s ince: [ X, , U] a X + a X + aX + a X + a X+ a X iZ 2 .. S '4 4 '" " '" " ,? ? [X 2, UI O [Xs ,U Ia X S2 2 [X 4,U I a X + a X 42 2 S2 S [X" , U] a X + a X + a X "2 242 S S2 4 [X" , UI a X + a X + a X + a X csz 2 "2 S 42 4S2 " [ X? ,U ]a X +a X + a X + a X + a X ?2 2 "2 S"2 4 42 "S2 " [Xs ,U ]a X .+ (a - o. a,?- a,,,- a,,,) X + (a -a a,,,) X + e2 2 ?2 e "2 ,? 4 + a X + a X + a X "2 "42 "S2 ? FLA o di o cnsion 8 wi h l aw i i s no de i cd o o Th c¡.le any LA, si nc c : aX+aX+ a X + a X+aX + a X '2 2 .. S '4 4 '" " '" " ,? ? O aX e2 2 aX + a X 42 2 e2 s a X + a X +aX "2 242 9 S2 4 aX+a X + a X + a X csz 2 "2 S 42 4 92 " a X + a X +aX+aX + a X ?2 2 "2 S"2 4 42 "S2 " aX + (a - a - a, 4 ~) X+ (a"2 -a,,,) X+ (a - a,,,) X + e2 2 ?2 ,? 94 "2 " + (a 42 - a )X+aX ,? "92 7 The FLA o di oen sion 8 wi h law ¡.I ~ ? i s no d e i e d o o any LA, since: 42 The FLA o di oension 8 wi h law ¡.I~s,o. is no de i ed o o any LA, since: a X+a X+a X .+a X+a X+a X+a X i2 2 iU Ss..... i!5 e id di i7 7 iB B O aX 92 2 aX+aX ""'z Z 82 a a"2 X2+ (a 4 2+ o. ale) X9+ a 92 X4 acsz X2+ (a"2 + a,? + a'9) Xs+ (a 42+ (1 +20.) a .. ) X4+ a 92x" . a?2 X2+ (a"2 - a,,,) X9+ (a"2 + a,? · + 2 a,e) X4+ + (a 42 + (2+30.) a. e) X" + a S2 x" aez X2+ (a?2- o. a,,,- a,,,) Xs+ (a"2- (2+0.) a,,,- a,?) X4+ + (a"2 + 2 a' 9- o. a,? ) X" + (a 42 + (2+ 30.) a u) X" + a 92 X? ( 2 . 1 2) ( 2.10) [X" U] [X 2, U] [Xs ,U] [X 4,U ] [ X",U I [X" , U] [X ?' UI ( 2. 9) [ X" UI [X2,UI [Xs ,U] [X4,UI [X" , U] [X",Uj [X?' UI [Xe,UI [Xl ,U] aX+aX+aX+ a X+ a X+aX+aX 12 2 18 8 14 4 115 15 ld d 1 7 7 18 8 [X 2, U] O [X 8,U] aX 92 2 [X 4,U] aX+aX 42 2 82 9 [X 15,U] aX+aX+ a X 152 2 42 9 92 4 [Xd,U] aX+ (a 15 2+a 18 )X+aX+aX dZ 2942 4 92 15 [X 7 ,U] aX+ (a + a 19 )X+ (a 15 2+2 a 19 )X+ a X+ a X 72 2 d2 94 42 15 92 a [X 9 ,U] aX+ (a 72 - a a1d )X+ (a + a - aH) X+ 92 2 17 9 d2 19 4 (a 15 2+2au)X+aX+ a X• 15 42 d 92 7 Co olla y 1.- The e exis exae ly 12 CNFLA o dimension 8 . These a e he ollowing: 1294 d,o. 9P,o. lO ,OI. /-18' /-18 '/-lo' /-l. , "« ' /-l., /-1 9'/-lo ' 11 18,a l l,a and /-1:7 /-l. ' /-l. ' /-l. P oo : I is an immedia e eonsequenee o Theo ems 1 and (1. 3) •• Theo em 2.- The eomplex FLA o dimension 8 wi h laws /-115, 7,0. U 8/-lo ' /-1 8' 14,0l ld 18 lP and /-120 a e de i ed om SLA o dimension 9. /-1 9'/-l. ,/-lo ' /-1 9• P oo : By applying he me hod men ioned aboye o ea eh o hem we ind ha eaeh o hem is espee i ely de i ed om a SLA o d imension 9, wi h basis {Xl' . .... ,X a, U}, e i ying [X" U] =a. X. (1 :S i:S 8 ) e e whe e he eoe ieien s a. o eaeh o hem a e he ollow ing: e FLA aaaaaa a a obs. 12o4 15 d7• 15 17654321 /-l. 7,0. 1 8 765432 /-l. 12 1 8 765432 /-l. :14 ,0. 19876543 /-lo ld 327 24 21 18 15 12 9( *) /-lo 18 110 987654 /-lo u' 1 11 10 98765 /-l. 20 18765432 /-l.