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A v e he 1 ' ernelUX TtheX(C algebra GRADED ; ) that The f ) ;U1( ( monomials Cv ( er free The 1 X ; and ( ; U(' U( eri cationxALGEBRAS ( Lie further )'1 ) ; as ; Jacobi fol a s ;Uo 0 n d )K'( i 1 maximal ) i =U=Uf )= )= the't LIE ense ( X o i s a owing ; ) f he ( Xl (see linear that )w )( lgebra ) denoted :> <> 8 U ;''1 s is ubsection =XXXp e 'pro lemen pace) ) ; '1 ; 0 s )= i h [V den teps i 1 +U;'only 0 ( art s ideal 1 2 n 0 ; c . U U=U;'( ]).C;'hi duct formulas A AND ) A )U;:( n 2 whic . Let y tit ; ('( 1 Uo ts 1 ) are ( f 2 b Axsc X ( ;'; y U;''UU; A ) ; 1 ( M ; a i U) whic ='n ( of n h ) u X 2 A t 0 n 2 i g l ; A s X . he ; a ; l Z , =UXU'1 s ( e i s 1 s ; t A ni0 c Tg TX'dC asso )'1 ) rue f h same ( ;U(U0 +1 1 hec ive ) ) n Th g = ollo ( ; U(' ) as iTxa ciativ D enerated ; 1 A : )Ut s ; 1 as r YNAMICAL ; +1 a ( the t ;K'=K' kXw ). 1 ) he n a 0 if 1n>=0 n>c ( tn( zero +1.'o e ; U)U Uhat'The 0 it ; n s gradedx1 i anonic S ; n ; ), 0) 0) p =( y mak a ( 1 1 in K (U(U)=U ([ 'fTr algebra b e act ; tersectionC( ; ; ac{Mo R 1 ( t 1 ( a 2 es t A ) bX i o YSTEMS ; he s =U'[K'])'dalCLie:: ) r 0 ' t sense) ( isomorphism ' hat X loC = U; ; o algebra n ) X ). 2 cal d X 0'U =0 ; y this ( with + a in )a = U[ 0 X 1 t K'; s X(U algebra algebra heory ) i d 0 the A ('1b with 0 C U]. )) .K artan ] [ lo It K]. ) the The ) and is b cal e- 5 is Uis a ; 1 (0,+) ( (+, The (+,+) part). forU emark. ( If Pr de nition kind the example, 2. but R 6 ; , ;XNon<o Lie ; nThis is this of. extension ) ) of ww = an algebra n Assume 0 If 0t the ( X consists (Xo o s 'en>completion -image is unitary cycle 1. 0 1. enough ) o h ( 'X Xc e Xan f c W ) ; n ; A et ( f e then = Xn'mn mrewriteX( of ( with T( n the A X of op n c ) ;' X( ( ; alculated ) ) e = b n functions = = =( ( e ) X=>+1Xerator).''of image ( ) of A :>= <> 8 ; X cause theXde ned T ) (X;'; e = 0. ';UmXXXm'the X( XAX ) ; ( +1 n X m ;X; nU 'initial ( 0 n ; ( ) ; ( ; n '; n n ; (1 ; n = U( '( X T1 n m m s ) m ; ; ; ;'=Ubrac Xa'not XA. ). ; ; X( ) Tof i it the ) A ; m e generates ( n ; )) )( ( Um = ; '1 UUU' '1 (U k 3. ; X form ; b ; =U'all n +' U; ; ; ( ( Tbrac ; M. nU; 1 o ( ; ets )) 'k ( n Using ; n ' Um'nU U'( 1 U1'mU v ; e )n0 ; VERSHIK ) A ) n ; ula )'m ( )(1 Umni R is n for R= n ; et m 1 and s X ( ; m all oUisomorphisms dense X only = ; X' ;cX U;U ; ( ; '1U;' ; all = m n w ; 0'U;UU m =Unur Talgebra. R n'; ) d )= e ) 0 Um U; 1 ; n 0 U U;XmX;X main ( n but for )= n 1 onomialsm+: n an (''Xc i ; 1 mU) m UobCdense 'n ;U+ m ( )= e ' n ( )Um' 1U; (Unnif ; ') ';nn ; consider 0 n lemen )U n +c m ( ; ( ; Uject. 1 ) 1UX n ) Un mU d nmw o ts ; ; U) nc; p =0 n m (notd ), n nly ='; ; )'e A art of 6 =0n>can X , o m 0nn>m>m for' X ) = X) ( <n = only:but , f lo n = (U U; TA cal K;<mm nnrewrite T), ( A 0) X R n R ( ; 'n a X 'X Ta 0 so lgebra, m s ) ' some l o ) )i cd d the for al s . algebra de ned subalgebra) Lie 2.1. General 2. (see the to W where (3). role 4) 3) 2) wher Tas e 1) A Theorem w (0, ( ell has can Supp W Lie 0 classical tral XXXeXXX; algebras 2 form nothing of 0 n 0 n ( ) This subsection f and e General m m algebra . ; ( ( cen ) ) no '(') (0,+)X; [ ) Kose '('the'K ulas arX= ) ) n manner X X Xsign Xw = = ] e H ones ; i and de nition. s, Lie with n t n 0 Lie ( o is extension or to ( n The ; A ( heU IGRADED ( ( is as bserv ; 1 HKa ) fol .1) lgebra d (( X ) nU general (Cartan ) ) = a \ a c giv con description. UUUn'the'n'; (0, ; algebras ; ; do + omm ) =0 formulas T= lowing: 1 : linear = e ; 1 H es with ) H ; tin n :>>>>>>>>X;''+ LIE <>>>>>>>> ( ( 8 "i =U; t . ). f hat us utativ m 1 m ; i o AXXXXn nUUUALGEBRAS 0-uous monomials ( andUU s e n 0 n the m ; simple ciated space, ot ; n n is asso ;K>fU H ; ( s ; + v +m+ the ( W examples ontinuous X ( 1 1 a or ro w Tery n e e ; 1 f ot ('1 ; orm U linear ciativ XU\' +1 )( asso system. Un'; ; form r ith algebras (+,+) ) '0 ; imp ecall ortan ; a asso b ulas UI ; m m ;U 1 ulas ''n UUn direct erator V2, omplicated c =( AND ; are ) r 0 c ( A ' ( op n ackets ; e') n [SV1, n . iate n crosspro R ( H of ; for ; 1 i e m t. sum S + + D "i 'H Lie a l A Twithn m mU)(1 ore ; o ; n f UoU UYNAMICAL nd ( c )) ; m X f 'X ;'; ( C Tc Kac{Mo al in duct n Cartan he - part asso<,'n ;' U'm a + ; algebra .'US nd a onomialsU) ; ) iV r U'mU1I new ; ] oot ) ; ; =0 ) t od ) 0 ( ciativ ) ; ( 1 f 1{2) YSTEMS s n , op )= terms, Xn ; n o ubsection 6 + + ( de nition + =0 m m> s n systems with d ) erator). e 1 y ; whic compare in a ; a t re algebras). o han h n m< lgebra : < 8 : < 8 : < 8 the unit n the n n n m< a K d m< sub' y and similar 7 0 0 T o re =0 0 1.1. ynamical f (Cartan he ame, ) c 0 algebr as g raded cycle lo The a n with B ew cal ut Haar consider of where New ondition Xmcan onding )b rational w ubalgebra Sine-algebra of sp sum op giv ( [SV2 not Lie example c 2.2. a is R appropriate w But A Then same the The with 8 emark. eh sequence Let d erator F e The The H The =G= Let unit v en ,F s Kw l a it T b is ( o brac (additiv additiv ( in o cal yZc and e ( d e could o X s ,i Z realized .I b H p ase 3 p n v L 2Ug of y o T, spirit k ectrum smG eigen eneral G algebras examples of algebras er to ets: algebra um is btain ie . T= sense) be orresp X it o c ^ e) r ) KL Ke is U)Eb e nition as e r b T -in measure A , e no i and translation on t e of example t v o ofX cause (see H n has n constan [V]), i alues o HXX' v f teresting o utativ s e 2Ufa V]). puts the arian sp A TSubsection ; comm i w is without'i g 1 ( of Sonds s ( e ectra roup HX) ell-kno = p It ) an rgo w from 1 c +1 0 2H i f e t essen of ompact X t Kin yp X = ctral on dic a ( to '0 w 0 s H X X) A. ( = R fs as '( Zof onUf as e lgebras ubalgebras ). v t (sa from -adic ) ) arian wn In tial , 1.2 o p system ( simple pZ=Bthe decomp imilar de ned done A X r Section . y some = +1 p more ( =0 VERSHIK X p e ; Zc a ( (v olynomial 2 particular, i restriction H M. t 1 1 0 our I ( is b Tunder algebra . ase t o generates ( ( on elian o 1. T where (and b )w the i n gelemen Neumann ), indepKHX 1 ro exactly ) ) ) 1 yp efore n with yp f p osition 0 Tx[GKL] ) e. as X tegers, i ots oin group n t = = =0 f as H 2 and = = t discrete e algebra ( T but t in rom '1' X 0 o subalgebra rational gro the xT:1g fnden f so e then o graded the ( f = f (if 1 A p+1 tly theory w 0 called of v with wth XEthe ( iew. considered andG=fs TA 2Gtheorem) is is set ) as di eren X 1 itK' a exists) 2H ( the op o Lie in H o prime, Cartan i T, The tp f X pEerator e ) f t sine-algebra. K2 . ctrum. hen f lgebra t s [SV2] is The s imple o he eigenfunction a ) with hat o )= of m H translation t :::. dimension o and i It i 1 erator. easure f LKM-algebras). i and hen n [Kop p r , T Haar AEs ( t rst functions) also sac view ositiv [ . o =1 A o Ts t uc meansE Ko i [FFZ ( n H is onE= e h i ystem A K haracter measure ( the trivial W system ill = adding n (in corre- on r e i ]. mc KH ). that ( w o W ase o ,i see a ts. ir- s in of ). ) is e s Ldep the space x2.3. Go A n^ algebras is description of direct form of alues Pr Cartan wher (5) GTheorem eigenfunction Uwith group existence discrete giv v the n + T is The This Supp W The is lgebr 1 of. ending A a = degree e ula a e U. The as Assume CI set s Y2Udense ose s of in g n g T ummand,T1 nUgon n ubalgebra )h ) ( is Z sub roup (UfGthep sp n ,( Zthis (1) ectrum. roup -in p case 0 1 ( , GRADED a = the e T of A Yg the unction ( on arro alid )= N algebr ). )1 f 3 v the a X is in with set n arian ws ( . c G Line o )( m is description re of TGv thirdY 1 Consider the ase ultiplicativ ALGEBRAS Section f of 2 c is natural annthe f n ab t, p eigenfunctions p so Apa haracters ofGtheG= 0A =0 =0 = with and ar o description ) n is for The ( 6 o elian soZ-adic v in w A , e s eh = Zfw ts nite G^ = ) p ; LIE . L Tn 3 fol b asis 1 ( !C lowinggers: , get discrete n o line the sp of ( ) eigen (additiv ofU(G in + Cl e) eci c !Ya ~ 1 = )i e in n v imaginary is g the e T iv ar o s simple an tegers. orm L n dimensional = y 1 f -gr v Zg Yu . is r p ), b c ^ the t + as ( ffZ b c ; alue quotien Return = m o haracters he s asisQ haracters n ade ( It 0 1 s 2 ; the compact rop general Q 2 p f ackets L ne) p of = o Z e 1 f 2 p op L1Gro ctrum ur ; d ie elemen X 2 in e AND n n c ) ( p ro enter ). ots n algebra algebr . c easy 2Gg algebr To 2 t -graded A 1 . = = 1 6 )CZ A rt erator 1 ula is ots.n(5) to to a n 1 1 UpZI f Y( ( subspaces Tk , on or o D m o nly 1 ! T o aGf aseUa YNAMICAL pp Eac b f n . g . ( lgebra ZG Ac:. f AGgroup ). a ac ( hec TUp the algebra. c osite A + = . AG .WQroup w =k ( N n ( pTXG sub n 1 hc 1 hen ote (as S n t + T ) Lo wTharacter hen e t Talgebr ) i . of and a A t t o ) that n here n t ) n o he nd x w i K hat . is X f + s Cartan Tthe ( p all YSTEMS ith ac{Mo c1 ts c \Dynkin" G Te unction rator the Ccase 1 lear the is f = elemen is ) o the set a ~gr ^ ( o inear Zl G p i that f brac n the ) subalgebra n cen o 1: = a o 2Tgt nfUon of dy c ter cYG= t his = Zcase diagram basis f c k subspace unctions s 9 haracter 2c n ounUZunit p ets ^ , gc T p i C ase Txour this s : 2 g is + table has with yo (see an ot in = is 2 ,Gi f arbitrary , ' [FFZ] dy [ZM] [D] [VSh] . [GKL] [K] [V] [SV2] [SV1] a ne con R Mo A Theorem 10 emark. p (1) 1 It tains Lie n o More U is Conse n algebras J. nite D d'automorphis . A. 1 Lie G. ao Sa (1989), br A. its M. V. 4 A. Lett. M. 126 M. . v where eliev, o con airlie, e no. ershik. Dixmier. the F V Kac. V ery r Golenishc Sa xactly 143 It epr ershik, dimensional ns fp Zeller{Mei -algebr tin eliev, A v 4 algebr 6 v quently, is esentatio . olynomials lgebr group (1993). i uous B In nite ' nstructiv , p The and i P nductiv are Fletc C Lie . ssible A. 367{378. as hev (1990), e functions as o mes. ershik. Continuum the S V our of er. cylindric hoikhet. de ne A. a a{Ku theory tudy ) dimensional V A ershik. T algebr nezo i p her, as e ( n with t J. 121{128. Za to e limit Z Pr one rm z A lgebr p Math. Tis L C. gener as. variable. er o ( whic a vertex , duits and v utations Gr functions Th. e s New ) con Zac its p Ph ade h a lo p References Lie tains Leb y analo ys. hos. epr o means lim ate ures a d D. op (c M. examples oking W n cr d ompletion) r L Lett. eyl ois is ie T lgebr on b ator. as. gues only the et esentation whose for VERSHIK rigonometric s algebr f es c t o appl. o d g anonic dev. a link B n ynamic i roup ind nd of m t linear F 218 c L d'une e W:of al ak Z co i unc. gr this ordinates as s Cam n Math. ly 47 e b e - ontinuum nough a no. of al example. s. ad c com P systems. Anal., trigonometric ie t (1968), b. completion. w C Ph ontr structur adien lgebra. 2 e een the Univ. aris, 27 g theory binations d isomorphic Cartan no. (1989), St. ys. t o agr this inductive 101{239. algebr 1969. 123 e r Press, xtend P ade in ec sub 203{206. Kac{Mo no. algebr onstants d Za of e 1 t L p (1993), etersburg 1991. monomials L the limit 2 . ie of to p ar sub the (2000), algebr algebr s as algebr et for un Group 12{24. is of n 345{352. . of as. as. Math. the algebr ew gr CMP ' o as Kac{ a f Ph a on t oup o in - e and yp lge- dy W ys. to e J

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