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I : The Root System of En . A Very I 't Brief - Moody Algebras - PDF document

Nov 24, 2022 •712 likes •1.03k views

- theory Kac - Moody Root M Systems and . Part O : Some comments motivating . - theory ? What M 0.1 is . Kac - Moody of Interest . theorists 0.2 . Why Algebras to M are . Part I : The Root System of En . A Very I 't

  1. - theory Kac - Moody Root M Systems and . Part O : Some comments motivating . - theory ? What M 0.1 is . Kac - Moody of Interest . theorists 0.2 . Why Algebras to M are . Part I : The Root System of En . A Very I 't Brief - Moody Algebras Introduction Kac to . 1.2 Decomposition of Dynkin Diagrams their Lie and Algebras . . ( Almost ) 1.3 A Construction of the root of En system . Part I . theory : M Solutions and En . 2.1 invariant . Constructing Lagrangian under coset symmetries . 5413¥ The 2.2 model i ) . sigma . 2.3 time Space - . . Parameter - theory Branes 1 2.4 Solutions of M . . : 2 Parameter Solutions unconventional 2.5 & spacetime signature . .

  2. Some comments motivating . - theory ? What M is - duality unified String theories 10 D U in by are . LIB IA - theory M in IID I . w Egxt theories eterotic 32 ) . . ' 95 ] [ Witten . The . theory low of the maximal 1113 boson description M is supergravity theory whose :c : energy in part is , 1=13*1 - HE ,n*E , the ,nF¢nA , where F+=4dA . . . KE , term . for E .H . . Self interaction - - Form a 3 term Field A . gauge } , ' ' term ? Chern Simons . Comment " it the of freedom elf field describes degrees of - bein and a 3- Form An an en , , gauge . , µµ ] which with Maxwell 's field will lead to Comparison electromagnetic An sources a point charge you brane ) ( that the An membrane MZ Surmise correctly sources - a , , µµ , - tzgwr Mti ' + 'ziIgnvFn µµµ F" MHM - kzt , Fun , µµ Fu" its of motion R~ =D 2 equations are . > FNSMAMOM " + { Emvrztynx fm+ µ sM6M " =D 2 Fun , µµ 3 , . M " .

  3. It has the the the KKG brane 3 solutions the MZ - brane MS - brane and simple : - wave - . pp . , , - theory 4 M with Su Gra at but must also include all the excitations must agree low string . in energies the Iod theories string . - theory ? Which . Moody algebras to Kac relevant M are The two arguments : pioneering Dimensional Reduction ( See the Klein 1. lectures Kaluza theory by Chris Pope ) - on . St The make of the spatial dimensions and let the shrink to small idea : compact , typically cycle rough one , ( For Excitations of the circle ) nX=2TR R volume example where is are standing waves satisfying . the radius of the circle X the of the NEZ Such have and wavelength wave waves energy . , E=Ku=hT#=X¥r R o = > E � - hence - if into . colliders ) So - energy ( too to have been Hence small radii excitations reached imply very high high in our one may . the the small compact coordinate In the of in the low effect neglect impact energy theory practise . on content the Field the to the consider the metric of theory is neglect index being reduced compact e.g. xs ) ( call it from SD to 4D : in one - dimension gms ± Am - and gss =D gnv gmn . , From of of vector ( electromagnetism ) scalar . This in SD a and a theory gravity a theory gravity a emerges ,

  4. ' 26 ' the observation Kaluza 21 of of and Klein in made to SD unification was original in a propose The scalar unwanted extra but it the scalars and electromagnetism is in gravity was an appearing . , the reduction Sugra the - Moody - theory dimensional of that first motivation For Kac M algebra a in . give the Sugra scalars that D= 10 have the the Lagrangian As HD the in , 9,8 one reduces appear symmetries in ... G- coset K( G) of reduction descends to of the fewer dimensions : a greater complexity as Gy KCG ) For D. diagram G Dynk :n . . 1 ÷ : . iii SLCZR ) ( × R µ 9 So (2) 0 . 0 ,R) × SL( 2,42 ) 543 Rn %( 8 31 × 5012 ) Esisoab o_O . SLCS ,R µ o Soto ) b- 7 . o_O . . o 5° € %( 6 0-150-0 sjxsocs ) . . . o ⇐ %p( 8) s taboo . . . E > o TUC 8) 4 -0-10-0-0 . . o_O . -0-0-150-0 ) . . o_O he For The observation of Julia that these hidden to continue ought and argued early was symmetries ' ' ' E Ea 85 ) and in D= 2 and D= 1 [ Julia 78 80 , . . , , The extension would be that Eu should appear Peter West that in D= 0 En of was a symmetry . argued an of Sugra should be of in IID extension 2001 - B Eu M - in N a symmetry theory . . , - Moody , E E all Kac Ea and algebras , , are , . .

  5. Billiards 2 . Cosmological . At the start of the millenium different of led Damour , Henneaux and Nicolai line to see the a investigation . They of E within HD Sugra the the of Fingerprints were physics in vicinity cosmological considering a ,o allowed the Sugra fields time The where they to only t equations singularity depend on greatly simplified . , E " € (E - geodesic motion had Solution which identical to hull motion of coset of E , ) of : a was a , , . . ¥ HE 'D - _ t ) Spacetime solution An ,m µ (t ) y gn ✓ ( , ¥ cosmological singularity to near a . = ⇒ x Later and Houart extended this to restore and time to Englert picture equal Footing space an ⇐ ¥Ed their construction called the model and the asset and brome 6 to was - symmetry was enlarged . This For . permitting ) the will work this talk and two Fold ( time : is in our aims setting we are - To root of E 1. construct the system , , . To the model to build Solutions M 2. brane of - theory use 6- .

  7. the if So if know construct the roots likewise know all all and we algebra we can we , roots construct the the algebra we can . It work the root to with is Frequently simpler system : . Root vectors while the must add algebra commute matrices : in one , ,E&z ] [ Ed that Ea = in then some Suppose algebra } - [ E. , ,[ , ,Ea ) ] [ Hi , ]=[ Hi ,[ Ex , ] . ,H ;D [ .CH , ( identity ) = - by the Jacobi . , Litdz , >[ Ea ,Ea > Ex Eg En Ea ;D ,t] Ea , ]+< , >[ =< xi xi ,x ,x , = < 2 ; . , > = 2 =L , tdz , . For - Moody Kac Matrix ( without a great inspiration be of - the will ) infinite representations algebras Dynkin diagram ) - dimensional while the roots will ( for finite finite vector rank reside a in space a . , Before the root , R ) the For For SLC 3 let introduce relations completing system us defining a - Moody ( xi )2=2 Kac the roots all have where algebra simple :

  8. A Very - Moody Algebras Brief Introduction to Kac . - Moody Given Cartan A ) matrix Kac Formed of ( an appropriate algebra is a ; , ¥ F H Ei and such that , 's i generators ; ; , H ;] ,x ; > Fj [ Ei [ [ Hi [ Hi . ]= F ; ]=8ijH = Hi 0 , E ; xi ,& ; >Ej < - xi , -5 , , , , , , Cheaney the Serve relations and : , E ;] ... ) ]=o [ Ei ,[Ei , . :[ Ei ]=< , } Ai ; ) there where A- commuters are . ... ]]=o [ Fi ,[ Fi . .a÷ , F , ] , , Comments . 1 If det ( A ;j ) the Lie det ( Aij )fO > 0 above relations define finite if then algebra a . , the Kac . Moody algebra is algebra a . The 2 Serre relations that the irreducible adjoint is guarantee representation . . The Serve relations worth exploring detail will to construct in as route are they a simple give - Moody the root of Kac system algebra a .

  9. The Serre Relations and Root Systems . all Recalling have that have limited focus to root where roots systems simple we our [ Such 2) called laced ] the - Squared ( normalised to length same simply algebras are - . Cartan three the matrix distinct entries in : are - Aij Serre Relation Aij I . . There [ E ; ,E ;] -1 Ci=j ) 2 =D . . . . [ E ]=o I , Ej 0 . ; . . > [ E [ Ei ,[ Ei ,E ;] ]=0 ; ,E ; ]=Ei+j -1 2 = . . . from Starting the roots the Serre simple 2 relations tell that root us is ; ditxj a ' - - + ( xj ) >= In that ( txj ) ( xi ) +2C Lj > if 4 - 1 this observe Ci = xi Xi case dj . we , , = 2 - 2 +2 = 2 . Consider third root to obtain This if the commentator 2 ; + Lk is root now adding a simple Ljt a . [ tie ,[ Ei ,E ;] ] = Ei+j+k not trivial is . have By Jacobi the identity : we - [ Ei ,[ E ; ,En ] ] , C Ei ,E ;] ]= [ En ;] ] [ E ; ,[ En , 't - The - 1 - hand trivial if < > = -1 both true side 2 > = ( Ln Li right is non . ,Ln or or even are . ; . ,

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