Induced Representation and Frobenius Reciprocity for Compact Quantum Groups By ARUPKUMAR PAL Indian Statistical Institute, Delhi Centre 7, SJSS Marg, New Delhi 110 016 e-mail: arup @ isid.ernet.in Abstract: Unitary representations of compact quantum groups have been de- scribed as isometric comodules. The notion of an induced representation for com- pact quantum groups has been introduced and an analogue of the Frobenius reci- procity theorem is established. Quantum groups, like their classical counterparts, have a very rich representation theory. In the representation theory of classical groups, induced representation plays a very important role. Among other things, for example, one can obtain families of irreducible unitary representations of many locally compact groups as representations induced by one dimensional representations of appropriate subgroups. Therefore it is natural to try and see how far can this notion be developed and exploited in the case of quantum groups. As a first step, we do it here for compact quantum groups. First we give an alternative description of a unitary representation as an isometric comodule map. This is trivial in the finite dimensional case, but requires a little bit of work if the comodule is infinite dimensional. Using the comodule description, the notion of an induced representation is defined. We then go on to prove that an exact analogue of the Frobenius reciprocity theorem holds for compact quantum groups. As an application of 2 this theorem, an alternative way of decomposing the action of SUq(2) on the Podle`ssphere Sq0 is given. Notations. , etc, with or without subscripts, will denote complex separable Hilbert spaces. H K ( ) and 0( ) denote respectively the space of bounded operators and the space of compact B H B H operators on . , , etc denote C -algebras. All the C -algebras used in this article have H A B C ∗ ∗ been assumed to act nondegenerately on Hilbert spaces. More specifically, given any C∗-algebra , it is assumed that there is a Hilbert space such that ( ) and for u , a(u) = 0 A K A ⊆ B K 2 K for all a implies u = 0. Tensor product of C -algebras will always mean their spatial 2 A ∗ tensor product. The identity operator on Hilbert spaces is denoted by I, and on C∗-algebras by id. For two vector spaces X and Y , X alg Y denote their algebraic tensor product. ⊗ 1 Let be a C -algebra acting on . The subalgebras a ( ): ab b and A ∗ K f 2 B K 2 A 8 2 A g a ( ): ab; ba b of ( ) are called respectively the left multiplier algebra f 2 B K 2 A 8 2 A g B K and the multiplier algebra of . We denote them by LM( ) and M( ) respectively. A good A A A reference for multiplier algebras and other topics in C∗-algebra theory is [4]. See [9] for another equivalent description of multiplier algebras that is often very useful. 1 Preliminaries 1.1 Let be a unital C -algebra. A vector space X having a right -module structure is A ∗ A called a Hilbert -module if it is equipped with an -valued inner product that satisfies A A i. x; y = y; x , h i∗ h i ii. x; x 0, h i ≥ iii. x; x = 0 x = 0, h i ) iv. x; yb = x; y b for x; y X, b , h i h i 2 2 A and if x := x; x 1=2 makes X a Banach Space. k k kh ik Details on Hilbert C∗-modules can be found in [1], [2] and [3]. We shall need a few specific examples that are listed below. Examples(a) Any Hilbert space with its usual inner product is a Hilbert C= -module. H (b) Any unital C -algebra with a; b = a b is a Hilbert -module. ∗ A h i ∗ A (c) , the `external tensor product' of and , is a Hilbert -module. H ⊗ A H A A (d) ( ; ), with S; T = S T is a Hilbert ( )-module. B H K h i ∗ B H 1.2 We have seen above that ( ) and ( ; ) both are Hilbert ( )-modules. It is H ⊗ B K B K H ⊗ K B K easy to see that the map # : ui ai ui ai(:) from alg ( ) to ( ; ) extends P ⊗ 7! P ⊗ H ⊗ B K B K H ⊗ K to an isometric module map from ( ) to ( ; ), i.e. # obeys H ⊗ B K B K H ⊗ K #(x);#(y) = x; y ; x; y ( ); h i h i 8 2 H ⊗ B K #(xb) = #(x)b; x ( ); b ( ): 8 2 H ⊗ B K 2 B K Thus # embeds ( ) in ( ; ). Observe two things here: first, if = C= , # is just H ⊗ B K B K H ⊗ K H the identity map. And, # is onto if and only if is finite dimensional. The following lemma, H the proof of which is fairly straightforward, gives a very useful property of #. Lemma Let #i be the map # constructed above with i replacing , i = 1; 2. Let S ( 1; 2) H H 2 B H H and x 1 ( ). Then #2((S id)x) = (S I)#1(x): 2 H ⊗ B K ⊗ ⊗ 2 1.3 For an operator T ( ), and a vector u , let Tu denote the operator v T (u v) 2 B H⊗K 2 H 7! ⊗ from to . It is not too difficult to show that Tu #( ( )) if T LM( 0( ) ( )). K H⊗K 2 H⊗B K 2 B H ⊗B K 1 Define a map Ψ(T ) from to ( ) by: Ψ(T )(u) = # (Tu). Then Ψ is the unique linear H H ⊗ B K − injective contraction from LM( 0( ) ( )) to ( ; ( )) for which #(Ψ(T )(u))(v) = B H ⊗ B K B H H ⊗ B K T (u v) u , v , T LM( 0( ) ( )). Here are a few interesting properties of this ⊗ 8 2 H 2 K 2 B H ⊗ B K map Ψ. Proposition Let Ψ: LM( 0( ) ( )) ( ; ( )) be the map described above. B H ⊗ B K !B H H ⊗ B K Then we have the following: i. Ψ maps isometries in LM( 0( ) ( )) onto the isometries in ( ; ( )). B H ⊗ B K B H H ⊗ B K ii. For any T LM( 0( ) ( )) and S 0( ), 2 B H ⊗ B K 2 B H Ψ(T (S I)) = Ψ(T ) S; Ψ((S I)T ) = (S id) Ψ(T ): ⊗ ◦ ⊗ ⊗ ◦ iii. If is any C -subalgebra of ( ) containing its identity, then T LM( 0( ) ) if and A ∗ B K 2 B H ⊗ A only if range Ψ(T ) . ⊆ H ⊗ A Proof : i. Suppose T LM( 0( ) ( )) is an isometry. By 1.2, Ψ(T )u; Ψ(T )v = 2 B H ⊗ B K h i 1 1 # (Tu);# (Tv) = Tu;Tv = u; v I for u; v . Thus Ψ(T ) is an isometry. h − − i h i h i 2 H Conversely, take an isometry π : ( ) and define an operator T on the product H ! H ⊗ B K vectors in by T (u v) = #(π(u))(v), # being the map constructed in 1.2. It is clear that H ⊗ K ⊗ T is an isometry. It is enough, therefore, to show that T ( u v S) 0( ) ( ) whenever j ih j ⊗ 2 B H ⊗ B K S ( ) and u; v are unit vectors in such that u; v = 0 or 1. 2 B K H h i Choose an orthonormal basis ei for such that e1 = u, er = v where f g H 0 if u; v = 0, r = 8 h i < 1 if u; v = 1. : h i Let πij = ( ei id)π(ej). Then T ( u v S) = ei er πi1S where the right hand side h j ⊗ j ih j ⊗ P j ih j ⊗ converges strongly. Since π(e1) ( ), it follows that πi1∗πi1 converges in norm. 2 H ⊗ B K Pi Consequently the right hand side above converges in norm, which means T ( u v S) j ih j ⊗ 2 0( ) ( ). B H ⊗ B K ii. Straightforward. iii. Take T = u v a, u; v , a . For any w , Ψ(T )(w) = v; w u a . j ih j ⊗ 2 H 2 A 2 H h i ⊗ 2 H ⊗ A Since Ψ is a contraction, and the norm closure of all linear combinations of such T 's is 0( ) , B H ⊗A we have range Ψ(T ) for all T 0( ) . ⊆ H ⊗ A 2 B H ⊗ A Assume next that T LM( 0( ) ). Then T ( u u I) 0( ) for all u . 2 B H ⊗ A j ih j ⊗ 2 B H ⊗ A 2 H Hence Ψ(T ( u u I))(u) , which means, by part (ii), that Ψ(T )(u) for all j ih j ⊗ 2 H ⊗ A 2 H ⊗ A u . Thus range Ψ(T ) . 2 H ⊆ H ⊗ A 3 To prove the converse, it is enough to show that T ( u v a) 0( ) whenever a j ih j ⊗ 2 B H ⊗ A 2 A and u; v are such that u; v = 0 or 1. Rest of the proof goes along the same lines as the 2 H h i proof of the last part of (i). 1.4 Let 1, 2 be two Hilbert spaces, i being a C -subalgebra of ( i) containing its identity. K K A ∗ B K Suppose φ is a unital *-homomorphism from 1 to 2. Then id φ : S a S φ(a) extends A A ⊗ ⊗ 7! ⊗ to a *-homomorphism from 0( ) 1 to 0( ) 2. Moreover ((id φ)(a))b : a B H ⊗ A B H ⊗ A f ⊗ 2 0( ) 1; b 0( ) 2 is total in 0( ) 2. Therefore id φ extends to an algebra B H ⊗ A 2 B H ⊗ A g B H ⊗ A ⊗ homomorphism by the following prescription: for all a LM( 0( ) 1), b 0( ) 1, 2 B H ⊗ A 2 B H ⊗ A c 0( ) 2, 2 B H ⊗ A ((id φ)a)(((id φ)b)c) := ((id φ)(ab))c: ⊗ ⊗ ⊗ Proposition Let φ be as above, and Ψi be the map Ψ constructed earlier with i replacing .