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 groups

joint with P. Bieliavsky, P. Bonneau, V. Gayral, Y. Maeda, Y. Voglaire

Francesco D'Andrea

International School for Advanced Studies (SISSA) Via Beirut 2-4, Trieste, Italy

09/09/2010

Workshop on Quantum Groups and Physics, Caen, 6­10 September 2010.

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Outline

1

Introduction Hopf algebras and compact quantum groups Drinfeld twist/cocycle quantization The operator algebra approach

2

From formal to non-formal deformations: Moyal The Moyal plane Moyal-Weyl quantization

3

Deformation of Kählerian Lie groups Definition and decomposition Formal deformations Multiplicative unitaries

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Hopf algebras & compact quantum groups

Motivating 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 ) D

S:AA S(f)(g) := f(g-1 ) D

satistying the axioms of a commutative Hopf algebra (HA):

( id) = (id ) D ( id) = (id ) = id D (S id) = (id S) = (ab)= (a)(b) D

Then [Drinfeld-Jimbo, Majid, `80s]:

(coassociativity) (counity) (coinverse)

D

(ab) = (a) (b) F

Tannaka-Kren duality: G can be reconstructed from its irreps., i.e. from R(G). i

quantum groups := non-commutative non-cocommutative Hopf algebras.

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Quantization of Poisson-Lie groups

Let 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 the

C[[h]]-module A[[h]] of the form f1 h f2 = f1 · f2 +

ih

2

{f1 D f2 } + O(h2 ) D

such 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) } D

for 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.

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Drinfeld twist/cocycle quantization

A 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 with

F (X) := F-1 (X)F D

XUF

for 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 product

a F b = m F(a b) D

aD b A F

F)

for any bialgebra O dual to U (e.g. R(G)[[h]]), a OF = (OD

dual to UF by

a

Here and

F

b := m(F (a b) F-1 ) F

are the left/right canonical actions. Rem.: the coproduct is undeformed!

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The operator algebra approach

Gelfand-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 & 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 & 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!

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Quantum groups via multiplicative unitaries

[Baaj-Skandalis `90s, later Woronowicz & 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 F

From W one can reconstruct C0 (G) (then the topological space G) as

C0 (G) = ( id)(W) B(H)

and the group structure from the property

norm cl.

W (I T )W = T T

T = Rg for some g G F

Idea: define a LCQG via a multiplicative unitary, i.e. a unitary W on some H H satisfying the Pentagon equation ().

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Outline

1

Introduction Hopf algebras and compact quantum groups Drinfeld twist/cocycle quantization The operator algebra approach

2

From formal to non-formal deformations: Moyal The Moyal plane Moyal-Weyl quantization

3

Deformation of Kählerian Lie groups Definition and decomposition Formal deformations Multiplicative unitaries

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The Moyal plane

Let {P1 D P2 } := basis of R2 and A := U(R2 )-module algebra with multiplication m. An associative product on A[[h]] is given by

a M b := m eihP1 P2 (a b) F h

Example: 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.

ih

2

{aD b} +

n 2h

n

Cn (aD b) D

From 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 formula

a h b = m Fh (a b) F

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Moyal-Weyl quantization

Non-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) := h

I (h)2

R2 ×R2

e h {(x,y)+(y,z)+(z,x)} f1 (y)f2 (z)d2 y d2 z D

i

with := standard symplectic form. Remark:

W can be extended to larger classes of functions: C - / Hilbert algebras, . . . h

[Gracia-Bondía & 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].

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2 [Wang, 1996],

Outline

1

Introduction Hopf algebras and compact quantum groups Drinfeld twist/cocycle quantization The operator algebra approach

2

From formal to non-formal deformations: Moyal The Moyal plane Moyal-Weyl quantization

3

Deformation of Kählerian Lie groups Definition and decomposition Formal deformations Multiplicative unitaries

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Normal j-algebras & Kählerian Lie groups

j-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]) > 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.

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Decomposition and formal extension lemma

Elementary 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.

Corollary

Any b can be constructed by iteration as semidirect product of elementary normal j-algebras and possibly an abelian factor.

Formal extension lemma

Let b = b0

b1 and Fi twists based on U(bi )[[h]]. If F0 U(b0 )b1 U(b0 )b1 [[h]] , then F := F0 F1

is a twist based on U(b)[[h]].

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Elementary normal j-algebras

Definition

Fix n N. Then i) b

R × 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 F

i vi Xi

ii) The map

bBD

x := (aD vD t) exp(aH)exp

+ tE

is a global Darboux diffeomorphism. iii) Within these coordinates the group law is

x · x = a + a D e-a v + v D e-2a t + t + 1 e-a (vD v ) 2

with (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.

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A star product on B

Within 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×B

Kh, (x1 D x2 )f1 (x0 x1 )f2 (x0 x2 )dx1 dx2 D

where 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 ) D

the canonical amplitude and phase are

i

Acan (x1 D x2 ) := (cosh a1 cosh a2 cosh (a1 - a2 ))n

cosh 2a1 cosh 2a2 cosh 2(a1 - a2 ) D

Scan (x1 D x2 ) := sinh (2a1 )t2 - sinh (2a2 )t1 + cosh a1 cosh a2 (v1 D v2 ) F

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The formal twist

-- n = 0, = 0 --

Define 0 (a) :=

1 2

log cosh 2a .

Proposition

For n = 0 and = 0 the formal twist is given by

Fh =

n 0

h ( i2 )n Fn D

Fn :=

(-I)l Bl,m Bj,k Hk El Hm Ej D

j,k,l,m 0 j+l=n

where

Bj,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 !

n

and n are the Taylor coefficients of arcsinh. Remark 1: since Bj,k = 0 for all j < k, Fn is a finite sum (i.e. Fn UEA). Remark 2: a completely explicit formula can be given for arbitrary .

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The formal twist

-- n and arbitrary --

Proposition

Denoting by Fh,

n=0

the twist in the n = 0 case, we have:

Fh, = Fh,

n n=0 i=0

Gi h

where the Gi 's are mutually commuting (so that their order in the product doesn't matter) h and are given by

k

Gi := h

k 0

-ihch (E) k3

(-1)|||| X1 F F F Xk X-1 F F F X-k D

{±}k

with the shorthand notation X- = Xi and X+ = Xn+i , with |||| the number of + signs in and with

ch (E) := (1 - h2 E2 1)- 2 (1-1 h2 E2 )- 2 F

1

1

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Deforming the Kac-Takesaki operator of B

The Kac-Takesaki operator of B is given on 2-point functions by

W = (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 define

Fh, = dx1 dx2 Kh, (x1 D x2 ) R 1 R 2 x x

Fh, = dx1 dx2 Kh, (x1 D x2 ) L -1 L -1 x x

1 2

where (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.

Proposition

Fh, and Fh, are unitary operators on L2 (BD dR x) and L2 (BD dL x) respectively.

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A common domain for the operators F

From 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 D

m = m F, F,() D

h )(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 (f1

W := (id m )( id)

¢ with domain D D , satisfies the pentagon equation.

Is it bounded? (unitary?) not on H = L2 (BD dR x) !

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On the unitarity of W

Lemma

The operator W can be rewritten as (*)

W = (1 2 )Yh, (1 - 2 )W Fh, D

1 1

where W is the (undeformed) Kac-Takesaki operator of B, is the modular function and

Yh, := dx1 dx2 Kh, (x1 D x2 )R 2 (x1 ) 2 L -1 F x x

1

1

Remarks: both R and x (x) 2 L -1 are unitary left actions of B on L2 (BD d2 x); x x

1

W , 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!

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Modifying the Hilbert space

Definition/Proposition

Let ,

: D × D D be the sesquilinear form: D = m (D )(x) dR x D

D =

(x)(x)dR x the L2 -inner product and Ph the pseudo-differential operator Ph := D +

1 + D2

(n+1)/2

with 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.

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Thank you for your attention.

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

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