ON NILPOTENCY IN NONASSOCIATIVE ALGEBRAS

Transcription

1 Jourl of Algebr Nuber Theory: Advces d Applctos Volue 6 Nuber 6 ges 85- Avlble t DOI: ON NILOTENCY IN NONASSOCIATIVE ALGERAS C. J. A. ÉRÉ M. F. OUEDRAOGO d M. OUATTARA Lbortore T. N. AGATA/UFR-SEA Déprteet de Mthétques Uversté de Ougdougou Adresse.. 7 Ougdougou ur Fso e-l: bere_e@yhoo.fr ofrcose@yhoo.fr outt_e@yhoo.fr Abstrct If o ssoctve lgebr A s rght lpotet (resp. left lpotet) of degree the t s strogly lpotet of degree less or equl to 4.. Itroducto It s well ow tht ssoctve lgebr A s lpotet f for soe postve teger every product of or ore eleets of A vshes. For ossoctve lgebr A we ust dcte whch order the products re de. It follows tht the products of eleets of A could be zero for choce of rrgeets of pretheses d dfferet fro zero for Mthetcs Subect Clssfcto: 7D. Keywords d phrses: (o)ssoctve lgebr left lpotecy rght lpotecy lpotecy strog lpotecy dex of lpotecy. Receved October Scetfc Advces ublshers

2 86 C. J. A. ÉRÉ et l. other choce. Ths fct led to the defto of vrous cocepts of lpotecy ely left lpotecy rght lpotecy lpotecy d strog lpotecy.... I [5 roposto p. 8] the uthors gve the followg: roposto.. Let A be couttve (or tcouttve) lgebr. The { } A A for y. I the other words f the lgebr A s rght lpotet of dex the t s lpotet of dex ot greter th. We gve geerlzto of ths result to ll ossoctve lgebrs d by the wy we show tht the boudry of the dex s polyol ely the boudry s less or equl to 4 whch s better th. The theore s lredy doe for Mlcev d Lebz lgebrs (see [ ]). There s Lebz lgebrs tht re left lpotet d ot rght lpotet see for exple [4 Exple.]. Usg the oto of Asso -rght l (resp. Asso -left l) we prove tht f del of ossoctve lgebr A s rght lpotet (resp. left lpotet) of degree the s strogly lpotet of degree less or equl to 4. I Secto we gve soe deftos tht we wll use log the pper the the followg Secto we prove soe results o rght products of legth. The Secto 4 s devoted to rght products of weght the del. I Secto 5 we gve the results.. relres Throughout ths pper F wll be feld of chrcterstc ot. All vector spces d lgebrs wll be fte desol over F. Let A be ossoctve lgebr d let us deote the bler product o A by

3 ON NILOTENCY IN NONASSOCIATIVE ALGERAS 87 A A A ( b) b. Let be oegtve teger d let us deote by I ( ) the set { ; ; ; }. If A s ossoctve lgebr over F the F-trler p { } : A A A A defed by ( x y z) ( xy) z x( yz) s clled the ssoctor of A. If for ll ( x y z) A we hve { x y z} the lgebr A s sd ssoctve. Let U V W be subspces of A. { U V W} s the subspce geerted by the eleets { u v w} where u U v V d w W. Note Assoc ( U ) the followg subspce: Assoc ( U ) { U A A} { A U A} { A A U}. A subspce H of ossoctve lgebr A s clled left del (resp. rght del) f for x H d A oe hs x H ( resp. x H ). A del s both left d rght del. Let Ide ( U ) be the del of A geerted by the sublgebr Assoc ( U ). For del of ossoctve lgebr we troduce the ottos d followg terologes: Let be product of fctors s s s tht hve bee ssocted rbtrry wy. We suppose tht or ore fctors belog to. We sy tht the product s of legth d of weght wth respect to the del or ore sply tht s of legth d of weght. The legth of wll be oted # ( ) d ts weght wll be oted # ( ). Whe (( (( ss ) s ) s ) s ) s where the ssocto s de lwys rght we sy tht s rght product d we wrte ss s s. Slrly f s ( s( s ( s ( ss )) )) where the ssocto s de lwys left we sy tht s left product.

4 88 C. J. A. ÉRÉ et l. Let S S S p be rght products. Oe c wrte the rght product (wth S s fctors) N S S. We cll N stdrd product. p p S Defto.. Let be subspce of the uderlyg vector spce A. A subspce s rght lpotet f {} for soe where d. y coveto we set A. Notce tht s geerted by rght products of legth d weght. A subspce s left lpotet f for soe where d ( ). y coveto we set A. Notce tht s geerted by left products of legth d weght. Defto.. Let be del of the ossoctve lgebr A. Let { } be the subspce of the uderlyg vector spce A geerted by ll the products of legth ssocted rbtrry wy. We sy tht the del s lpotet f there exsts teger such tht { } {}. Let be the subspce geerted by ll products of eleets A wth t lest eleets. A del s strogly lpotet f {} for soe. Obvously for ll oegtve teger s del of A d oe hs d A ( resp. A ) s rght del (resp. left del) of A. We hve lso tht for ll oegtve tegers.

5 ON NILOTENCY IN NONASSOCIATIVE ALGERAS 89 Defto.. Let D be subspce of the uderlyg vector spce A d be oegtve teger. D ( L ) s the vector subspce geerted by ll rght products: d where d belogs to D d belogs to A for y teger I( ). ( L ) D s the vector subspce geerted by ll left products: ( ( ( ( d ))) ) where d belogs to D d belogs to A for y teger I( ). A. Defto.4. Let { } be del of the ossoctve lgebr We wll sy tht s Asso -rght l f there s teger such tht the del Assoc ( ) stsfes Assoc( ) ( L ) Assoc( ) LLL L { }. -tes We wll sy tht s Asso -left l f there s teger such tht the del Assoc ( ) stsfes ( L ) Assoc( ) L LLL Assoc( ) { }. -tes Defto.5. Let A be ossoctve lgebr d be del A. Let b be two eleets of A. If b we wll sy tht b (odulo ).. Rght roducts the Algebr A Le.. Let A be ossoctve lgebr be del of A d U be subspce of A. For ll x U b d for ll A xb x( b) { U A} Assoc( U ). roof. Trvl sce xb x( b) { x b } Assoc( U ). Le.. For y teger Assoc( ) d Assoc( ).

6 9 C. J. A. ÉRÉ et l. roof. ( ) ( ) Assoc( ) ( ) Assoc( ) ( ) Assoc( ) ( ) ( ) Assoc( ) ( ) Assoc( ) Assoc( ) ( ) Assoc( ) ( ) ( ) Assoc( ) ( ) ( ) Assoc( ) ( ) Assoc( ) Assoc( ) ( ) Assoc( ) ( ) ( ) Assoc( ) ( ) Assoc( ) Assoc( ) ( ) Assoc( ) ( ) ( ) Assoc( ) ( ) Assoc( ) ( ) Assoc( ) Assoc( ). Slrly we c deduce tht Assoc( ). Let us otce tht Ide( ) Ide( ) Ide( ) sce Assoc( ) Ide( ). Le.. Let A be ossoctve lgebr d be del of A. For ll teger d stsfy: A Ide( ) A d A Ide( ) A.

7 ON NILOTENCY IN NONASSOCIATIVE ALGERAS 9 roof. For oegtve teger we hve A ( ) A ( A) { A} Ide( ) Ide( ) d A lso we hve tht A ( ) ( A ) { A } Ide( ) Ide( ) A A ( Ide( )) A Ide( ) Ide( ) Ide( ) d A ( Ide( )) A A Ide( ) Ide( ) Ide( ). Clerly we hve d fro Le. we c deduce the followg le: Le.4. Let A be ossoctve lgebr d be del of A. Let us defe A d Ide( ) Ide( ) for ll teger ; s del of A whch stsfes. roof. Nturlly we hve A. A.

8 C. J. A. ÉRÉ et l. 9 roposto.5. For gve rght product d rbtrry product of legth. Let us set for y teger ( ) I. The T { } { }. roof. Frst of ll let us defe the followg products: Let be the legth of the product. For ( ) I let us defe. So we c wrte ( ). Now we c copute the proof: If oe hs ( ) { } { } ; f d T ( ) T ( ) { } ( ) ( ) { } { } ( ) { } { } { }. Wth the ssupto tht for rght product of legth oe hs T { } { }. (.)

9 ON NILOTENCY IN NONASSOCIATIVE ALGERAS 9 The let us cosder rght product of legth sy : T { }. (.) y the ssupto (sce the legth of s ) c be wrtte s: T { } { }. Ths to Equto (.) we hve { } T { } { } { } { } { } { }.

10 C. J. A. ÉRÉ et l. 94 It follows tht { } T { } { } { } { } { } { }. The proof s doe. Le.6. Ay product T wth legth ossoctve lgebr A s ler cobto of rght products of legth odulo the del ( ). A Ide roof. y ducto o the legth we hve: If equls or there s othg to do. If oe c otce tht bc T or ( ) bc { } c b bc for ll c b A. The le s lso obvous. Let us suppose tht the le s true for product whch legth s strctly less th. 4 Now for gve product (wth legth ) T where s rght product of legth such tht. > Ths to roposto.5 T { } { }

11 ON NILOTENCY IN NONASSOCIATIVE ALGERAS 95 ( ) { }. Let I( ). The legth of the products ( ) re less or equl to so they re ler cobtos of rght products odulo the del Ide ( A). Ths let us ow tht T s ler cobto of rght products of legth odulo the del Ide ( A). 4. Rght roducts of Weght the Idel Le 4.. Ay product T wth legth d weght wth regrd to the del of ossoctve lgebr A s ler cobto of rght products of legth d weght odulo the del Ide ( ). roof. y ducto o the legth we hve The le s obvous f. If the there re eleets A such tht s oe of the followg: d ( ) { }. So for s ler cobto of rght products odulo the del Ide ( ). Let us suppose tht the le s true for product whch legth s strctly less th 4. Now for gve product (wth legth d weght ) T where s rght product of legth such tht >. Ths to the roposto.5 T { } { }

12 96 C. J. A. ÉRÉ et l. ( ) { }. Let I( ) we ow tht ( ) d re products whch legth re less or equl to. they re ler cobtos of rght products odulo the del Ide ( ). So by ducto Ths let us ow tht T s ler cobto of rght products odulo the del Ide ( ). Through the clculto t s cler tht the ters ( for I( ) ): ( ) d eep vrt the legth d weght of T. Le 4.. Let A be ossoctve lgebr d { } be del A. Let be rght product wth legth d weght. The belogs to. roof. Let σ be ectve p of I ( ) to ds I ( ) such tht < ples tht σ () < σ( ) d for ll I ( ) σ( ). Let us lso defe for ll teger I( ) the followg products: σ( ) σ( ) σ( ) σ( ) σ(). Clerly belogs to Ide( { A A} ) d lso belogs to.

13 ON NILOTENCY IN NONASSOCIATIVE ALGERAS 97 Sce s del σ( ) belogs to Ide( ). y ducto let us suppose tht for teger such tht < we hve belogs to Ide( ). The let us show tht s eleet of Ide( ). Ideed we hve σ( ) σ( ) s eleet of the del d so o σ( ) belogs to Ide ( ). The we hve proved tht s eleet of. Le 4.. Let l be tegers such tht l d let A be ossoctve lgebr d be del of A whch s Es -rght l. A rght product of legth d weght ( ) greter or equl to l belogs to l ( L ). roof. The rght product s of weght greter or equls to l deed let be the weght of d be the weght of. We hve d the equlty ples tht. So l l. The Le 4. tells tht l. Ad so o ( ) ( ) l L l ( ( { } )) ( ) ( l Ide A A L L ) sce s del Assoc -rght l. Le 4.4. Let be teger such tht the del s Es -rght l. Let be product of weght t 4 wth regrd to the del the s ler cobto of rght products such tht for y we hve belogs to ( L ) or hs t lest oe fctor ( L ).

14 98 C. J. A. ÉRÉ et l. roof. Let > d t 4. Th s to the Le 4. y product of weght greter or equls to t s ler cobto of rght products of weght greter or equls to t. Let µ where s rght product of weght greter or equls to t. For y we hve s ps p s where s L (p s the legth of ). If there s oe eleet s such tht t s weght s greter or equls to the Ad so o s belogs to hs fctor Else every fctor the uber of fctors ( L ) by pplcto of the Le 4.. ( L ). s hs weght strctly less th. Let q be s wth weght greter or equls to. The oe hs q ( ) t 4 d the q >. Whe the weght of s s greter or equls to we hve s ; belogs to. So s of weght greter or equls to q. Sce q we hve ( L ) th s to Le 4.. y buse we wll sy tht hs t lest oe fctor ( L ). 5. M Theore Theore 5.. Let F be couttve feld wth chrcterstc ot A be F-ossoctve lgebr d be del of A whch s Es -rght-l. The the followg ssertos re equvlet: () s rght lpotet; () s lpotet; () s strogly lpotet.

15 ON NILOTENCY IN NONASSOCIATIVE ALGERAS 99 roof. Ideed for y teger the vectors spces s clusos {} tell us tht () () (). Furtherore suppose tht there s teger l l such tht {}. Let us defe x{ l } the for teger A such tht l 4 the Le 4.4 tells us tht y product wth weght greter or equls to l 4 s ler cobto of rght L L products whch hve t lest oe fctor ( ) ( ) ( ) ( ) {}. l Ad so o. The. The plcto () () s doe. Corollry 5.. Let F be couttve feld wth chrcterstc ot d A be ossoctve F-lgebr. The followg ssertos re equvlet: () A s rght lpotet; () A s lpotet; () A s strogly lpotet. roof. Clerly we hve () () (). Let us otce tht Assoc ( L) s subset of l A d f there s teger l such tht A {} we do hve tht Assoc ( L) s del Asso l -rght-l. The Theore 5. tells us tht () (). Corollry 5.. Let F be couttve feld wth chrcterstc ot d A be ossoctve F-lgebr. The followg ssertos re equvlet: () A s left lpotet; () A s lpotet; () A s strogly lpotet.

16 C. J. A. ÉRÉ et l. roof. Fro the lgebr A wth bler product o A by φ : A A A ( b) φ ( b) b we c obt the lgebr A op where bler product of d b s φ ( b ). The left lpotecy of A s equvlet of op rght lpotecy of A. Rer 5.4. Rght lpotecy of del of A y ot ples left lpotecy of the del d vce-vers. For the (left) Lebz lgebr A defed Exple. of [] the del I Ess( L) geerted by ll squres s Es -rght-l. ut for ll teger ( A ) Assoc( I ) A AAA Assoc( I ) { }. - tes Refereces [] C. J. A. éré N.. lbré d M. Outtr Nlpotece ds les lgebres de Mlcev sous à publcto. [] C. J. A. éré M. F. Ouedrogo d M. Outtr O lpotecy Lebz lgebrs sous à publcto. [] C. J. A. éré A. Kobbye d A. Koobo O clss of Lebz lgebrs Itertol Jourl of Advced Mthetcl Sceces () (5) Scece ublshg Corporto do:.449/s.v.59 [4] C. J. A. éré N.. lbré d A. Kobbye Le s theores o soluble Lebz lgebrs rtsh Jourl of Mthetcs & Coputer Scece 4(8) (4) [5] K. A. Zhevlov A. M. Sl o I.. Shestov d A. I. Shrshov Rgs tht re Nerly Assoctve Trslted by Hrry F. Sth Acdec ress New Yor 98 x 7 pp. ISN g