Deformation of Kählerian Lie groups

Transcription

1 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, 6 10 September / 22

2 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 2 / 22

3 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 : A A A ɛ : A C S : A A (f)(g 1, g 2 ) := f(g 1 g 2 ), ɛ(f) := f(1 G ), S(f)(g) := f(g -1 ), satistying the axioms of a commutative Hopf algebra (HA): ( id) = (id ), (coassociativity) (ɛ id) = (id ɛ) = id, (counity) (S id) = (id S) = ɛ, (coinverse) (ab)= (a) (b), ɛ(ab) = ɛ(a)ɛ(b). Tannaka-Kreĭn duality: G can be reconstructed from its irreps., i.e. from R(G). Then [Drinfeld-Jimbo, Majid, 80s]: quantum groups := non-commutative non-cocommutative Hopf algebras. 3 / 22

4 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 f 1 h f 2 = f 1 f 2 + i h 2 {f 1, f 2 } + O( h 2 ), such that (A[[ h]], h,,...) is still an Hopf algebra. The necessary condition at the leading order in h gives (f 1 h f 2 ) = (f 1 )( h h) (f 2 ) ( ) {a, b} = {a (1), b (1) } a (2) b (2) + a (1) b (1) {a (2), b (2) }, for all a, 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]], h ) of Poisson-Lie groups with undeformed coproduct. 4 / 22

5 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 1) = (id )(F) (1 F), (ɛ id)(f) = (id ɛ)(f) = 1 1. F -1 is a counital Hopf 2-cocycle (cf. e.g. [Majid, 1995]). With F we can define: a new bialgebra U F = (U, F ) by replacing of U with F (X) := F -1 (X)F, X U. for any left U-module algebra A with product m(a b) = ab (e.g. R(G)[[ h]]), a left U F -module algebra A F = (A, F ) with product a F b = m F(a b), a, b A. for any bialgebra O dual to U (e.g. R(G)[[ h]]), a O F = (O, F ) dual to U F by a F b := m(f (a b) F -1 ). Here and are the left/right canonical actions. Rem.: the coproduct is undeformed! 5 / 22

6 The operator algebra approach Gelfand-Naĭmark thm: the top. space G can be reconstructed from Q := C 0 (G). A compact quantum group (CQG) is a pair (Q, ) given by a complex unital 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: Hopf C -algebras [Vaes & Van Daele, 2001]: similar to HAs but Im( ) Q Q. Multiplier Hopf algebras [Van Daele, 1994]: HA with no 1, Im( ) M(Q Q). LCQG [Kusterman & Vaes, 2000]: (Q,, ϕ, ψ) with : Q M(Q Q) and ϕ (resp. ψ) left (resp. right) invariant faithful KMS weight. Modelled on C 0 (G). Bornological QGs [Voigt, 2005]: (A,, ϕ), modelled on A := C c (G), uses bornological algebras. A pair (Q, ) can be constructed from a multiplicative unitary! 6 / 22

7 Quantum groups via multiplicative unitaries [Baaj-Skandalis 90s, later Woronowicz & Sołtan] All informations about a locally compact group G are encoded in the Kac-Takesaki operator, that is the op. W B(H H) H = L 2 (G, d R µ) closure of (Wψ)(g, g ) = ψ(gg, g ). W is unitary... and multiplicative: ( ) W 12 W 13 W 23 = W 23 W 12. From W one can reconstruct C 0 (G) (then the topological space G) as C 0 (G) = { (ω id)(w) } norm cl. ω B(H) and the group structure from the property W (1 T)W = T T T = R g for some g G. Idea: define a LCQG via a multiplicative unitary, i.e. a unitary W on some H H satisfying the Pentagon equation ( ). 7 / 22

8 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 8 / 22

9 The Moyal plane Let {P 1, P 2 } := basis of R 2 and A := U(R 2 )-module algebra with multiplication m. An associative product on A[[ h]] is given by a M h b := m e i hp 1 P 2 (a b). Example: if A = C (R 2 ) and P i = i we get a deformation quantization of R 2. Memorandum A deformation quantization of (M, {, }) is an ass. product h on C (M)[[ h]] given by: where C n are bidifferential operators. a h b := ab + i h 2 {a, b} + n 2 h n C n (a, b), From deformation quantizations to twists and back Left invariant h s on G are in bijection with twisting elements (Hopf 2-cocycles) F h based on U(g)[[ h]] via the formula a h b = m F h(a b). 9 / 22

10 Moyal-Weyl quantization Non-formal deformations: if f 1, f 2 S(R 2 ) then f 1 M h f 2 is convergent (to a Schwartz function) for any h R, and can be rewritten as an oscillatory integral: (f 1 W h f 2 )(x) := 1 e h i {ω(x,y)+ω(y,z)+ω(z,x)} f (π h) 2 1 (y)f 2 (z)d 2 y d 2 z, R 2 R 2 with ω := standard symplectic form. Remark: W h can be extended to larger classes of functions: C - / Hilbert algebras,... [Gracia-Bondía & Várilly, 1988] Weyl: formulation of quantum mechanics on phase-space. Rieffel: strict deformation quantization for actions of R d. Examples: SU q (2) [Sheu, 1991], G θ for any compact G with rank 2 [Wang, 1996], Connes-Landi spheres,... Using the integral kernel of W h one can deform the Kac-Takesaki operator of any locally compact group G with rank 2 [Vaes et al., Lecture Notes, 2001]. 10 / 22

11 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 11 / 22

12 Normal j-algebras & Kählerian Lie groups j-algebras were introduced in the study bounded homogeneous domains in C n. Definition [Piatetski-Shapiro] A normal j-algebra (b, α, j) is given by i) a solvable Lie algebra b split over R (i.e. Ad X has only real eigenvalues X b); ii) j End(b) s.t. j 2 = 1 and j[x, Y] = [jx, Y] + [X, jy] + j[jx, jy] for all X, Y b; iii) a linear form α : b R s.t. α([jx, X]) > 0 X 0 and α([jx, jy]) = α([x, Y]). One can associate a (b, α, 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 dα 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 = 1) one can attach a normal j-algebra, with b := a n. These are called elementary. 12 / 22

13 Decomposition and formal extension lemma Elementary normal j-algebras are building blocks for normal j-algebras. Theorem [Piatetski-Shapiro] For any (b, α, j) a split exact sequence 0 b 0 b b 1 0 where b 0 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 = b 0 b 1 and F i twists based on U(b i )[[ h]]. If F 0 U(b 0 ) b 1 U(b0 ) b 1 [[ h]], then is a twist based on U(b)[[ h]]. F := F 0 F 1 13 / 22

14 Elementary normal j-algebras Definition Fix n N. Then i) b R R 2n R has basis H, {X i } i=1,...,2n, E and relations ii) The map [X i, X j ] = (δ i+n,j δ i n,j )E, [E, X i ] = 0, [H, E] = 2E, [H, X i ] = X i. b B, x := (a, v, t) exp(ah)exp ( i v ix i + te ) is a global Darboux diffeomorphism. iii) Within these coordinates the group law is x x = ( a + a, e a v + v, e 2a t + t e a Ω(v, v ) ) with Ω(v, v ) = n i=1 (v iv i+n v i+nv i ) standard symplectic structure on R2n. Remark 1: B is a 1D extension of the Heisenberg group H n := {x = (a, v, t) B : a = 0}. Remark 2: there is extra structure on B, namely it is a symplectic symmetric space. 14 / 22

15 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 τ : R C satisfying some technical conditions, we define (f 1 h,τ f 2 )(x 0 ) = K h,τ(x 1, x 2 )f 1 (x 0 x 1 )f 2 (x 0 x 2 )dx 1 dx 2, B B where dx is the left-invariant Haar, K h,τ(x 1, x 2 ) := 2 2n (π h) 2(n+1) A can (x 1, x 2 ) e i h Scan(x 1,x 2 ) e τ(a 1)+τ( a 2 ) τ(a 1 a 2 ), the canonical amplitude and phase are A can (x 1, x 2 ) := (cosha 1 cosha 2 cosh(a 1 a 2 )) n cosh2a 1 cosh2a 2 cosh2(a 1 a 2 ), S can (x 1, x 2 ) := sinh(2a 1 )t 2 sinh(2a 2 )t 1 + cosha 1 cosha 2 Ω(v 1, v 2 ). 15 / 22

16 The formal twist n = 0, τ = τ 0 Define τ 0 (a) := 1 2 logcosh2a. Proposition For n = 0 and τ = τ 0 the formal twist is given by F h = n 0( i h 2 )n F n, F n := ( 1) l B l,m B j,k H k E l H m E j, where B j,k := j,k,l,m 0 j+l=n n 1 +n n j =k n 1 +2n jn j =j and β n are the Taylor coefficients of arcsinh. β n 1 1 βn β n j j n 1! n 2!... n j! Remark 1: since B j,k = 0 for all j < k, F n is a finite sum (i.e. F n UEA). Remark 2: a completely explicit formula can be given for arbitrary τ., 16 / 22

17 The formal twist n and τ arbitrary Proposition Denoting by F h,τ n=0 the twist in the n = 0 case, we have: n F h,τ = F h,τ n=0 i=0 Gi h where the Gi h s are mutually commuting (so that their order in the product doesn t matter) and are given by G i h := k 0 ( i hc h(e) ) k k! σ {±} k (-1) σ X σ1... X σk X σ1... X σk, with the shorthand notation X = X i and X + = X n+i, with σ the number of + signs in σ and with c h(e) := (1 h 2 E 2 1) 1 2 (1-1 h 2 E 2 ) / 22

18 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(ψ)(g) = ψ(g, g) extends the multiplication map and (ψ)(g, g ) = ψ(gg ). The pentagon equation follows from (f 1 f 2 ) = (f 1 ) (f 2 ). 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 F h,τ = dx 1 dx 2 K h,τ(x 1, x 2 ) R x 1 R x 2 F h,τ = dx 1 dx 2 K h,τ(x 1, x 2 ) L L x -1 1 x -1 2 where (R xf)(g) := f(gx) and (L xf)(g) := f(xg). The operators F h,τ and F h,τ are the non-formal analogue of F h,τ and F h,τ -1, with / the left/right canonical actions. Proposition F h,τ and F h,τ are unitary operators on L 2 (B, d R x) and L 2 (B, d L x) respectively. 18 / 22

19 A common domain for the operators F From now on assume τ(a) = τ(a) = τ( a). One of the main technical points: Proposition [P. Bieliavsky, V. Gayral] Let T : (a, v, t) (sinh2a, (cosha) 1 v, t). The Fréchet space D := T ( S(R 2n+2 ) ) is stable under left/right translations and F h,τ and F h,τ are continuous operators on D D. As a corollary, the doubly twisted product m : D D D, m = m F θ,τ F θ,σ(τ), is a well defined associative product on D. It satisfies (f 1 h f 2 ) = (f 1 )( h h) (f 2 ) where : D M(D D) (bornological quantum group??). The operator W := (id m )( id) with domain D D, satisfies the pentagon equation. Is it bounded? (unitary?) not on H = L 2 (B, d R x)! 19 / 22

20 On the unitarity of W Lemma The operator W can be rewritten as (*) W = (1 χ 1 2 )Y h,τ (1 χ 1 2 )W F h,τ, where W is the (undeformed) Kac-Takesaki operator of B, χ is the modular function and Y h,τ := dx 1 dx 2 K h,τ(x 1, x 2 )R x 2 χ(x 1 ) 1 2 L x Remarks: both R x and x χ(x) 1 2 L x -1 are unitary left actions of B on L 2 (B, d 2 x); W, F h,τ and Y h,τ in (*) are unitary on L 2 (B, d 2 x) but χ is not. a minimal modification of W in order to get a unitary operator is U := Y h,τ W F h,τ U does not satisfy the pentagon equation. change Hilbert space! 20 / 22

21 Modifying the Hilbert space Definition/Proposition Let, : D D D be the sesquilinear form: ψ, ϕ = m (ψ, ϕ)(x) d R x, ψ, ϕ = ψ(x)ϕ(x)d R x the L 2 -inner product and P h the pseudo-differential operator P h := ( D D 2) (n+1)/2 with D := i h / t. Then i) ψ, ϕ = P hψ, P hϕ ; ii), is an inner product (P h 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 = L 2 (R 2n+1, e 2(n+1)a dad 2n v) H (n+1)/2 (R), where H s (R) is the s-th Sobolev space in the variable t. 21 / 22

22 Thank you for your attention. 22 / 22