Abstract tensor systems and diagrammatic representations

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1 Astrt tensor systes n grt representtons Jāns Lzovss Septeer 28, 2012 Astrt The grt tensor lulus use y Roger Penrose ost notly n [7]) s ntroue wthout sol thetl grounng. We wll ttept to erve the tools of suh syste, ut n roer settng. We show tht Penrose s wor oes fro the grston of the syetr lger. Le lger representtons n ther extensons to not theory re lso susse. Contents 1 Astrt tensors n erve strutures Astrt tensor notton Soe s opertons Tensor grs A grse strt tensor syste Generton Tensor onepts Representtons of lgers The syetr lger Le lgers The tensor lger Tg) The syetr Le lger Sg) The unversl envelopng lger Ug) The etrze Le lger Dgrston wth lner for Dgrston wth syetr lner for Dgrston wth syetr lner for n n orthonorl ss Dgrston uner -nvrne The unversl envelopng lger Ug) for etrze Le lger g Ensung onnetons 32 A Appenx 35 Note: Ths wor reles hevly upon the text of Chpter 12 of rft of An Introuton to Quntu n Vsslev Invrnts of Knots, y Dv M.R. Json n In Mofftt, yet-unpulshe oo t the te of wrtng. 1

2 1 Astrt tensors n erve strutures Our pproh s to egn wth the oon tensor struture, generlze to strt tensors where ssoton wth unque vetor spe s relxe), n fnlly to nterpret the strt tensors s rete grphs of spef sort - we shll ll these tensor grs. Let us egn wth soe nottonl rers. 1.1 Astrt tensor notton Let V e n n-ensonl vetor spe wth ss {v 1,..., v n }, n V ts ul vetor spe, wth ul ss {v 1,..., v n }. We use the onventon of representng vetor spe V tensore wth tself tes y V ) V V. }{{} tes Defnton Astrt tensor. An strt tensor s n eleent w W V r1 V s1 V r V s, whh s tensor prout of the spes V, V n soe fxe onton n orer. The rrngeent of the nees s prt of strt nex notton, n entfes the orer of the tensore vetor spes. The followng presentton of suh generl tensor shoul e seen s eng on sngle lne. w α 1,...,r f 1 1,...,f r 1,..., r1 : g 1,...,g s 1,..., s1 g 1,...,g s v 1 v r1 v 1 v s 1 vf1 v fr v g1 v gs 1.1) suh tht r 1,..., r, s 1,..., s N {0}. Low-ensonl tensors re qute oon n thets, n they hve een gven oon nes: A tensor wth no upper nor lower nees s tere slr A tensor wth one nex upper or lower) s tere vetor A tensor wth two nees upper or lower or ny onton of the two) s tere trx Conventons. Beuse of ths lengthy presentton, even for sple tensors, we use the nexe oeffent n the sun of w equvlently s w, notng tht one s esly reoverle fro the other. Tht s, w α 1,...,r 1 f 1,...,f r 1,..., s1 g 1,...,g s v 1 v gs α 1,...,r 1 f 1,...,f r 1,..., s1 g 1,...,g s Here we norporte Ensten s shorthn of elntng the suton syol. For longer lultons esrous of ore thorough ount, we wll rng the suton syol. It s unerstoo tht the suton tes ple over ll repete nees. Reovng the ss tensors whh potentlly) repet nees, wll pose no prole, s we then su over ll nees pperng n the upper n lower postons of the strt tensor syol, tng re to note ultpltes. In 2.1 we prove tht ths uses no onfuson. We lso follow syol onventon. Tensors wll e represente y Gree letters α, β,... ) n tensor nees wll e gven y possly nexe) Ltn letters,,... ). An exepton s slrs, whh we wll soetes represent y λ, λ 1, λ 2,.... In generl, no one spef syol s ssote to one spef oet, ut gn we e n exepton for the Kroneer-elt syol n tensor) δ, n the Le ret tensor γ, eh of whh we use wth spef enng. Any slrs susse re ssue to e over oon fel, usully ut not neessrly) C. Any fel y e use. 2

3 1.2 Soe s opertons Let us onser soe strt tensor opertons. Together they e up wht s nown s tensor lulus, whh s essentlly the forlzton of nex npulton. All these opertons n e erve fro the proper exhustve efnton of n strt tensor n strt nex notton. Renexton. Tensor nees y e relelle s esre, s long s the lel s not lrey n use y nother nex of the tensor. For exple, α α v v α z v v z α z Slr ultplton. Tensors y e ultple y slrs over our fel, s well s groupe together y slrs. For exple, gven slr λ, λ α λ α v v ) Aton. Aton of tensors nee not our over oon spe, s ths s ust forl su. If tensors re of the se type, then t y e possle to one the, though we wll not o tht, s nforton s lost y ong so. Exples of generl ton n spel se re β e + χ gh β e v v v e + χ gh v g v h v α + α 2α v v Tensor prout. The tensor prout lso oes not neessrly our over oon spe, ut even f t oes, there s no spel splfton. The nees etween the tensors re requre to not one. For exple, β e α β e v v v v v e α Contrton. We y entfy one upper nex wth one lower nex y the ontrton operton, therey ssgnng the the se lel. We use the opertor syol κ, to enote entfton of the th n th nees of ρ. For exple, κ 2,4 α β ) e α β e α β e v v v v v e The generl e of opertons wthn tensor lulus s tht we o not hve uh leewy n splfton. By ovng to strt tensor systes, we ntroue outtvty of the prout, ong other reutons. Aopnyng strt tensors s nother syste tht grses strt tensors n n effetve wy. Ths entfton wll e etve y onstruton. We shll now esre ths syste. 1.3 Tensor grs Defnton Prtve tensor gr. A prtve tensor gr s rete grph wth n 1 vertes n 1 onnete oponent suh tht n 1 vertes hve egree 1 every ege s orente n one of two retons every ege s gven unque lel there s preefne lowse orerng of the eges The vertex wth the lrgest egree s the stngushe vertex of the grph. If n 1, the stngushe vertex s the sole vertex; for n 2 eson s e; n for n 3 t s the only vertex wth egree 2. 3

4 The prtve tensor gr elow hs 4 vertes, wth eges rrnge,,, n s entfe y to the strt tensor α. α 1.2) The stngushe vertex s enote y syol grphlly lrger thn the other noes. To fx the orerng of the eges, we ple the nhor syol n the spe tht ves the frst fro the lst orere ege. The frst ege n the orerng s then retly n front of, wth respet to the lowse reton, the nhor. Prtve tensor grs re llowe to hve loops, n whh se we reove the lel of the ege. y x β xz y z 1.3) Defnton Copoun tensor gr. A opoun tensor gr s grph wth 1 onnete oponent resultng fro entfyng eges of fferent prtve tensor grs wth opposte orentton. Eges tht re entfe etween stngushe vertes lso hve ther lels entfe, n thus reove n the gr. For exple, α g h βh χ l l g 1.4) Defnton Tensor gr. A tensor gr s rete grph suh tht every onnete oponent s ether prtve tensor gr or opoun tensor gr An exple of tensor gr wth 2 onnete oponents s f α β e γf g h χ l l 1.5) g h e 4

5 Note tht the hoe of prout orer of the strt tensor on the rght-hn se s not portnt, s the prout s outtve see 2.1). The grphl shpe of eh stngushe vertex only ntes tht ll tensors n gven gr re not the se. We nee not eep wth, sy, enotng α y rle, n nee we shll not. Exeptons re e for tensors tht wll e entrl to our nlyss, s prevously entone. The grphl rrngeent of the eges n vertes s not portnt, s long s the orer n whh the eges onnet to the stngushe vertex s respete. Therefore, we y ple stngushe vertes n lelle eges wherever we wsh n gr. In other wors, pplyng sooth ps to the gr oes not hnge the tensors t grses. For exple, e 1.6) e We prefer to eep ege lels rrnge n two rows, ross the top n otto of the gr, though the rrngeent s rtrry. Ths herene to ertn spet of presentton s ept throughout for ese of reng. Rer Wth regrs to notton, s lrey use, we eploy the followng syols: entfton etween n strt tensor n tensor gr equvlene of strt tensors n vetor spes uner soorphs equlty of slrs, strt tensors, vetor spes, or tensor grs For the syol, we l tht there s 1-1 orresponene etween strt tensors n tensor grs. Ths s ler fro the followng retves. Go roun the ounry of every stngushe vertex, strtng fro the nhor, n lowse reton, rng eges wth unque lels, notng whether eges re rete towr or wy fro t. The lels of the eges rete towr the stngushe vertex eoe susrpts, n the ones rete wy eoe supersrpts. Eges tht hve oth ens nent wth stngushe vertex hve ther lels entfe wth eh other. The orerng of the strt tensors oes not tter, s strt tensor systes re outtve. Astrt tensor systes re outtve n the sense tht there exst soorphss etween V V n V V, s well s etween V V V n V V V, n so on, s there s no hnge n enson. Thus for ss {v 1,..., v 6 } of V n ul ss {v 1,..., v 6 } of the ul spe V wth λ 1, λ 2 C, we wrte λ 1 v 1 v 3 λ 1 v 3 v 1 n λ 2 v 4 v 6 v 1 λ 2 v 4 v 1 v 6 It s portnt to note tht ths use of soorphss s llowle only f V s fnte-ensonl, whh t wll lwys e for our susson. We wll oupy ourselves n lrge prt wth the forulton of slr soorphss nto equltes etween tensor grs. 5

6 Proposton Twst soorphs. Let us efne for lter use, the splest soorphs. Let U e n n-ensonl vetor spe wth ss {u 1,..., u n }, n V e n -ensonl vetor spe wth ss {v 1,..., v }, n λ C. Then there exsts p T suh tht T : U V V U λ u v T λ) v u for ll 1 n n 1. Ths p s generte y entfyng the ss {u 1 v 1,..., u n v } of U V to {v 1 u 1,..., v u n }, whh s the ss of V U. Sne T s etve orphs, we n fn ts nverse p T 1 : V U U V. 2 A grse strt tensor syste An strt tensor syste s set of strt tensors long wth opertons tng on the strt tensors. 2.1 Generton If the eleents n grt tensor syste re presely the eleents whh rse through the opertons elow tng on gven set of S of grs, then we sy tht S genertes the grt tensor syste. Renexton. By ressgnng fferent syols s nees, we re not hngng the vlue of the tensor, only ts presentton. Renexton s slly renng wth new syol tht oes not lrey pper s n nex, thus ny new lels ssgne re rtrry. α α v v v v α g f v f v v g v α g f f 2.1) α α g f g Slr ultplton. Slr ultplton of the strt tensor s slr ultplton of the forl tensor. In the followng tensor gr, the slr λ C s ple to the rght of the stngushe vertex t ffets. Ths n lwys e one n nner tht oes not ntroue onfuson, s eges y e ove roun n the gr. λ α λα v v v v 2.2) λ α λ 6

7 Aton. As forl operton, ton oes not requre eleents of the se type. g α + β ef α v v v v + β g ef v e v f v g α + β g ef + e 2.3) g f Every tensor gr tht represents ter n the su shoul e seprte fro the the other tensor grs. If the types of the two tensors re the se, then we y one the. Ths loses the entty ssote wth the strt tensor tht we egn wth, ut oes splfy presentton. For exple, e f γ + β γ v v f v + β v v e v f e γ v v v + β v v v 2.4) γ + β ) v v v χ v v v 2.5) χ + 2.6) f e By nex renng we get 2.4), n y ton 2.5). Here we hve tht the su of two tensors n gven spe s nother unnown) tensor n tht se spe, whh s wht 2.6) grses - note the hnge fnl shpe. Thus ton wth slr ultplton fors vetor spe of strt tensors. Tensor prout. The operton : V V V V nvolves onng two tensors nto sngle tensor. For lrty, we requre tht the two seprte tensors o not hve ny oon nees, esly ensure y renexton. Any slrs tht the tensors ght hve re ultple together. α βef α g β g ef β g ef α e f e f e f 2.7) g g g 7

8 We nnot stngush extly whh vetor spe the ove eleent s n, so nste we vew t s eng n oth V ) 2 V V ) 3 V n V ) 2 V V ) 2 V V onurrently. The prout operton s strutve over ton. We wll further ot the syol, when tensor ultplton s unerstoo. ) θ h α + β g ef θ h α + θ h β g ef Contrton. We efne n operton tht sets equl the lels of n upper-lower pr of nees. For tensor ρ, the ontrton κ, ρ) wll set the th n th nees equl to eh other. Extly one of the nees to e ontrte ust e lower nex, n extly one ust e n upper nex. After entfton, we y ove to lower-ensonl vetor spe, rellng tht ths ove wll lose us nforton. κ 1,3) α ) κ 1,3) α v v v v ) α v v v v β v v 2.8) β f α then κ 1,3 α ) 2.9) When pplyng the ontrton p to tensor tht s the prout of other tensors, re ust e ten, s the nees nte n the susrpt of κ refer to the orer of nees n the rguent. Thus ) ) κ 4,7) α β g ef α β ef κ 4,7) β g ef α β g ef α Through ths operton we y reue the rn of the tensor y two. Ientl upper-lower nees re tere uy nees. Thus we y onser the tensor n lower ensonl spe. β β v v v v v β 1 1 v 1 v v v 1 v + + β n n v n v v v n v 2.10) β 1 1 v 1 v 1 v v v + + β n n v n v n v v v 2.11) γ v v v + + χ v v v φ v v v φ The orgnl tensor s n eleent n V V ) 3 V. By expnng n 2.10) over, the repete nex n eh ter oes not ontrute to the type of tht ter. In 2.11) we use s we re pplyng the T soorphs fro Ths lso llows us to fn pproprte tensors γ,..., χ V ) 3 V to put n ple of the prevous ones. Ths lulton s grse y 8

9 2.12) The entfton of the entl eges s nturl, though we y leve the tensor s s wth loope ege, s suh loops re worth onserng when nlyzng nots wth the of these grs. So we see tht we y nsert n reove n upper-lower pr of entl nees n ny esre postons of n strt tensor. Dgrston of strt tensors. The grt tensor syste s generte y equvlenes of strt tensors n tensor opertons wth tensor grs. Eleents of ths syste re tere T V )-grs. 2.2 Tensor onepts Here we suss soe notons unque to tensor grs. The Kroneer elt tensor. Generlly t s efne s funton on two vrles. { 1 δ 0 We wll vew t s tensor. Penrose n [7] uses ths syol wthout tng nto ount the orer of the nees. Sne ther orer n ths exposton s t tes prount, we wll e no suh splfton. We wll hve δ δ v v δ δ v v δ δ 2.13) When the nees te on onrete vlues, we llow the syol to e evlute s slr vlue. For exple, δ1 2 δ3 1 δ n δ1 1 δ2 2 δ Ths proves forl wy of relellng nees. If we wsh to relel the lower nex n gven tensor 9

10 α to nother nex, sy e, we pply the tensor prout to α n δ e. For exple, α δ e α δ e e α δ e + e α δ e 2.14) α δ e 2.15) e α1 δ α2 δ αn δ n n α1 + α2 + + αn αe As Kroneer s elt tensor y e evlute s slr, rerrngent of the su n 2.14) shows tht ost of the ters reue to zero n 2.15), resultng n the se tensor tht we egn wth, now relelle. In grt presentton, e 2.16) e The ove ntes tht we y ept the Kroneer elt tensor s lne,.e. grph wth sngle ege ut no stngushe vertex. Dgrtlly, δ δ 2.17) Ths nterpretton s useful n npultng grs. It shoul lso e note tht elt-tensor y e splt up nto ore elt tensors, nto s ny s esre: δ δ δ δ δ l δ l δ δ δ l δ l... Here we hve only susse the Kroneer elt syol wth n upper-lower pr, so t ght e teptng to onser the syol wth ust n upper pr or lower pr, δ n δ. We put off ths teptton to when the nee for suh syol wll neesstte ts use n 3.6). Equlty. The equlty relton es stteent tht two tensors re entl. When elrng equlty, l s e out he slrs lso. For exple, Ths s equvlently stte s α β 2.18) α v v β v v 10

11 Fully expne, we hve α α v v,, β, β v v Then 2.18) sserts tht, one n re fxe, the followng equton hols for the slrs. α β Suonfgurtons. Ths ter pples to grs generte y our syste. Gven tensor ρ, soltng suset of the stngushe vertes n ll ther ssote legs, we get suonfgurton of the orgnl tensor. For exple, s suonfgurton of g 3 Representtons of lgers The phrse the lger s generte y wll en tht the tensor opertons, esre n the prevous seton, wth the nlue lst of tensors, onstrne y ny nlue restrtons, rng forth every tensor n the lger uner susson. Rell frst soe s fts out the nture n onstruton of lgers. Defnton Alger. Let V e vetor spe over fel F. Defne p p : V V V to e prout on V. Note tht p s lner n eh of ts rguents. Then the pr A V, p) s tere n lger of the vetor spe V enowe wth p. Defnton Tensor lger. Gven vetor spe V over fel F, the tensor lger TV ) over V s efne to e the lger of the vetor spe T V ) n 0 V n F V V V ) V V V ) enowe wth n pproprte prout. Every eleent of ths lger s strtly wthn soe V n for n N {0}, so the prout p s gven y p : V p V q V p+q). We y lso hve tensor lger TA) over n lger A, n whh se the se onstruton, wth A nste of V, hols. The prout on the tensor lger n ths se woul e the ultlner equvlent of the lger prout of A. Proposton Vetor spe soorphs. Let U e n n-ensonl vetor spe wth ss {u 1,..., u n }, n the ul spe U wth ul ss {u 1,..., u n }. Let V e n -ensonl vetor spe wth ss {v 1,..., v }, n the ul spe V wth ul ss {v 1,..., v }. Then there exsts n soorphs esre y T U,V : HoU, V ) U V f f u v The oet f s p, n f represents the slr oeffent of the tensor n the ge. The ton of the p f s gven y: f : U V u f v It s portnt to note tht the oeffents n the ges of f n T U,V re entl. Ths gves us wy to nterpret ps etween vetor spes s eleents n tensor prout of vetor spes. 11

12 3.1 The syetr lger Defnton Syetr lger. The syetr group, or perutton group of orer n s enote y S n. The syetr lger s efne, over vetor spe V over fel F, to e the spe SV ) 0 S ) V ) 0 T ) V ) S where T ) V ) s the tensor lger of the -ensonl vetor spe T V ) enowe wth the tensor prout. The quotent of ths spe y the syetr group llows us to reue the struture of the lger n the followng nner. Proposton Every strt tensor, n equvlently, every tensor gr, esres unque eleent of SV ). Conser n strt tensor. If t hs prtve gr, the entfton s ler. Suppose we hve opoun gr. As note prevously, tensor tht s the prout of two tensors nnot e entfe wth ertnty s elongng to sngle spe, thus to sngle eleent of T V ) euse of outtvty of the prout). In SV ) tensor ppers n ll possle peruttons of ts prtve prts, therefore uulo ths group, they ll esre the se eleent n SV ). Tht eng s, n ths lger the orer of spes n the onstruton of gven tensor s not portnt. For exple, onser the strt tensor β β v v v T 3) V ) In the syetr group we fn s peruttons ll of the yles Therefore we hve s strng of equltes n s tensor grs, β ) ) ) ) ) ) β β β β β 3.1) Rell tht the lels hve no sgnfne pst eng pleholers. So n 3.1) we hve tensor wth equlty ong ll the peruttons of ts eges. Ths les to the onluson tht the nhor n the pose orer of vetor spes n e sregre. Therefore we y reove nhors of tensors n lels of eges n the grston of the syetr lger, eepng only the reton n whh eh ege ponts, even sregrng the ege lels. Therefore n SV ), Ths llows us to grse ths lger. V V V V V V V V V 12

13 Dgrston of the syetr lger SV ). The syetr lger s generte y unlelle T V )- grs wth no nhors. The grt tensor lulus tht Roger Penrose uses n [6],[7],[8]) s n exellent teplte for suh grston, s hs syste es no use of nhors, ut fully opts the other tensor opertons presente n the prevous seton. In hs tensor grs, rete eges re stngushe y where they re lote on the gr, n those tht on two stngushe vertes hve no efnng fetures. 3.2 Le lgers The grston of Le lgers proves soe otvton for Vsslev nvrnts n not theory see [4]), s we shll prtly sover. A Le lger g, [, ]) s n lger g ssote wth lner p [, ]. We wll enote ths pr, n soetes the vetor spe, sply y g. Ths wll hopefully not use ny onfuson s t wll e e ler wht oet we re referrng to t eh nstne. Defnton Le ret. The lner p [, ] Hog g, g) of Le lger g, tere the Le ret, s efne y [, ] : g g g x y [x, y] stsfyng the followng reltons, for ll x, y, z g. [x, y] [y, x] [x, [y, z]] + [z, [x, y]] + [y, [z, x]] 0 R1) R2) The relton R1) s nown s the nt-syetr relton, n R2) s the Jo relton. In ths seton we wll suss the n-ensonl vetor spe V s g wth ss {v 1,..., v n } n ts ul spe V s the ul Le lger g wth ul ss {v 1,..., v n }. An eleent n the vetor spe g y e represente y α v s tensor. Then we ssote the strt tensor α wth α v, n grse ths eleent y α 3.2) Anlogously, ny eleent n the vetor spe g y e represente n the for α v for soe slrs α 1,..., α n. Then we ssote the tensor α wth α v, n grse ths eleent y α 3.3) As [, ] Hog g, g), for soe slrs β we hve [v, v ] β v 3.4) [v, [v, v ]] [v, l β l v l ] l β l [v, v l ] l, β l κ v 3.5) 13

14 It sees tht the onstnts β shoul soehow epen on v n v, otherwse they woul e the se no tter wht tensors we put n, ounter-ntutve result. Rell the vetor spe soorphs We wsh to pply t to 3.4) n 3.5), so onserng ther ent spes, we see tht T g g,g [v, v ] ) g g) g g g g. Applyng T g g,g to 3.4), T g g,g [v, v ]),, γ v v v Now the slr shows the ontruton of the nees n. Slrly we hve T g g g,g [v, v, v ] ) g g g) g g g g g, n pple to 3.5), T g g g,g [v, [v, v ]]),,, γl γ l v v v v Now we now enough to grse the Le ret. Ths s one s [, ] γ 3.6) Fro now on we wll represent the Le ret s the strt tensor γ n wth rulr gr n ts tensor grston. We n now restte the reltons R1) n R2) n ters of strt tensors n grs, rther thn ps. γl γ l + γ l γ + γ 0 + γ 0 γ l l γ l restteent of R1) restteent of R2) The grston of these tensors follows etely. The nt-syetr relton R1) eoes ) An the Jo relton R2) eoes ) So fr we hve only nspete how the Le ret opertes on prtve tensors, so now we generlze for rtrry tensors ρ, κ to onser [ρ, κ]. But how o we get to n rtrry tensor? Let us onser the splest of eleents n ths lger. These re presente n 3.6) n 3.2). Thus we onser only 14

15 tensor grs retly rsng fro these strt tensors. Whle t s possle to proue other very sple tensors through the prout operton n wth nforton loss), we woul then no longer e le to grse the wth the genertor tensor. In orer of nresng oplexty, they woul e,,,,,, et. 3.9) These grs orrespon to the strt tensors v, [v, v ], [v, [v, v ]], [[v, v ], v ], [[v, v ], [v, v ]], [[v, [v, v ]], v ] So, gven n opes of g,.e. g n, there s ertn nuer of wys tht the Le ret y e opose wth tself to yel prtulr tensor fro ths spe. In generl, gven two spef nestng orers of Le rets ρ [ ] : g p g n κ [ ] : g q g, we wll hve tht [ρ, κ] [[ ], [ ]] : g p+q) g For exple, suh onton oul e gven y: l l [ρ, κ] 3.10) h h The pttern hs now eoe ler. It s portnt to note tht n the grston of the Le lger g y the genertor 3.2) n the Le ret 3.6), every prtve tensor hs only sngle lelle ege rete wy fro the tensor. Wth tht n n, we y now hrterze ths grston. Dgrston of the Le lger g. The eleents of g re generte y Every gr hs extly one lelle ege rete wy fro stngushe vertex. The syste s suet to 15

16 the nt-syetr relton R1) n the Jo relton R2) The Le ret s useful for reung vlene of tensor gr, s t s the only tensor gr wth eges of oth types. For exple, onser the strt tensor α β e To ssote t grtlly wth Le ret, we pply the ontrton operton. )) κ 4,7 κ 1,6 α β e γ α β e γ We strt wth the tensor gr 3.11) e Ientfyng lels, ontrtng the, n rerrngng the whole gr yels the followng result, showng tht the Le ret hs n effet turne n upper nex nto lower nex. e 3.12) e 3.3 The tensor lger Tg) Fro we hve tht every eleent n T ) V ) s lner onton of eleents n V. Any prtulr eleent of V s then gven y the strt tensor α 1...p α 1...p v 1 v p for ll p N. Ths s grse y 16

17 ) p The lels n orresponng eges 3,..., p 1 hve een otte n reple y otte lne. Beuse of the ent spe, ll eges re rete wy fro the stngushe vertex. Note tht for p 1, ths s equvlent to the Le lger genertor 3.2). By 3.1.1, the tensor lger s to e opne y prout, n n ths se we hve ultplton. In the generl settng, Multplton of two fferent tensors s ther sont unon n gr the ts of onnetng the s unerten y the ontrton operton). Sne we hve lrey nlyze Le lger g, y tng the tensor lger of g the struture s unhge, n only new funentl tensors re ntroue. In ontrst wth the Le lger g, t s not true tht there s extly one lele ege rete outwr n eh prtve tensor gr, ue to the ntrouton of these tensors. Dgrston of the tensor lger Tg). The eleents of Tg) re generte y 1 2 p N p The syste s suet to the nt-syetr relton R1) n the Jo relton R2) Eleents of ths syste re tere TV )-grs. 3.4 The syetr Le lger Sg) As note n 3.1), use of the syetr lger s the quotent n quotent spe reoves the lels n nhors ut preserves ege orentton. Therefore we e the ret nlogy to the spler syetr lger n the prevous seton, whle eepng es fro the Le lger. 17

18 Dgrston of the syetr lger Sg). The eleents of Sg) re generte y The syste s suet to the nt-syetr relton R1) n the Jo relton R2) The unversl envelopng lger Ug) Ths lger gves n nterpretton of the Le ret. Ths llows for splfton, whle stll retnng the s struture of g. Gven Le lger g wth ret opertor [, ] : g g g n prout : g g g g, the unversl envelopng lger of g s efne s Tg) Ug) [x, y] x y y x g /[x, y] 0 Our next ts s to fn proper wy to grse x y y x. Bse on our nlyss so fr, t s nturl to hoose the tensor prout s the prout. Rell tht n 2.1 outtve prout ws onstrute, suh tht for tensors ρ, κ we hve ρ κ κ ρ. Although then [ρ, κ] ρ κ κ ρ ρ κ ρ κ 0 Suh stteent woul rener useless the potentl lger struture. However, the Le ret opertes not on n rtrry pr of tensors, ut on pr of tensors n the vetor spe g 1. Dgrtlly speng, ths s equvlent to pr of lele eges rete wy fro ther respetve stngushe vertes. So for ρ, κ Tg). the expresson [ρ, κ] s etter nterprete s [x, y], where x s n outwr ege of the gr of ρ n y s n outwr ege of the gr of κ, n z s the outwr rete ege of the Le ret gr. In vew of ths, t s nturl to nterpret the ntroue efnton of [x, y] s [x, y] xy yx z 3.14) x y x y We hve solte seton of the gr [ρ, κ] on the left-hn se, oservng only the outwr eges x, y n how they onnet to γxy z. Everythng else outse ths rle stys the se n oth ters on the rght-hn se, whh s lner onton of two tensors. Essentlly, eoposton of the Le ret proues two opes of the orgnl gr, one wth nterhnge lels. 18

19 If nste of tensors, we onser ust the Le ret opertor n ts gr, we get the equlty 3.15) Ths llows for renterpretton of grs, s we y reue or nrese the nuer of trvlent vertes n gven grph. Now we y grse ths lger. Dgrston of the unversl envelopng lger Ug). The eleents of Ug) re generte y 1 2 p N p The syste s suet to the nt-syetr relton R1) n the Jo relton R2) s well s n nterpretton of the Le ret gr 3.15) The Le ret eoposton y e represente n nother slr nner, y orerng the lels long lne, or fxng the long rle. Lels re then reove fro the grs, s ther orer hs een fxe. 3.16) 19

20 For tensors hvng the Le ret s suonfgurton, the se eoposton pples. Representng the on suh rles s lso nlogous, ut re ust e ten to / reove noes n eges n n entl nner for ll grs n n equlty - no e eges y en n the shortest seton etween eges on the gr ove. If, for exple, we wsh to eopose the Jo relton, we y esre the ters n fxe rulr grs n expn y 3.16). For eh ter seprtely, ) ) ) Conng the three equtons results n oplete reuton to zero on the rght-hn se, showng the Jo relton s true stteent. 3.6 The etrze Le lger Ths lger hs rh struture, wth roo for ore extensons thn expne upon here Dgrston wth lner for Here we shll ntroue n strt etr through type of stne funton. The etrze Le lger g, [, ],, ) s Le lger g opne y non-egenerte lner for,. See [3] for ore extensve esrpton of lner fors n wht y e oplshe wth the. If g s over fel F, then forlly, : g g F v, v ) f We woul le the ge f of the nput tensors v n v to epen upon n. Conserng the lner p y tself n rellng the vetor spe soorphs we prevously h n 3.0.3, Thus we hve for soe slrs h, Hog g, F) g g) g g, h v v 3.20) 20

21 Fro ths we y grze the lner p. We ssgn to t unque tensor gr shpe., h 3.21) To proure n nlogous tensor wth nwr rther thn outwr rete eges, we note tht y the se vetor spe soorphs, we hve, g g Hog, g ), gvng the p, for the se slrs h s ove, : g g v h v 3.22) Sne we hve non-egenerte lner for,, t s nvertle. Therefore there exsts p, 1 Hog, g) g g As suh we y hrterze t, usng the letter h for ontnuty, y, 1 h v v Fro ths we y grze the nverte lner p., 1 h 3.23) We shll ll these the h-tensors. Sne, 1 s hooorphs fro g to g, we hve nother esrpton wth the se slrs h of ths p, 1 : g g v h v 3.24) These two ps re relte through the Kroneer elt syol, whh we enountere erler n 2.2. Conser v v ), 1, v ), 1 h v ) h, 1 v ) h h v Coprng oeffents, t s ete tht s slrs, h h δ. Ipleentng the struture fro 2.2 wth ths tensor, we now hve tht 21

22 3.25) Slrly perforng n nlogous seres of oves on v, we fn tht h h δ, n grse, 3.26) Whle 3.25) n 3.26) see slr, t shoul e note tht the pleent of the nhor ffers. To one the grstons, t shoul e tht the pleent of the sepont e rrelevnt. No suh splfton n e e yet. We proee wth the grston of ths lger. Dgrston of the etrze Le lger. The eleents of g, [, ],, ) re generte y 1 2 p N p The syste s suet to the nt-syetr relton R1) n the Jo relton R2) s well s two reltons on the h-tensors, 3.25) n 3.26) 22

23 3.6.2 Dgrston wth syetr lner for If the lner for s syetr,.e.,, for ny pproprte rguents,, then the nhor on h y e reove, s,, 3.27) Therefore h h n nlogously h h. Ths llows us to reove one of the h-tensor reltons fro the grston of ths lger. Dgrston of the etrze Le lger wth syetr lner for. The eleents of g, [, ],, ) re generte y 1 2 p N p The syste s suet to the nt-syetr relton R1) n the Jo relton R2) s well the h-tensor relton To vew the Le ret tensor lso wthout n nhor, t s neessry to ssue tht we hve n orthonorl ss, t lest y our pproh Dgrston wth syetr lner for n n orthonorl ss Conser n orthonorl ss {e 1,..., e n }. Ths wll llow us to eonstrte tht the grston of the Le ret s nvrnt uner oveent of the nhor - tht γ γ γ. Begn y splfyng 23

24 ,,l soe tensor expressons. γ h l γ,,,l,,l,,l,,,,l h l γ h l e e e e e l γ 1 h 1l e e e 1 e 1 e l + + γ n h nl e e e n e n e l) γ 1 h 1l e e e l + + γ n h nl e e e l) 3.28) γ 1 h 11 e e e γ n h nn e e e n) 3.29) γ 1 e e e γ n e e e n) γ l e e e l,,l γ l e e e l 3.30) γ l 3.31) In 3.28) we use the ft tht the lner for s syetr, n n 3.29) we expne over l to show tht ny ters reue to zero. For 3.30) we e the swth of upper nex l to lower nex l to oote the ft tht the tensor s n g ) 3 now. The equlty n 3.30) lso ens tht, s slrs, γ γ l for l, whh we use n 3.32) elow. We ontnue n the se nner s ove. γ l h γ l h h γ l h γ l h γ l e e e e e l,,,l,,l h 1 γ 1l e e 1 e 1 e e l + + h n γ nl e e n e n e e l) 3.32) h 1 γ 1l e e e l + + h n γ nl e e e l) h 11 γ 1l e 1 e e l + + h nn γ nl e n e e l) 3.33),l,l γ1l e 1 e e l + + γ nl e n e e l),,l γ l e e e l 3.34),,l γ l e e e l 3.35) γ l γ l 3.36) 24

25 Equton 3.32) uses the ft tht γ γ l s slrs for l, n the nt-syetr relton. The se hnge s e n 3.34) s ove, to oote for the hnge of spe, eepng n n tht slrs hve not hnge. Tht s, s slrs γ l γ l γ l for, whh llows us to reorer the nees n 3.36), ontnung fro the propertes of the relelle Le ret ove. In 3.33) the se hnge s e s n 3.29). Now we one the ove results, n pply the Kroneer elt, s tensor. γ γ γ γ δ δ δ δ δ δ l l δ l δ l γ γ γ l δ l γ δ l δ δ l δ l δ l h l h δ l γ l h δ l 3.37) γ l h δ l γ l δ l 3.38) In 3.37) we use the onluson fro 3.31), n n 3.38) we use 3.36). The rest oes fro npulton wth the Kroneer elt lel ressgnent) n the h-tensor ege reretonng). A slr rguent, gven n full n Appenx A, shows tht γ γ, leng to the onluson tht γ Ths y e grtlly expresse s γ γ 3.39) 3.40) Ths gves us ustfton for reovng the nhor opletely fro the gr of the Le ret, s grse further elow. Now we onser other enefts of hvng n orthonorl ss. It woul see nturl for the h-tensors swth the reton of eges, s they re relte to the Kroneer-elt tensor. Ths ntuton turns out to e orret. Frst note tht e, e e, e δ, where δ the Kroneer-elt syol. Fro 3.20) we hve the entfton, h e e 3.41) In the se the for hs n orthonorl ss, t follows tht h δ, where δ s now the Kroneerelt tensor. Tht s, renterpretng 3.22) n pplyng the prevous efnton, we hve, : g g e δ e e 3.42) 25

26 Ths yels p tht hnges reton of tensor oponent. Inee, gven n rtrry tensor n n rtrry nex fro t, onng t wth the pproprte h-tensor: α 1,...,r 1 f 1,...,f r 1,..., s1 g 1,...,g s h α 1,...,r 1 f 1,...,f r 1,..., s1 g 1,...,g s 3.43) Inste of provng ths n the generl se, we wll show the ehns n n exple. Frst rell tht fro 1.3.5, we lrey hve n soorphs T : U V V U for fnte-ensonl vetor spes U, V. Let U g n V g, gvng ps T : g g g g λ e e T λ) e e T 1 : g g g g λ e e T 1 λ) e e 3.44) By ng the orresponene n ss vetors etween g n the ul ss of g, the nverse p y lso e vewe s T 1 : g ) g ) g ) g ) λ e e T 1 λ) e e 3.45) So then T s n nvoluton,.e. T 1 T n so T 2. Now onser the tensor α h g, whh s equvlent to α. To prove ths we use the long notton of tensor. g α h g e e e e e e g T α h g e e e e e e g T α h g e e e e e e g α g e e e e g T α g e e e g e T αg e e g e e The one oposton of the ps pple ove reues to the entty p. Copose the frst two ps: T ) T ) T ) T ) Apply the next p to the enson-reue p: T ) T ) T )) T ) T ) T ) T )) Apply the lst p: T 2 ) T )) ) T )) T )) T ) T )) T ) T )) T 2 )) ) 4 Ths npulton nvolves, n n rtrry tensor, ovng the nex to e hnge postons forwr resultng n ppltons of T ), reung egree, then ovng postons wr ore ppltons of T ). In totl, T s pple 2 tes, n sne the orer n whh t s pple s reverse gong, every 26

27 pr of ps reues to the entty p, gvng the esre result. Therefore the h-tensors t to sply hnge the reton of n ege. Wth grs n the orresponng strt tensors, 3.46) β f h f β g h g β β h In generl, through the grt reltons 3.47) The otte lnes n the nte n rtrry ege n the esre reton of n rtrry tensor, eepng the se tensor n eh respetve equlty. We y now grse ths lger. Dgrston of the etrze Le lger n n orthonorl ss wth syetr lner for. The eleents of g, [, ],, ) re generte y 1 2 p N p The syste s suet to the nt-syetr relton R1) n the Jo relton R2) s well s the ege-reversng ton 3.47) of the h-tensors 27

28 3.6.4 Dgrston uner -nvrne Defnton Aont p. For x g, efne the ont p x y x : g g y [x, y] Defnton A-nvrne. The lner for, s tere -nvrnt f, for x, y, z g, the followng equlty hols: Or, equvlently, As suh, we y vew [, ], n, [, ] s ps, n the sense tht x y), z y, x z) 3.48) [x, y], z y, [x, z] 3.49) [, ], : g g g F e, e, e ) [e, e ], e γ l e l, e γ l e l, e γ l h l, [, ] : g g g F e, e, e ) e, [e, e ] e, γ e γ e, e γ γ We hoose not to reue ths to elt tensor, ut rther eep the h-tensors s they re. Ths nterpretton llows us to grze 3.49). In eepng wth the progresson of results, we ontnue to ot nhors for the grs of the Le n h-tensors. h h 3.50) Agn, t y see nturl to pply 3.47) to elnte the h-tensors. We hoose not to, s ths oul potentlly use onfuson s to how the Le ret gr opertes. Now we grse ths lger. 28

29 Dgrston of the etrze Le lger n n orthonorl ss wth n -nvrnt syetr lner for. The eleents of g, [, ],, ) re generte y 1 2 p N p The syste s suet to the nt-syetr relton R1) n the Jo relton R2) s well s the ege-reversng ton 3.47) of the h-tensors n the -nvrnt relton 3.50) In the next seton, y onng 3.50) wth 3.16), we re le to proue well nown relton. Ths s the 4-ter relton use n oputton of not nvrnts. 3.7 The unversl envelopng lger Ug) for etrze Le lger g The struture onng the unversl envelopng lger n etrze Le lger wth n orthonorl ss n n -nvrnt syetr lner for wll e enowe wth the Le ret eoposton n the -nvrne relton, whh wll, eploye together, le nely nto stuy of hor grs. In not theory, the 4-ter relton s ost often gven s n equlty of hor grs, ) 29

30 More hors y e e, to eh gr n n entl nner, ut none y en n the shortest gps etween hor ens on the ounry rle. Note tht there re no rete eges, s the lnes n these grs o not nte reton, sne reton s ounter-lowse long the rle. The lnes nte whh ponts re to e entfe together - they re the sngulrtes of the not gr. Now te the relton 3.50) n ssgn fxe postons to the free ens of eges, eeng eh ter n rulr gr. Then 3.16) yels ) As ove, ore tensor grs y e e entlly to the equton, ut none y en n the shortest gps etween ege ens, s tht oul hnge ny grs lyng outse the restrte rle. The hoe of orerng the eges strtng fro the top of the rle s rtrry, n the 4-ter relton s n the ove expnson. So unfor hnge of ntl pont of oservton wll only result n vsul hnge, wth no nfortonl hnge. So f we loo t the nner prt of the ove equton n rotte t 120 egrees ounter-lowse, we get ) An rerrnge, ) Ths now n e seen s the orgnl 4-ter relton n 3.51). The h-tensors n the grs wor to elnte noton of reton, whh s onsstent wth the ft tht hors n hor grs nte only sngulrtes, s entone. We y now grse ths lger. Dgrston of the unversl envelopng lger Ug) for etrze Le lger g. The eleents of Ug) re generte y 1 2 p N p 30

31 The syste s suet to the nt-syetr relton R1) n the Jo relton R2) s well s the ege-reversng ton 3.47) of the h-tensors n the -nvrnt relton 3.50) n n nterpretton of the Le ret gr 3.15) 4 Ensung onnetons Here g wll enote etrze Le lger wth n orthonorl ss n n -nvrnt syetr lner for. In the prevous seton we sw the lger Ug) wth n nhor-less Le tensor n n nterestng nterpretton of the Le ret. Conser the spe U Ug) of tensors generte y unvlent tensors, the h-tensors representng the etr, ), n the Le ret. Then ll grs n U hve vertes of egree 1 or 3, tere untrvlent grs. By the relton n 3.15), trvlent vertes y e elnte fro the gr ltogether. It woul see nturl to onser ths spe s spe of hor grs, whh wll le to the stuy of Vsslev nvrnts. Ths spe y e onsely grse. 31

32 Dgrston of the set U Ug). The eleents of U re generte y The syste s suet to the nt-syetr relton R1) n the Jo relton R2) s well s the ege-reversng ton 3.47) of the h-tensors n the -nvrnt relton 3.50) n n nterpretton of the Le ret gr 3.15) For exple, onser the followng tensor gr 32

33 We now nterpret t n rulr gr orng to 3.7, n e soe splftons to express suh gr wth hor grs. The elow nterpretton s not unque, s the unvlent vertes y e proete n ny esre orer on the rulr seleton. Bvlent vertes the h-tensors) hve sppere, s hor wth vlent vertex hs the se struture s the se hor wthout vlent vertex. Drete eges hve lost reton, s hors n hor grs hve no preeterne reton. By pplyng 3.16, the gr on the rght y e eopose further. A repete pplton ontnues eoposton. + A fnl pplton gves oplete expresson of the tensor gr n ters of hor grs The -nvrnt relton n rulr grs fro 3.51 gves the 4-ter relton of hor grs, gn hghlghtng the slrtes. Therefore ler ssoton y e e etween eleents of U n hor grs, n ssoton tht wll not e pursue further here. 33

34 ,,l A Appenx In we showe tht γ γ h l γ,,,l,,l,,l,,,,l h l γ γ, n here we wll show tht γ γ. We proee ntlly s efore. h l e e e e e l γ 1 h 1l e e e 1 e 1 e l + + γ n h nl e e e n e n e l) γ 1 h 1l e e e l + + γ n h nl e e e l) A.1) γ 1 h 11 e e e γ n h nn e e e n) A.2) γ 1 e e e γ n e e e n) γ l e e e l,,l γ l e e e l A.3) γ l A.4) As prevously, A.1) follows s the lner for s syetr. In A.2) we expn over l to reue zero ters, n A.3) es the swth of upper to lower nex l to oote the hnge of spe. As slrs, γ l for l fro A.3). γ γ l h h γ l h γ l e e e e e,,,l,,l h 1 γ 1l e e 1 e 1 e e l + + h n γ nl e e n e n e e l) A.5) h 1 γ 1l e e e l + + h n γ nl e e e l) h 11 γ 1l e 1 e e l + + h nn γ nl e n e e l),l γ1l e 1 e e l + + γ nl e n e e l),l γ l e e e l,,l γ l e e e l,,l γ l A.6) In A.5) we use the propertes of the h-tensor, n the rest s s efore. Now one the ove results, 34

35 n pply the Kroneer elt tensor. γ γ γ γ γ δ l l δ l δ l δ l δ δ l h l h δ γ l h δ γ l h δ γ l δ γ l δ l γ δ l δ l δ l δ l A.7) A.8) In A.7) we use A.4), n n A.8) we use A.6). The rest oes fro npulton wth the Kroneer elt n the h-tensor. Ths opletes the rguent to prove tht γ γ γ A.9) 35

36 Referenes [1] Ry M. Bowen n C.C. Wng. Introuton to vetors n tensors. Multlner lger Volue 1). Plenu Press, [2] Prerg Cvtnov. Group Theory: rtrs, Le s, n exeptonl groups. Prneton Unversty Press, [3] John Mlnor n Dle Huseoller. Syetr Blner fors. Sprnger-Verlg, [4] Toot Ohtsu. Quntu Invrnts: A stuy of nots, 3-nfols, n ther sets. Worl Sentf, [5] Roger Penrose. Tensor ethos n lger geoetry. PhD thess, Unversty of Crge, [6] Roger Penrose. Struture of spe-te. In Btelle Reontres, pges W.A. Benn, In., [7] Roger Penrose. Appltons of negtve ensonl tensors. In Appltons of Contorl Mthets, pges Ae Press, [8] Roger Penrose. The Ro to Relty: A oplete gue to the lws of the unverse. Jonthn Cpe, Deprtent of Contors n Optzton, Fulty of Mthets, Unversty of Wterloo, Wterloo, Ontro, Cn E-l ress: ns.lzovss@uwterloo. 36