Degenerate Clifford Algebras and Their Reperesentations

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1 BAÜ Fn Bl. Enst. Dgs t 8-8 Dgnt ffod Algbs nd Th Rsnttons Şny BULUT * Andolu Unvsty Fculty of Scnc Dtmnt of Mthmtcs Yunum Cmus Eskşh. Abstct In ths study w gv n mbddng thom fo dgnt ffod lgb nto nondgnt on. By usng th snttons of non-dgnt ffod lgb w dvlo mthod fo th snttons of th dgnt ffod lgbs. W gv som lct constuctons fo low dmnsons. Kywods: Dgnt ffod lgb sntton of ffod lgbs udtc fom nlotnt dl. Djn ffod Cbl v Tmsll Özt Bu çlışmd djn ffod cblndn djn olmyn ffod cbln b gömm tom vdk. Djn olmyn ffod Cblnn tmslln kullnk djn ffod cblnn tmsll çn b mtod glştdk. Düşük boyutl çn bzı çık tmsll vdk. Anht klml: Djn ffod cb ffod cblnn tmsl kudtk fom nlotnt dl. Intoducton It s wll known tht non-dgnt l ffod lgb s somohc to mt lgb n o dct sum n n of mt lgbs wh RC o H s [ ]. Th stuctus nd snttons bng studd by mthmtcns nd hyscsts. On th oth hnd dgnt l ffod lgb cn not b somohc * Şny BULUT skz@ndolu.du.t Tl: 8. 8

2 BULUT Ş. to full mt lgbs s thy contn nlotnt dl. Th som mks bout th dgnt ffod lgb nd dgnt sn gous n []. Eto lgbs nd dul numbs scl css of th dgnt ffod lgbs. Dfnton Th ffod lgb V Q ssoctd to vcto sc V ov F wth udtc fom Q s dfnd s th uotnt lgb V Q T V / I Q wh TV s tnso lgb T V F V V V L nd I Q s th two-sdd v v Q v. dl n TV gntd by th lmnts { } As n ml whn Q thn th ffod lgb V Q s just th Gssmnn lgb lso known s th to lgb Λ V. A non-dgnt udtc fom Q on th l vcto sc VR n s gvn by Q L L n. Th ffod lgb on R n ssoctd to Q s dnotd by. Th tbl of nondgnt l ffod lgbs s s follows s [ ]: mod8 m R m R R m C 6 m H m H H Th lct somohsm Φ fom th ffod lgb C l to th ltd mt lgb s gvn n [] fo ll. Dfnton Lt A b l lgb nd W l vcto sc thn n lgb homomohsm ρ : A End W s clld sntton of th lgb A. Fo ml th cnoncl sntton of th mt gou Rn on R n s odny oduct of mt wth vcto. Th cnoncl sntton of th lgb of coml numbs C on R s gvn ρ : C R EndR y ρ y y fo ll y C. On cn obtn th cnoncl sntton of th lgb Cn n by n mtcs wth coml nts on th vcto sc R n s follows: Lt A [ z j ] n n Cn b mt of coml numbs z j. Usng th cnoncl sntton of C w obtn n lgb homomohsm ρ : Cn Rn EndR n n 9

3 BAÜ Fn Bl. Enst. Dgs t 8-8 whby z z L z n z z L zn ρ n A M M L M zn zn L znn Smlly th cnoncl sntton of th utnon lgb H on R s gvn by ρ : H R EndR y u v y v u ρ y ju kv u v y v u y fo ll y ju kv H. Lkws th coml cs usng th cnoncl sntton ρ : H R w obtn n lgb homomohsm whby ρ : Hn Rn EndR n n ρn A M n M n L L L L M whch s th cnoncl sntton of Hn th lgb of n n mtcs wth utnon nts. By usng th cnoncl snttons of mt lgbs on cn obtn sntton of non-dgnt ffod lgbs C l fo ll. Th snttons of th C l bng studd by vous uthos s [6 8]. Th vous lctons of th mt snttons of ffod lgbs [ 9 ]. Fo th lct sson of Dc oto th sntton of th ffod lgb s lso ctcl s []. n n nn Dgnt ffod lgbs on R n In ths scton w nvstgt ffod lgbs ssoctd to dgnt udtc fom Q on R n. Fom Sylvst thom t s nough to consd th followng dgnt udtc fom Q L L n >. Th ffod lgb ssoctd to th dgnt udtc fom Q s dnotd by. W cn dtmn th dgnt ffod lgb s follows: Lt b n othonoml bss fo R n { K K } K wth sct to th dgnt udtc fom

4 BULUT Ş. Q.Thn w hv y y fo th bss lmnts nd y nd y. Also th followng dntts hold: Q fo Q fo j j j Q fo k k k Dfnton An dl I of th lgb A s clld nlotnt f ts ow I k s { } fo som ostv ntg k. Such lst k s clld od of I. All lmnts of th fom I I bjkj K clm L M I PRS J K L M P R S d consttut n dl of wh I bjk clm dprs l numbs nd w us multl nd notton.. K I K k nd I fo { } { } k P K < < L < k. W dnot ths dl by I som uthos cll t th Jcobson dcl s bss fo th dl I nd lso not tht th ll []. Th st { } I J K L M P R S lmnts nlotnt. Thn I s nlotnt dl s.8 Thom n [] on g 9. Nlotnt dls motnt to th chctzton of n lgb. Thom Lt A b n lgb wth unt. Thn A s sm-sml f nd only f th s no nlotnt dl dffnt fom zo. cn not b mt lgb fo > bcus I s nlotnt dl. By th Thom th ffod lgb s not sm-sml lgb. On th oth hnd cn b mbddd nto th mt lgbs. R S. Embddngs of dgnt ffod lgbs nto mt lgbs Now lt us ln thom concnng th dgnt ffod lgbs. Thom Th dgnt ffod lgb cn b mbddd nto th nondgnt ffod lgb. Poof It s nough to gv on to on lgb homomohsm fom to. It s ossbl to tnd t to whol lgb whn th homomohsm s dfnd on th bss lmnts. Ths homomohsm s dfnd on th bss lmnts n th followng wy:

5 BAÜ Fn Bl. Enst. Dgs t 8-8 Ψ : M M M M M M Moov th followng ults: Ψ Ψ Ψ Ψ [ ] Ψ Ψ [ ] Ψ Ψ [ Ψ ] [ ] Fom ths thom dgnt ffod lgb cn b mbddd nto th mt lgbs such s Rm Cm Hm Rm Rm Hm Hm. Rmk 6 Th s gnl mbddng thom of n lgb nto n ndomohsm lgb s [] Thom. on g 9. In ths cs th dmnson of th sntton sc bcm lg nd t s not usful fo lct constuctons. Lt us gv som mls n low dmnsons: Eml Th full dgnt ffod lgb whch s uvlnt to th Gss- mnn lgb Λ R cn b mbddd nto th non-dgnt ffod lgb n th followng wy: f : Λ R Snc s somohc to th mt lgb R on cn lso gv n mbddng fom nto th mt lgb R s follows:

6 BULUT Ş. R Eml 8 Dgnt ffod lgb cn b mbddd nto th nondgnt ffod lgb. W know tht s somohc to th mt lgb R R. Th mbddng f cn b gvn on th bss lmnts s follows: : f Th m f dfnd bov cn b sn tht t s n on to on lgb homomohsm. If w us th comoston of th homomohsm f nd th somohsm Ψ thn w gt n on to on lgb homomohsm s follows: R R wh. Any cn b wttn s. By wtng th vlus of w gt th followng dntty:

7 BAÜ Fn Bl. Enst. Dgs t 8-8 Hnc th dgnt ffod lgb s somohc to th followng sublgb of th mt lgb R R. R Eml 9 Lt Q b dgnt udtc fom on V R nd ts ffod lgb. Th dgnt ffod lgb s mbddd nto nondgnt ffod lgb. Moov w know tht th ffod lgb somohc to th mt lgb C. Th mbddng m fom to s dfnd on bss lmnts s follows: C It cn b sn tht ths m s on to on lgb homomohsm. By comosng ths homomohsm nd th somohsm Ψ w gt th followng homomohsm: C wh. Any cn b wttn s. Thn by wtng th vlus of w hv th followng dntty: Hnc R C.

8 BULUT Ş. Eml Lt Q b dgnt udtc fom on V R nd ts ffod lgb. In sml wy t s ossbl to b mbddd nto sutbl mt lgb. As bov mls th dgnt ffod lgb s mbddd nto non-dgnt ffod lgb. Th mbddng s dfnd on th bss lmnts s follows: R By comosng ths homomohsm nd th somohsm Ψ w gt homomohsm fom to th mt lgb R s follows: R W hv. Thn ny lmnt n cn b wttn n th followng wy: 6

9 BAÜ Fn Bl. Enst. Dgs t W know th vlus of. If ths vlus usd thn w gt 6. Thn 6 6. Hnc th dgnt ffod lgb s somohc to th followng sublgb of th mt lgb R. 6 6 K R If w contnu n sml wy w hv th followng mbddngs: H H R H C C H H H H H H R R W cn sly dtmn th dgnt ffod lgbs fo th hgh vlus of n th sml wy.

10 BULUT Ş. Sno Rsnttons of Dgnt ffod Algbs A sntton of dgnt ffod lgb cn b gvn s follows: Lt : V End ρ b th sno sntton of whch s wdly known nd Ψ : b th mbddng dfnd bov thn th comoston : V End o Ψ ρ s n lgb homomohsm so t s sntton of th dgnt ffod lgb nd w cll t sno sntton of. Now w gv som ml n low dmnson. Eml Eml Eml R R R

11 BAÜ Fn Bl. Enst. Dgs t 8-8 Eml R Rfncs [] T. Fdch Dc Otos n Rmnnn Gomty AMS. [] R. Hvy Snos nd Clbtons Acdmc Pss 99. [] H.B. Lwson nd M.L. Mchlsohn Sn Gomty Pncton Unv [] R. Ablmowcz Stuctu of sn gous ssoctd wth dgnt ffod lgbs Jounl of Mthmtcl Physcs 986. [] N. Dğmnc nd Ş. Kz Elct Isomohsms of Rl ffod Algbs Intntonl Jounl of Mthmtcs nd Mthmtcl Scncs 6. [6] M. F. Atyh R. Bott nd V.K. Sho ffod Moduls Toology [] A.Dmsks A nw sntton of ffod Algbs Jounl of Physcs A: Mthmtcl nd Thotcl [8] A.Tutmn ffod Algbs nd Th Rsnttons Encyclod of Mthmtcl Physcs. [9] N. Dğmnc nd Ş. Koçk Gnlzd Slf-Dulty of -Foms Advnc n Ald ffod Algbs -. [] N. Dğmnc nd N. Özdm Th Constucton of Mmum Indndnt St of Mtcs v ffod Algbs Tuksh Jounl of Mthmtcs 9-. [] H. Bum nd I. Kth Plll snos nd holonomy gous on sudo- Rmnnn sn mnfolds Annls of Globl Anlyss nd Gomty [] C. Fth Algb I: Rngs Moduls nd Ctgos Sng-Vlg 9. [] D. R. Fnck Algbs of Ln Tnsfomtons Sng-Vlg 8

Chapter 2 Reciprocal Lattice. An important concept for analyzing periodic structures

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