#### Read 20pt Deformation of Kählerian Lie groups - (to.joint with P. Bieliavsky, P. Bonneau, V. Gayral, Y. Maeda, Y. Voglaire)to. text version

`Deformation of Kählerian Lie groupsjoint with P. Bieliavsky, P. Bonneau, V. Gayral, Y. Maeda, Y. VoglaireFrancesco D'AndreaInternational School for Advanced Studies (SISSA) Via Beirut 2-4, Trieste, Italy09/09/2010Workshop on Quantum Groups and Physics, Caen, 6­10 September 2010.1 / 22Outline1Introduction Hopf algebras and compact quantum groups Drinfeld twist/cocycle quantization The operator algebra approach2From formal to non-formal deformations: Moyal The Moyal plane Moyal-Weyl quantization3Deformation of Kählerian Lie groups Definition and decomposition Formal deformations Multiplicative unitaries2 / 22Hopf algebras &amp; compact quantum groupsMotivating example: if G = compact topological group, the group structure is encoded in the algebra A := R(G) of complex-valued representative functions. Dually to the group operations one can define co-operations:AAA (f)(g1 D g2 ) := f(g1 g2 ) D:AC (f) := f(IG ) DS:AA S(f)(g) := f(g-1 ) Dsatistying the axioms of a commutative Hopf algebra (HA):(  id)  = (id  )   D (  id)   = (id  )   = id D (S  id)   = (id  S)   = (ab)= (a)(b) DThen [Drinfeld-Jimbo, Majid,  `80s]:(coassociativity) (counity) (coinverse)D(ab) = (a) (b) FTannaka-Kren duality: G can be reconstructed from its irreps., i.e. from R(G). iquantum groups := non-commutative non-cocommutative Hopf algebras.3 / 22Quantization of Poisson-Lie groupsLet G := is both a compact group and a Poisson manifold and A := R(G). It is natural to look for formal deformations of A, i.e. associative products h on theC[[h]]-module A[[h]] of the form f1 h f2 = f1 · f2 +ih2{f1 D f2 } + O(h2 ) Dsuch that (A[[h]]D h D D F F F) is still an Hopf algebra. The necessary condition(f1 h f2 ) = (f1 )(h  h )(f2 )at the leading order in h gives (){aD b} = {a(1) D b(1) }  a(2) b(2) + a(1) b(1)  {a(2) D b(2) } Dfor all aD b  A , and with (u) = u(1)  u(2) the Sweedler notation. A Lie group with a Poisson structure satisfying () is called Poisson-Lie group. Most concrete examples of quantum groups are of this type: deformations(R(G)[[h]]D h ) of Poisson-Lie groups with undeformed coproduct.4 / 22Drinfeld twist/cocycle quantizationA twist based on a bialgebra U (e.g. U = U(g)[[h]]) is an invertible F  U  U s.t.(  id)(F) · (F  I) = (id  )(F) · (I  F) D (  id)(F) = (id  )(F) = I  I F  F -1 is a counital Hopf 2-cocycle (cf. e.g. [Majid, 1995]). With F we can define:a new bialgebra UF = (UD F ) by replacing  of U withF (X) := F-1 (X)F DXUFfor any left U-module algebra A with product m(a  b) = ab (e.g. R(G)[[h]]), a left UF -module algebra AF = (AD F ) with producta F b = m  F(a  b) D aD b  A FF)for any bialgebra O dual to U (e.g. R(G)[[h]]), a OF = (ODdual to UF byaHere andFb := m(F (a  b) F-1 ) Fare the left/right canonical actions. Rem.: the coproduct is undeformed!5 / 22The operator algebra approachGelfand-Namark thm: the top. space G can be reconstructed from Q := C0 (G) . i¢ C -algebra Q and a coassociative unital C -algebra morphism  : Q  Q  Q satisfying certain density properties [Woronowicz,  `80s].Theorem: commutative compact quantum groups = compact topological groups. Other possible definitions/generalizations to the non-compact case:A compact quantum group (CQG) is a pair (QD ) given by a complex unital¢ Hopf C -algebras [Vaes &amp; Van Daele, 2001]: similar to HAs but sm()  Q  Q. ¢ Multiplier Hopf algebras [Van Daele, 1994]: HA with no I, sm()  M(Q  Q). ¢ LCQG [Kusterman &amp; Vaes, 2000]: (QD D D ) with  : Q  M(Q  Q) and (resp. ) left (resp. right) invariant faithful KMS weight. Modelled on C0 (G). Bornological QGs [Voigt, 2005]: (A D D ), modelled on A := C (G), uses c bornological algebras. A pair (QD ) can be constructed from a multiplicative unitary!6 / 22Quantum groups via multiplicative unitaries[Baaj-Skandalis  `90s, later Woronowicz &amp; Soltan]All informations about a locally compact group G are encoded in the Kac-Takesaki operator, that is the op. W  B(H  H) -- H = L2 (GD dR µ) -- closure of(W)(gD g ) = (gg D g ) F W is unitary . . . and multiplicative:()W12 W13 W23 = W23 W12 FFrom W one can reconstruct C0 (G) (then the topological space G) asC0 (G) = (  id)(W)   B(H)and the group structure from the propertynorm cl.W  (I  T )W = T  TT = Rg for some g  G FIdea: define a LCQG via a multiplicative unitary, i.e. a unitary W on some H  H satisfying the Pentagon equation ().7 / 22Outline1Introduction Hopf algebras and compact quantum groups Drinfeld twist/cocycle quantization The operator algebra approach2From formal to non-formal deformations: Moyal The Moyal plane Moyal-Weyl quantization3Deformation of Kählerian Lie groups Definition and decomposition Formal deformations Multiplicative unitaries8 / 22The Moyal planeLet {P1 D P2 } := basis of R2 and A := U(R2 )-module algebra with multiplication m. An associative product on A[[h]] is given bya M b := m  eihP1 P2 (a  b) F hExample: if A = C (R2 ) and Pi = i we get a deformation quantization of R2 . Memorandum A deformation quantization of (MD {D }) is an ass. product h on C (M)[[h]] given by:a h b := ab +where Cn are bidifferential operators.ih2{aD b} +n 2hnCn (aD b) DFrom deformation quantizations to twists and back Left invariant h 's on G are in bijection with twisting elements (Hopf 2-cocycles) Fh based on U(g)[[h]] via the formulaa h b = m  Fh (a  b) F9 / 22Moyal-Weyl quantizationNon-formal deformations: if f1 D f2  S(R2 ) then f1 M f2 is convergent (to a Schwartz h function) for any h  R× , and can be rewritten as an oscillatory integral:(f1 W f2 )(x) := hI (h)2R2 ×R2e h {(x,y)+(y,z)+(z,x)} f1 (y)f2 (z)d2 y d2 z Diwith  := standard symplectic form. Remark:W can be extended to larger classes of functions: C - / Hilbert algebras, . . . h[Gracia-Bondía &amp; Várilly, 1988] Weyl: formulation of quantum mechanics on phase-space. Rieffel: strict deformation quantization for actions of Rd . Examples: SUq (2) [Sheu, 1991], G for any compact G with rank Connes-Landi spheres, . . . Using the integral kernel of W one can deform the Kac-Takesaki operator of any locally h compact group G with rank 2 [Vaes et al., Lecture Notes, 2001].10 / 222 [Wang, 1996],Outline1Introduction Hopf algebras and compact quantum groups Drinfeld twist/cocycle quantization The operator algebra approach2From formal to non-formal deformations: Moyal The Moyal plane Moyal-Weyl quantization3Deformation of Kählerian Lie groups Definition and decomposition Formal deformations Multiplicative unitaries11 / 22Normal j-algebras &amp; Kählerian Lie groupsj-algebras were introduced in the study bounded homogeneous domains in Cn .Definition [Piatetski-Shapiro]A normal j-algebra (bD D j) is given by i) a solvable Lie algebra b split over R (i.e. AdX has only real eigenvalues  X  b); ii) j  End(b) s.t. j2 = -I and j[XD Y] = [jXD Y] + [XD jY] + j[jXD jY] for all XD Y  b; iii) a linear form  : b  R s.t. ([jXD X]) &gt; H  X = H and ([jXD jY]) = ([XD Y]) . One can associate a (bD D j) to any bounded homogeneous domain. The Lie group B is a Kähler manifold with an invariant Kähler structure (j gives the complex structure and the Chevalley coboundary d gives the Kähler form). We call this a Kählerian Lie group. To each semisimple g = k  a  n of Hermitian type (i.e. rank k = I) one can attach a normal j-algebra, with b := a  n. These are called elementary.12 / 22Decomposition and formal extension lemmaElementary normal j-algebras are building blocks for normal j-algebras.Theorem [Piatetski-Shapiro]For any (bD D j)  a split exact sequence 0  b0  b  b1  0 where b0 is an elementary normal j-ideal.CorollaryAny b can be constructed by iteration as semidirect product of elementary normal j-algebras and possibly an abelian factor.Formal extension lemmaLet b = b0b1 and Fi twists based on U(bi )[[h]]. If F0  U(b0 )b1  U(b0 )b1 [[h]] , then F := F0  F1is a twist based on U(b)[[h]].13 / 22Elementary normal j-algebrasDefinitionFix n  N. Then i) bR × R2n × R has basis HD {Xi }i=1,...,2n D E and relations [Xi D Xj ] = (i+n,j - i-n,j )E D [HD E] = PE D [ED Xi ] = H D [HD Xi ] = Xi Fi vi Xiii) The mapbBDx := (aD vD t)  exp(aH)exp+ tEis a global Darboux diffeomorphism. iii) Within these coordinates the group law isx · x = a + a D e-a v + v D e-2a t + t + 1 e-a (vD v ) 2with (vD v ) =n i=1 (vi vi+n- vi+n vi ) standard symplectic structure on R2n .Remark 1: B is a 1D extension of the Heisenberg group Hn := {x = (aD vD t)  B : a = H}. Remark 2: there is extra structure on B, namely it is a symplectic symmetric space.14 / 22A star product on BWithin the theory of symplectic symmetric spaces [Bieliavsky-Cahen-Gutt, 1997] it is possible to define an associative product (on suitable test functions) in the form of an oscillatory integral. For h  R× and  : R  C satisfying some technical conditions, we define(f1 h, f2 )(x0 ) =B×BKh, (x1 D x2 )f1 (x0 x1 )f2 (x0 x2 )dx1 dx2 Dwhere dx is the left-invariant Haar,Kh, (x1 D x2 ) := P-2n (h)-2(n+1) Acan (x1 D x2 ) e h Scan (x1 ,x2 ) e(a1 )+(-a2 )-(a1 -a2 ) Dthe canonical amplitude and phase areiAcan (x1 D x2 ) := (cosh a1 cosh a2 cosh (a1 - a2 ))ncosh 2a1 cosh 2a2 cosh 2(a1 - a2 ) DScan (x1 D x2 ) := sinh (2a1 )t2 - sinh (2a2 )t1 + cosh a1 cosh a2 (v1 D v2 ) F15 / 22The formal twist-- n = 0,  = 0 --Define 0 (a) :=1 2log cosh 2a .PropositionFor n = 0 and  = 0 the formal twist is given byFh =n 0h ( i2 )n Fn DFn :=(-I)l Bl,m Bj,k Hk El  Hm Ej Dj,k,l,m 0 j+l=nwhereBj,k :=n1 +n2 +...+nj =k n1 +2n2 +...+jnj =j n1  n2 F F F  j j 1 2 D n1 ! n2 ! F F F nj !nand n are the Taylor coefficients of arcsinh. Remark 1: since Bj,k = 0 for all j &lt; k, Fn is a finite sum (i.e. Fn  UEA). Remark 2: a completely explicit formula can be given for arbitrary .16 / 22The formal twist-- n and  arbitrary --PropositionDenoting by Fh,n=0the twist in the n = 0 case, we have:Fh, = Fh,n n=0 i=0Gi hwhere the Gi 's are mutually commuting (so that their order in the product doesn't matter) h and are given bykGi := hk 0-ihch (E) k3(-1)|||| X1 F F F Xk  X-1 F F F X-k D{±}kwith the shorthand notation X- = Xi and X+ = Xn+i , with |||| the number of + signs in  and withch (E) := (1 - h2 E2  1)- 2 (1-1  h2 E2 )- 2 F1117 / 22Deforming the Kac-Takesaki operator of BThe Kac-Takesaki operator of B is given on 2-point functions byW = (id  m)(  id)where m()(g) = (gD g) extends the multiplication map and ()(gD g ) = (gg ) . The pentagon equation follows from (f1 f2 ) = (f1 )(f2 ). Idea: deform the product m into m such that  is still an algebra morphism. Formally this is obtained by twisting the product. In the framework of bounded operators we proceed as follows. . . We defineFh, = dx1 dx2 Kh, (x1 D x2 ) R 1  R 2 x xFh, = dx1 dx2 Kh, (x1 D x2 ) L -1  L -1 x x1 2where (R f)(g) := f(gx) and (L f)(g) := f(xg) . The operators Fh, and Fh, are the x x `non-formal analogue' of Fh, and-1 Fh, , with / the left/right canonical actions.PropositionFh, and Fh, are unitary operators on L2 (BD dR x) and L2 (BD dL x) respectively.18 / 22A common domain for the operators FFrom now on assume (a) = -(a) = (-a) . One of the main technical points:Proposition [P. Bieliavsky, V. Gayral]¢ under left/right translations and Fh, and Fh, are continuous operators on D  D.As a corollary, the doubly twisted product Let T : (aD vD t)  (sinh 2aD (cosh a)-1 vD t). The Fréchet space D := T S(R2n+2 ) is stable¢ m : DD  D Dm = m  F,  F,() Dh )(f2 )h f2 ) = (f1 )( h  ¢ D) (bornological quantum group??). The operator where  : D  M(D is a well defined associative product on D. It satisfies (f1W := (id  m )(  id)¢ with domain D  D , satisfies the pentagon equation.Is it bounded? (unitary?) not on H = L2 (BD dR x) !19 / 22On the unitarity of WLemmaThe operator W can be rewritten as (*) W = (1   2 )Yh, (1  - 2 )W Fh, D1 1where W is the (undeformed) Kac-Takesaki operator of B,  is the modular function andYh, := dx1 dx2 Kh, (x1 D x2 )R 2  (x1 ) 2 L -1 F x x11Remarks: both R and x  (x) 2 L -1 are unitary left actions of B on L2 (BD d2 x); x x1W , Fh, and Yh, in (*) are unitary on L2 (BD d2 x) but  is not. a minimal modification of W in order to get a unitary operator is U := Yh, W Fh,U does not satisfy the pentagon equation.change Hilbert space!20 / 22Modifying the Hilbert spaceDefinition/PropositionLet ,: D × D  D be the sesquilinear form: D  = m (D )(x) dR x DD  =(x)(x)dR x the L2 -inner product and Ph the pseudo-differential operator Ph := D +1 + D2(n+1)/2with D := -ih/t . Then i) ii)D ,= Ph D Ph  ;is an inner product (Ph is positive and invertible);iii) W estends to a unitary operator on H  H , where H is the Hilbert space completion of D w.r.t. the inner product , .Remark: H = L2 (R2n+1 D e2(n+1)a dad2n v)  H(n+1)/2 (R), where Hs (R) is the s-th Sobolev space in the variable t.21 / 22Thank you for your attention.22 / 22`

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