Quantum isometry group of dual of finitely generated discrete groups- II
aa r X i v : . [ m a t h . OA ] F e b Quantum isometry group of dual of finitely generated discretegroups- II
Arnab Mandal
Indian Statistical Institute203, B. T. Road, Kolkata 700108Email: [email protected]
Abstract
As a continuation of the programme of [19], we carry out explicit computations of Q (Γ , S ), thequantum isometry group of the canonical spectral triple on C ∗ r (Γ) coming from the word length functioncorresponding to a finite generating set S, for several interesting examples of Γ not covered by theprevious work [19]. These include the braid group of 3 generators, Z ∗ n etc. Moreover, we give analternative description of the quantum groups H + s ( n,
0) and K + n (studied in [6], [4]) in terms of freewreath product. In the last section we give several new examples of groups for which Q (Γ) turns out tobe a doubling of C ∗ (Γ). It is a very important and interesting problem in the theory of quantum groups and noncommutative geom-etry to study ‘quantum symmetries’ of various classical and quantum structures. S.Wang pioneered this bydefining quantum permutation groups of finite sets and quantum automorphism groups of finite dimensionalmatrix algebras. Later on, a number of mathematicians including Wang, Banica, Bichon and others ([24], [1],[12]) developed a theory of quantum automorphism groups of finite dimensional C ∗ -algebras as well as quan-tum isometry groups of finite metric spaces and finite graphs. In [17] Goswami extended such constructionsto the set-up of possibly infinite dimensional C ∗ -algebras, and more interestingly, that of spectral triples ala Connes [14], by defining and studying quantum isometry groups of spectral triples. This led to the studyof such quantum isometry groups by many authors including Goswami, Bhowmick, Skalski, Banica, Bichon,Soltan, Das, Joardar and others. In the present paper, we are focusing on a particular class of spectraltriples, namely those coming from the word-length metric of finitely generated discrete groups with respectto some given symmetric generating set. There have been several articles already on computations and studyof the quantum isometry groups of such spectral triples, e.g [11], [23], [16], [6], [4] and references therein.In [19] together with Goswami we also studied the quantum isometry groups of such spectral triples in asystematic and unified way. Here we compute Q (Γ , S ) for more examples of groups including braid groups, Z ∗ Z · · · ∗ Z | {z } n copies etc.The paper is organized as follows. In Section 2 we recall some definitions and facts related to compact quan-tum groups, free wreath product by quantum permutation group and quantum isometry group of spectraltriples defined by Bhowmick and Goswami in [9]. This section also contains the doubling procedure of acompact quantum group, say Q , with respect to an order 2 CQG automorphism θ . The doubling is denotedby D θ ( Q ). In Section 3 we compute Q (Γ , S ) for braid group with 3 generators. Its underlying C ∗ -algebraturns out to be four direct copies of the group C ∗ -algebra. In fact, it is precisely a doubling of doubling ofthe group C ∗ -algebra. Section 4 contains an interesting description of the quantum groups H + s ( n,
0) and K + n (studied in [6], [4]) in terms of free wreath product. Moreover, Q (Γ , S ) is computed for Γ = Z ∗ Z · · · ∗ Z | {z } n copies .In the last section we present more examples of groups as in [16], [23], Section 5 of [19] where Q (Γ , S ) turnsout to be a doubling of C ∗ (Γ). First of all, we fix some notational conventions which will be useful for the rest of the paper. Throughout thepaper, the algebraic tensor product and the spatial (minimal) C ∗ -tensor product will be denoted by ⊗ and1 ⊗ respectively. We’ll use the leg-numbering notation. Let Q be a unital C ∗ -algebra. Consider the multiplieralgebra M ( K ( H ) ˆ ⊗Q ) which has two natural embeddings into M ( K ( H ) ˆ ⊗Q ˆ ⊗Q ). The first one is obtainedby extending the map x x ⊗ ω and ω for the images of an element ω ∈ M ( K ( H ) ˆ ⊗Q ) under these twomaps respectively. We’ll denote the Hilbert C ∗ -module by H ¯ ⊗Q obtained by the completion of H ⊗ Q withrespect to the norm induced by the Q valued inner product << ξ ⊗ q, ξ ′ ⊗ q ′ >> := < ξ, ξ ′ > q ∗ q ′ , where ξ, ξ ′ ∈ H , q, q ′ ∈ Q . Let us recall the basic notions of compact quantum groups, then actions on C ∗ -algebra and free wreathproduct by quantum permutation groups. Definition 2.1
A compact quantum group (CQG for short) is a pair ( Q , ∆) , where Q is a unital C ∗ -algebraand ∆ : Q → Q ˆ ⊗Q is a unital C ∗ -homomorphism satisfying two conditions : . (∆ ⊗ id )∆ = ( id ⊗ ∆)∆ (co-associativity ). . Each of the linear spans of ∆( Q )(1 ⊗ Q ) and that of ∆( Q )( Q ⊗ is norm dense in Q ˆ ⊗Q . A CQG morphism from ( Q , ∆ ) to another ( Q , ∆ ) is a unital C ∗ -homomorphism π : Q
7→ Q such that( π ⊗ π )∆ = ∆ π . Definition 2.2 ( Q , ∆ ) is called a quantum subgroup of ( Q , ∆ ) if there exists a surjective C ∗ -morphism η from Q to Q such that ( η ⊗ η )∆ = ∆ η holds. Sometimes we may denote the CQG ( Q , ∆) simply as Q , if ∆ is understood from the context. Definition 2.3
A unitary (co) representation of a CQG ( Q , ∆) on a Hilbert space H is a C -linear map from H to the Hilbert module H ¯ ⊗Q such that . << U ( ξ ) , U ( η ) >> = < ξ, η > Q where ξ, η ∈ H . . ( U ⊗ id ) U = ( id ⊗ ∆) U.
3. Span { U ( ξ ) b : ξ ∈ H , b ∈ Q} is dense in H ¯ ⊗Q . Given such a unitary representation we have a unitary element ˜ U belonging to M ( K ( H ) ˆ ⊗Q ) given by˜ U ( ξ ⊗ b ) = U ( ξ ) b, ( ξ ∈ H , b ∈ Q ) satisfying ( id ⊗ ∆)( ˜ U ) = ˜ U ˜ U .Here we state Proposition 6.2 of [20] which will be useful for us. Proposition 2.4
If a unitary representation of a CQG leaves a finite dimensional subspace of H , then it’llalso leave its orthogonal complement invariant. Remark 2.5
It is known that the linear span of matrix elements of a finite dimensional unitary represen-tation form a dense Hopf *- algebra Q of ( Q , ∆) , on which an antipode κ and co-unit ǫ are defined. Definition 2.6
We say that CQG ( Q , ∆) acts on a unital C ∗ -algebra B if there is a unital C ∗ -homomorphism(called action) α : B → B ˆ ⊗Q satisfying the following : . ( α ⊗ id ) α = ( id ⊗ ∆) α . . Linear span of α ( B )(1 ⊗ Q ) is norm dense in B ˆ ⊗Q . Definition 2.7
The action is said to be faithful if the ∗ -algebra generated by the set { ( f ⊗ id ) α ( b ) ∀ f ∈ B ∗ , ∀ b ∈ B } is norm dense in Q , where B ∗ is the Banach space dual of B. Remark 2.8
Given an action α of a CQG Q on a unital C ∗ -algebra B, we can always find a norm-dense,unital ∗ -subalgebra B ⊆ B such that α | B : B B ⊗ Q is a Hopf-algebraic co-action. Moreover, α isfaithful if and only if the ∗ -algebra generated by { ( f ⊗ id ) α ( b ) ∀ f ∈ B ∗ , ∀ b ∈ B } is the whole of Q . Q , Q the free product Q ⋆ Q admits the natural CQG structure equipped with thefollowing universal property (for more details see [25]): Proposition 2.9 (i) The canonical injections, say i , i , from Q and Q to Q ⋆ Q are CQG morphisms.(ii) Given any CQG C and morphisms π : Q
7→ C and π : Q
7→ C there always exists a unique morphismdenoted by π := π ∗ π from Q ⋆ Q to C satisfying π ◦ i k = π k for k = 1 , . Definition 2.10
The C ∗ -algebra underlying the quantum permutation group, denoted by C ( S + N ) is the uni-versal C ∗ -algebra generated by N elements t ij such that the matrix (( t ij )) is unitary with t ij = t ∗ ij = t ij ∀ i, j, X i t ij = 1 ∀ j, X j t ij = 1 ∀ i,t ij t ik = 0 , t ji t ki = 0 ∀ i, j, k with j = k. It has a coproduct ∆ is given by ∆( t ij ) = Σ Nk =1 t ik ⊗ t kj , such that ( C ( S + N ) , ∆) becomes a CQG. For further details see [24]. We also recall from [13] the following:
Definition 2.11
Let Q be a compact quantum group and N > . The free wreath product of Q by thequantum permutation group C ( S + N ) , is the quotient of Q ∗ N ⋆ C ( S + N ) by the two sided ideal generated by theelements ν k ( a ) t ki − t ki ν k ( a ) , ≤ i, k ≤ N, a ∈ Q , where (( t ij )) is the matrix coefficients of the quantum permutation group C ( S + N ) and ν k ( a ) denotes the naturalimage of a ∈ Q in the k-th factor of Q ∗ N . This is denoted by Q ⋆ w C ( S + N ) . Furthermore, it admits a CQG structure, where the comultiplication satisfies∆( ν i ( a )) = N X k =1 ν i ( a (1) ) t ik ⊗ ν k ( a (2) ) . Here we have used the Sweedler convention of writing ∆( a ) = a (1) ⊗ a (2) . First of all, we are defining the quantum isometry group of spectral triples defined by Bhowmick and Goswamiin [9].
Definition 2.12
Let ( A ∞ , H , D ) be a spectral triple of compact type (a la Connes). Consider the category Q ( D ) ≡ Q ( A ∞ , H , D ) whose objects are ( Q , U ) where ( Q , ∆) is a CQG having a unitary representation Uon the Hilbert space H satisfying the following:1. ˜ U commutes with ( D ⊗ Q ) .2. ( id ⊗ φ ) ◦ ad ˜ U ( a ) ∈ ( A ∞ ) ′′ for all a ∈ A ∞ and φ is any state on Q , where ad ˜ U ( x ) := ˜ U ( x ⊗
1) ˜ U ∗ for x ∈ B ( H ) .A morphism between two such objects ( Q , U ) and ( Q ′ , U ′ ) is a CQG morphism ψ : Q → Q ′ such that U ′ = ( id ⊗ ψ ) U . If a universal object exists in Q ( D ) then we denote it by ^ QISO + ( A ∞ , H , D ) and thecorresponding largest Woronowicz subalgebra for which ad ˜ U is faithful, where U is the unitary representationof ^ QISO + ( A ∞ , H , D ) , is called the quantum group of orientation preserving isometries and denoted by QISO + ( A ∞ , H , D ) . .
23 of [9] which gives a sufficient condition for the existence of
QISO + ( A ∞ , H , D ). Theorem 2.13
Let ( A ∞ , H , D ) be a spectral triple of compact type. Assume that D has one dimensionalkernel spanned by a vector ξ ∈ H which is cyclic and separating for A ∞ and each eigenvector of D belongsto A ∞ ξ . Then QISO + ( A ∞ , H , D ) exists. Let ( A ∞ , H , D ) be a spectral triple satisfying the condition of Theorem 2.13 and A = Lin { a ∈ A ∞ : aξ is an eigenvector of D} . Moreover, assume that A is norm-dense in A ∞ . Let ˆ D : A
7→ A be definedby ˆ D ( a ) ξ = D ( aξ )( a ∈ A ). This is well defined as ξ is cyclic and separating vector for A ∞ . Let τ be thevector state corresponding to the vector ξ . Definition 2.14
Let A be a C ∗ -algebra and A ∞ be a dense *-subalgebra such that ( A ∞ , H , D ) is a spectraltriple as above. Let ˆ C ( A ∞ , H , D ) be the category with objects ( Q , α ) such that Q is a CQG with a C ∗ -action α on A such that1. α is τ preserving, i.e. ( τ ⊗ id ) α ( a ) = τ ( a ) . for all a ∈ A .2. α maps A into A ⊗ Q .3. α ˆ D = ( ˆ D ⊗ I ) α. The morphisms in ˆ C ( A ∞ , H , D ) are CQG morphisms intertwining the respective actions. Proposition 2.15
It is shown in Corollary 2.27 of [9] that
QISO + ( A ∞ , H , D ) is the universal object in ˆ C ( A ∞ , H , D ) . C ∗ r (Γ) Now we discuss the special case of our interest. Let Γ be a finitely generated discrete group with generatingset S = { a , a − , a , a − , ·· a k , a − k } . We make the convention of choosing the generating set to be symmetric,i.e. a i ∈ S implies a − i ∈ S ∀ i . In case some a i has order 2, we include only a i , i.e. not count it twice.The corresponding word length function on the group defined by l ( g ) = min { r ∈ N , g = h h · · · h r } where h i ∈ S i.e. for each i, h i = a j or a − j for some j . Notice that S = { g ∈ Γ , l ( g ) = 1 } , using this length functionwe can define a metric on Γ by d ( a, b ) = l ( a − b ) ∀ a, b ∈ Γ. This is called the word metric correspondingto the generating set S. Now consider the algebra C ∗ r (Γ), which is the C ∗ -completion of the group ring C Γviewed as a subalgebra of B ( l (Γ)) in the natural way via the left regular representation. We define a Diracoperator D Γ ( δ g ) = l ( g ) δ g . In general, D Γ is an unbounded operator. Dom ( D Γ ) = { ξ ∈ l (Γ) : X g ∈ Γ l ( g ) | ξ ( g ) | < ∞} . Here, δ g is the vector in l (Γ) which takes value 1 at the point g and 0 at all other points. Natural generatorsof the algebra C Γ (images in the left regular representation ) will be denoted by λ g , i.e. λ g ( δ h ) = δ gh . Letus define Γ r = { δ g | l ( g ) = r } , Γ ≤ r = { δ g | l ( g ) ≤ r } . Moreover, p r and q r be orthogonal projections onto Sp (Γ r ) and Sp (Γ ≤ r ) respectively. Clearly D Γ = X n ∈ N np n , where p r = q r − q r − and p = q . The canonical trace on C ∗ r (Γ) is given by τ ( P c g λ g ) = c e . It is easy tocheck that ( C Γ, l (Γ) , D Γ ) is a spectral triple. Now take A = C ∗ r (Γ) , A ∞ = C Γ , H = l (Γ) and D = D Γ asbefore, δ e is the cyclic separating vector for C Γ. Then QISO + ( C Γ, l (Γ) , D Γ ) exists by Theorem 2.13. As4he object depends on the generating set of Γ it is denoted by Q (Γ , S ). Most of the times we denote it by Q (Γ) if S is understood from the context. As in [11] its action α (say) on C ∗ r (Γ) is determined by α ( λ γ ) = X γ ′ ∈ S λ γ ′ ⊗ q γ,γ ′ , where the matrix [ q γ,γ ′ ] γ,γ ′ ∈ S is called the fundamental representation in M card ( S ) ( Q (Γ , S )). Note that wehave ∆( q γ,γ ′ ) = P β q β,γ ′ ⊗ q γ,β . Q (Γ , S ) is also the universal object in the category ˆ C ( C Γ, l (Γ) , D Γ ) by Proposition 2.15 and observe thatall the eigenspaces of ˆ D Γ , where ˆ D Γ as in Definition 2.14 are invariant under the action. The eigenspaces ofˆ D Γ are precisely the set Span { λ g | l ( g ) = r } with r ≥ C τ of CQG’s consisting of the objects ( Q , α ) such that α is an action of Q on C ∗ r (Γ)satisfying the following two properties:1. α leaves Sp (Γ ) invariant.2. It preserves the canonical trace τ of C ∗ r (Γ).Morphisms in C τ are CQG morphisms intertwining the respective actions. Lemma 2.16
The two categories C τ and ˆ C ( C Γ , l (Γ) , D Γ ) are isomorphic.Proof: Let ( Q , α ) ∈ ˆ C ( C Γ, l (Γ) , D Γ ) then clearly ( Q , α ) ∈ C τ . Consider any ( Q , α ) ∈ C τ . Then the action α leaves Sp (Γ ≤ r ) invariant ∀ r ≥ Sp (Γ ) invariant. Considerthe linear map U ( x ) := α ( x ) from C ∗ r (Γ) ⊂ H = l (Γ) to H ¯ ⊗Q is an isometry by the invariance of τ . Thusit extends to H and in fact it becomes a unitary representation. Now, observe that Sp (Γ r ) is the orthogonalcomplement of Sp (Γ ≤ r − ) inside Sp (Γ ≤ r ). By the Proposition 2.4, Sp (Γ r ) is invariant under U too, i.e. α leaves Span { λ g | l ( g ) = r } invariant for all r. Thus ( Q , α ) ∈ ˆ C ( C Γ, l (Γ) , D Γ ). Clearly any morphism in thecategory C τ is in the category ˆ C ( C Γ, l (Γ) , D Γ ) and vice-versa. This completes the proof. ✷ Corollary 2.17
It follows from Lemma 2.16 that there is a universal object, say ( Q τ , α τ ) in C τ and ( Q τ , α τ ) ∼ = Q (Γ , S ) . We now identify Q (Γ , S ) as a universal object in yet another category. Let us recall the quantum freeunitary group A u ( n ) introduced in [25]. It is the universal unital C ∗ -algebra generated by (( a ij )) subjectto the conditions that (( a ij )) and (( a ji )) are unitaries. Moreover, it admits co-product structure withcomultiplication ∆( a ij ) = Σ nl =1 a lj ⊗ a il . Consider the category C with objects ( C , { x ij , i, j = 1 , · · · , k } )where C is a unital C ∗ -algebra generated by (( x ij )) such that (( x ij )) as well as (( x ji )) are unitaries and thereis a unital C ∗ - homomorphism α C from C ∗ r (Γ) to C ∗ r (Γ) ⊗ C sending e i to P kj =1 e j ⊗ x ij , where e i − = λ a i and e i = λ − a i ∀ i = 1 , ·· , k . The morphisms from ( C , { x ij , i, j = 1 , · · · , k } ) to ( P , { p ij , i, j = 1 , · · · , k } ) areunital ∗ -homomorphisms β : C 7→ P such that β ( x ij ) = p ij .Moreover, by definition of each object ( C , { x ij , i, j = 1 , · · · , k } ) we get a unital ∗ -morphism ρ C from A u (2 k )to C sending a ij to x ij . Let the kernel of this map be I C and I be intersection of all such ideals. Then C U := A u (2 k ) / I is the universal object generated by x U ij in the category C . Furthermore, we can show,following a line of arguments similar to those in Theorem 4.8 of [18], that it has a CQG structure with theco-product ∆( x U ij ) = P l x U lj ⊗ x U il . Proposition 2.18 ( Q τ , α τ ) and C U are isomorphic as CQG. α is of the form α ( λ a ) = λ a ⊗ A + λ a − ⊗ A + λ a ⊗ A + λ a − ⊗ A + · · · + λ a k ⊗ A k − + λ a − k ⊗ A k ) ,α ( λ a − ) = λ a ⊗ A ∗ + λ a − ⊗ A ∗ + λ a ⊗ A ∗ + λ a − ⊗ A ∗ + · · · + λ a k ⊗ A ∗ k ) + λ a − k ⊗ A ∗ k − ,α ( λ a ) = λ a ⊗ A + λ a − ⊗ A + λ a ⊗ A + λ a − ⊗ A + · · · + λ a k ⊗ A k − + λ a − k ⊗ A k ) ,α ( λ a − ) = λ a ⊗ A ∗ + λ a − ⊗ A ∗ + λ a ⊗ A ∗ + λ a − ⊗ A ∗ + · · · + λ a k ⊗ A ∗ k ) + λ a − k ⊗ A ∗ k − , ... ... α ( λ a k ) = λ a ⊗ A k + λ a − ⊗ A k + λ a ⊗ A k + λ a − ⊗ A k + · · · + λ a k ⊗ A k (2 k − + λ a − k ⊗ A k (2 k ) ,α ( λ a − k ) = λ a k ⊗ A ∗ k + λ a − ⊗ A ∗ k + λ a ⊗ A ∗ k + λ a − ⊗ A ∗ k + · · · + λ a k ⊗ A ∗ k (2 k ) + λ a − k ⊗ A ∗ k (2 k − . From this we get the unitary representation U ≡ (( u ij )) = A A A A · · · A k − A k ) A ∗ A ∗ A ∗ A ∗ · · · A ∗ k ) A ∗ k − A A A A · · · A k − A k ) A ∗ A ∗ A ∗ A ∗ · · · A ∗ k ) A ∗ k − ... ... A k A k A k A k · · · A k (2 k − A k (2 k ) A ∗ k A ∗ k A ∗ k A ∗ k · · · A ∗ k (2 k ) A ∗ k (2 k − . From now on, we call it as fundamental unitary. The coefficients A ij and A ∗ ij ’s generate a norm densesubalgebra of Q (Γ , S ). We also note that the antipode of Q (Γ , S ) maps u ij to u ∗ ji . Remark 2.19
Using Corollary 2.17 and Proposition 2.18, Q (Γ , S ) is the universal unital C ∗ -algebra gen-erated by A ij as above subject to the relations that U is a unitary as well as U t and α given above is a C ∗ -homomorphism on C ∗ r (Γ) . Q (Γ) as a doubling of certain quantum groups In this subsection we briefly recall from [16], [22] the doubling procedure of a compact quantum group whichis just a particular case of a smash co-product, a well-known construction of Hopf-algebra theory introducedin [21]. Let ( Q , ∆) be a CQG with a CQG-automorphism θ such that θ = id . The doubling of this CQG,say ( D θ ( Q ) , ˜∆) is given by D θ ( Q ) := Q ⊕ Q (direct sum as a C ∗ -algebra), and the coproduct is defined bythe following, where we have denoted the injections of Q onto the first and second coordinate in D θ ( Q ) by ξ and η respectively, i.e. ξ ( a ) = ( a, , η ( a ) = (0 , a ) , ( a ∈ Q ) . ˜∆ ◦ ξ = ( ξ ⊗ ξ + η ⊗ [ η ◦ θ ]) ◦ ∆ , ˜∆ ◦ η = ( ξ ⊗ η + η ⊗ [ ξ ◦ θ ]) ◦ ∆ .
6t is known from [22] that, if there exists a non trivial automorphism of order 2 which preserves the generatingset, then D θ ( C ∗ (Γ)) ([22], [16]) will be always a quantum subgroup of Q (Γ). Below we give some sufficientconditions for the quantum isometry group to be a doubling of some CQG. For this, it is convenient touse a slightly different notational convention: let U i − ,j = A ij for i = 1 , . . . , k, j = 1 , . . . , k and U i, l = A ∗ i (2 l − , U i, l − = A ∗ i (2 l ) for i = 1 , . . . , k, l = 1 , . . . , k . Proposition 2.20
Let Γ be a group with k generators { a , a , · · a k } and define γ l − := a l , γ l := a − l ∀ l =1 , , · · · , k . Now σ be an order 2 automorphism on the set { , , ·· , k − , k } and θ be an automorphism ofthe group given by θ ( γ i ) = γ σ ( i ) ∀ i = 1 , , ·· , k . We assume the following :1. B i := U i,σ ( i ) = 0 ∀ i, and U i,j = 0 ∀ j
6∈ { σ ( i ) , i } ,2. A i B j = B j A i = 0 ∀ i, j such that σ ( i ) = i, σ ( j ) = j, where A i = U i,i ,3. All U i,j U ∗ i,j are central projections,4. There are well defined C ∗ -isomorphisms π , π from C ∗ (Γ) to C ∗ { A i , i = 1 , , ·· , k } and C ∗ { B i , i =1 , , ·· , k } respectively such that π ( λ a i ) = A i , π ( λ a i ) = B i ∀ i. Then Q (Γ) is doubling of the group algebra (i.e. Q (Γ) ∼ = D θ ( C ∗ (Γ)) ) corresponding to the given automorphism θ . Moreover, the fundamental unitary takes the following form A · · · B A · · · B
00 0 A · · · A · · · ... ... B k − · · · A k − B k · · · A k . The proof is presented in Lemma 2.26 of [19], the case σ ( i ) = i for some i, is also taken care in the proof.Now we give a sufficient condition for Q (Γ) to be D θ ′ ( D θ ( C ∗ (Γ))), where θ ′ is an order 2 CQG automorphismof D θ ( C ∗ (Γ)). Proposition 2.21
Let Γ be a group with k generators { a , a , · · a k } and define γ l − := a l , γ l := a − l ∀ l =1 , , · · · , k . Now σ , σ , σ are three distinct automorphisms of order 2 on the set { , , ·· , k − , k } and θ , θ , θ are automorphisms of the group given by θ j ( γ i ) = γ σ j ( i ) for all j = 1 , , and i = 1 , , ·· , k . Weassume the following :1. B ( s ) i := U i,σ s ( i ) = 0 ∀ i, and s = 1 , , also U i,j = 0 ∀ j
6∈ { σ s ( i ) , i } , A i B ( s ) j = B ( s ) j A i = 0 ∀ i, j, s such that σ t ( i ) = i, σ t ( j ) = j ∀ t where A i = U i,i , B ( s ) i B ( k ) j = B ( k ) j B ( s ) i = 0 ∀ i, j, s, k with s = k and σ t ( i ) = i, σ t ( j ) = j ∀ t ,4. All U i,j U ∗ i,j are central projections,
5. There are well defined C ∗ -isomorphisms π , π ( s )2 from C ∗ (Γ) to C ∗ { A i , i = 1 , , ·· , k } and C ∗ { B ( s ) i , i =1 , , ·· , k } respectively where s = 1 , , such that ( λ a i ) = A i , π ( s )2 ( λ a i ) = B ( s ) i ∀ i. Furthermore, assume that using the group automorphisms we have two CQG automorphisms θ and θ ′ oforder 2 from C ∗ (Γ) and D θ ( C ∗ (Γ)) respectively defined by θ ( λ x ) = λ θ ( x ) ,θ ′ ( λ x , λ θ ( y ) ) = ( λ θ ( x ) , λ θ ( y ) ) ∀ x, y ∈ Γ . Then Q (Γ) will be D θ ′ ( D θ ( C ∗ (Γ))) corresponding to the given automorphisms. Moreover, the fundamentalunitary takes the following form A B (1)1 · · · B (2)1 B (3)1 B (1)2 A · · · B (3)2 B (2)2 A B (1)3 · · · B (1)4 A · · · ... ... B (2)2 k − B (3)2 k − · · · A k − B (1)2 k − B (3)2 k B (2)2 k · · · B (1)2 k A k . The proof is very similar to the Proposition 2.20, thus omitted. We end the discussion of Section 2 with thefollowing easy observation which will be useful later.
Proposition 2.22 If U V = 0 for two normal elements in a C ∗ -algebra then U ∗ V = V U ∗ = 0 ,V ∗ U = U V ∗ = V U = 0 . Its proof is straightforward, hence omitted.
In this section we will compute the quantum isometry group of the braid group with 3 generators. The grouphas a presentation Γ = < a, b, c | ac = ca, aba = bab, cbc = bcb > . Here S = { a, b, c, a − , b − , c − } . Theorem 3.1
Let Γ be the braid group with above presentation. Then Q (Γ , S ) ∼ = D θ ′ ( D θ ( C ∗ (Γ))) with thechoices of automorphisms as in Proposition 2.21 given by: θ ( a ) = a − , θ ( b ) = b − , θ ( c ) = c − ,θ ( a ) = c, θ ( b ) = b, θ ( c ) = a,θ ( a ) = c − , θ ( b ) = b − , θ ( c ) = a − . Proof:
Let the action α of Q (Γ , S ) be given by α ( λ a ) = λ a ⊗ A + λ a − ⊗ B + λ b ⊗ C + λ b − ⊗ D + λ c ⊗ E + λ c − ⊗ F,α ( λ a − ) = λ a ⊗ B ∗ + λ a − ⊗ A ∗ + λ b ⊗ D ∗ + λ b − ⊗ C ∗ + λ c ⊗ F ∗ + λ c − ⊗ E ∗ , ( λ b ) = λ a ⊗ G + λ a − ⊗ H + λ b ⊗ I + λ b − ⊗ J + λ c ⊗ K + λ c − ⊗ L,α ( λ b − ) = λ a ⊗ H ∗ + λ a − ⊗ G ∗ + λ b ⊗ J ∗ + λ b − ⊗ I ∗ + λ c ⊗ L ∗ + λ c − ⊗ K ∗ ,α ( λ c ) = λ a ⊗ M + λ a − ⊗ N + λ b ⊗ O + λ b − ⊗ P + λ c ⊗ Q + λ c − ⊗ R,α ( λ c − ) = λ a ⊗ N ∗ + λ a − ⊗ M ∗ + λ b ⊗ P ∗ + λ b − ⊗ O ∗ + λ c ⊗ R ∗ + λ c − ⊗ Q ∗ . Then, the fundamental unitary is of the form
A B C D E FB ∗ A ∗ D ∗ C ∗ F ∗ E ∗ G H I J K LH ∗ G ∗ J ∗ I ∗ L ∗ K ∗ M N O P Q RN ∗ M ∗ P ∗ O ∗ R ∗ Q ∗ . We need a few lemmas to prove the theorem.
Lemma 3.2
All the entries of the above matrix are normal.Proof:
First, using the condition α ( λ ac ) = α ( λ ca ) comparing the coefficients of λ a , λ a − , λ b , λ b − , λ c , λ c − onboth sides we have, AM = M A, BN = N B, CO = OC, DP = P D, EQ = QE, F R = RF (1)Applying the antipode we get, AE = EA, BF = F B, GK = KG, HL = LH, M Q = QM, N R = RN (2)Similarly, from the relation α ( λ ac − ) = α ( λ c − a ) following the same argument as above, one can deduce thefollowing AF = F A, BE = EB, GL = LG, HK = KH, N Q = QN, M R = RM (3)We observe AE ∗ + F B ∗ = 0 by comparing the coefficient of λ ac − in the expression of α ( λ a ) α ( λ a − ). Thisshows that AE ∗ A ∗ = 0 as B ∗ A ∗ = 0. Thus, ( AE )( AE ) ∗ = AEE ∗ A ∗ = E ( AE ∗ A ∗ ) = 0. Similarly, all theterms of the equations (1),(2) and (3) are zero.Further using the condition α ( λ a ) α ( λ a − ) = α ( λ a − ) α ( λ a ) = λ e ⊗ Q one can deduce, AC ∗ = AD ∗ = CA ∗ = C ∗ A = DA ∗ = D ∗ A = 0 ,A ∗ C = A ∗ D = BD ∗ = D ∗ B = BC ∗ = B ∗ C = C ∗ B = 0 . Applying the antipode we have, AG ∗ = G ∗ A = AH = HA = BG = BH ∗ = H ∗ B = GB = 0 . Similarly from α ( λ b ) α ( λ b − ) = α ( λ b − ) α ( λ b ) = λ e ⊗ Q one obtains CJ = JC = CI ∗ = I ∗ C = C ∗ I = IC ∗ = J ∗ C ∗ = C ∗ J ∗ = 0 ,DI = ID = DJ ∗ = J ∗ D = 0 . Again using α ( λ c ) α ( λ c − ) = α ( λ c − ) α ( λ c ) = λ e ⊗ Q we have, EL = LE = EK ∗ = K ∗ E = 0 ,F K = KF = F L ∗ = L ∗ F = 0 . α ( λ aba ) = α ( λ bab ) we obtain α ( λ ab ) = α ( λ baba − ). From α ( λ ab ) = α ( λ baba − )comparing the coefficients of λ b and λ b − on both sides we obtain CI = DJ = 0. Now applying the antipodewe get I ∗ G ∗ = JH = 0. This implies GI = JH = 0. Again from α ( λ ab − ) = α ( λ b − a − ba ) and applyingprevious arguments we can deduce CJ ∗ = DI ∗ = 0. Applying antipode we get GJ = IH = 0. Now fromthe unitarity condition we know GG ∗ + HH ∗ + II ∗ + JJ ∗ + KK ∗ + LL ∗ = 1. This shows that G G ∗ = G as we have already got GH = GI = GJ = GK = GL = 0. In a similar way, it follows that G ∗ G = G .Thus we can conclude that G is normal. Using the same argument as before we can show that H, I, J, K, L are normal, i.e. all elements of 3rd row are normal. Using the antipode the normality of
C, D, O, P follows.Now we are going to show that
A, B, E, F, M, N, Q, R are normal too. Using AA ∗ + BB ∗ + CC ∗ + DD ∗ + EE ∗ + F F ∗ = 1 we can write, A = A ( AA ∗ + BB ∗ + CC ∗ + DD ∗ + EE ∗ + F F ∗ )= A A ∗ + ACC ∗ + ADD ∗ ( as AB = AE = AF = 0)= A A ∗ + ( AC ∗ ) C + ( AD ∗ ) D ( as C, D are normal )= A A ∗ ( as AC ∗ = AD ∗ = 0) . Similarly A ∗ A = A , hence A is normal. Following exactly a similar line of arguments one can show thenormality of the remaining elements. ✷ Lemma 3.3 C = D = G = H = K = L = O = P = 0 . Proof:
From the relation α ( λ ac ) = α ( λ ca ) equating the coefficients of λ ba , λ ab , λ ab − , λ b − a on both sides we get AO = M C, CM = OA, AP = M D, CN = OB . This implies that CM M ∗ = OAM ∗ = 0 , CN N ∗ = OBN ∗ = 0 as AM ∗ = BN ∗ = 0. Similarly one can obtain CQQ ∗ = CRR ∗ = 0. Now using ( AA ∗ + BB ∗ + GG ∗ + HH ∗ + M M ∗ + N N ∗ ) = 1 we have, C = C ( AA ∗ + BB ∗ + GG ∗ + HH ∗ + M M ∗ + N N ∗ )= C ( AA ∗ + BB ∗ + GG ∗ + HH ∗ ) ( as CM M ∗ = CN N ∗ = 0)= C ( GG ∗ + HH ∗ ) ( as CAA ∗ = CA ∗ A = 0 , CBB ∗ = CB ∗ B = 0) . Moreover, we have C = C ( EE ∗ + F F ∗ + KK ∗ + LL ∗ + QQ ∗ + RR ∗ )= C ( KK ∗ + LL ∗ + QQ ∗ + RR ∗ ) ( as CE ∗ = CF ∗ = 0)= C ( KK ∗ + LL ∗ ) ( as CQQ ∗ = CRR ∗ = 0) . Using the above equations we get that C ( KK ∗ + LL ∗ )( GG ∗ + HH ∗ ) = C ( GG ∗ + HH ∗ ) = C = 0 (as KG = KH = LG = LH = 0). Similarly, we can find D = 0. Then we have G = H = 0 by using the antipode.Moreover, AO = M C = 0 , AP = OB = BO = 0. This gives us O = ( A ∗ A + B ∗ B + M ∗ M + N ∗ N ) O = 0.Similarly, we get P = 0 , K = L = 0. ✷ Applying the above lemma, the fundamental unitary is reduced to the form
A B
E FB ∗ A ∗ F ∗ E ∗ I J J ∗ I ∗ M N
Q RN ∗ M ∗ R ∗ Q ∗ . emma 3.4 AIA = IAI, BJB = JBJ, AQ = QA,QIQ = IQI, RJR = JRJ, BR = RB,AJ = BI = AR = BQ = IR = JQ = 0 ,EIE = IEI, F JF = JF J, EM = M E,M IM = IM I, N JN = JN J, F N = N F,EJ = F I = EN = F M = IN = JM = 0 . Proof:
First of all we deduce the following relations among the generators, aba = bab, a − b − a − = b − a − b − , ab − a − = b − a − b,a − ba = bab − , ba − b − = a − b − a, b − ab = aba − . We also get same relations replacing a by c. Using the condition α ( λ aba ) = α ( λ bab ) and comparing on bothsides the coefficients of λ aba , λ a − b − a − , λ ab − a − , λ a − ba , λ ba − b − , λ b − ab one can get AIA = IAI, BJB = JBJ, AJB = JBI,BIA = IAJ, IBJ = BJA, JAI = AIB.
Moreover, comparing the coefficients of λ ab − a , λ a − ba − , λ ba − b , λ b − ab − on both sides we have AJA = BIB = IBI = JAJ = 0 . Similarly, equating the coefficients of λ cbc , λ c − b − c − , λ cb − c − , λ c − bc , λ bc − b − , λ b − cb we also find EIE = IEI, F JF = JF J, EJF = JF I,F IE = IEJ, IF J = F JE, JEI = EIF.
Furthermore, comparing the coefficients of λ cb − c , λ c − bc − , λ bc − b , λ b − cb − on both sides we have EJE = F IF = IF I = JEJ = 0 . Now our aim is to show JA = IB = 0. We have JAI = AIBI as JAI = AIB , this implies
JAI = 0because of IBI = 0. This shows that
JAI = 0 as we proved before I I ∗ = I . Thus we can deduce JA = JA ( II ∗ + JJ ∗ )= ( JAI ) I ∗ + ( JAJ ) J ∗ ( as JAI = JAJ = 0)= 0 . Similarly, it follows that IB = 0. Now using Proposition 2.22 we get JA = AJ = IB = BI = 0 . In a similarway one can prove that EJ = JE = IF = F I = 0 , and M J = IN = IR = JQ = 0 as well. Now AR = A ( II ∗ + JJ ∗ ) R = ( AI ∗ )( IR ) + ( AJ )( J ∗ R )= 0 ( as IR = AJ = 0) . We get BQ = EN = F M = 0 applying similar arguments as above. The only remaining part of the lemmais to prove AQ = QA, BR = RB, EM = M E, F N = N F . Using Lemma 4 . ✷ Proof of Theorem 3.1: It follows by combining Lemmas 3.2, 3.3, 3.4 and Proposition 2.21. ✷ We can also prove the obvious analogue of Theorem 3.1 for the braid group with 2 generators.11 heorem 3.5
Let Γ be the braid group with 2 generators. It has a presentation Γ = < a, b | aba = bab > where, S = { a, b, a − , b − } . Then Q (Γ , S ) ∼ = D θ ′ ( D θ ( C ∗ (Γ))) with the choices of automorphisms as inProposition 2.21 given by θ ( a ) = a − , θ ( b ) = b − ,θ ( a ) = b, θ ( b ) = a,θ ( a ) = b − , θ ( b ) = a − . The proof is omitted because it involves very similar computations and arguments as in Theorem 3.1. H + s ( n, , K + n andcomputing the QISO of free copies of Z H + s ( n, , K + n which are discussed in [6], [4] and [5]. K + n is the universal C ∗ -algebra generated by the unitary matrix (( u ij )) which is described in Subsection 2.3 subject to the conditionsgiven below.1. Each u ij is normal, partial isometry.2. u ij u ik = 0 , u ji u ki = 0 ∀ i, j, k with j = k . H + s ( n,
0) is the universal C ∗ -algebra satisfying the above conditions and moreover, u ∗ ij = u s − ij . In thissection we are giving another description of these objects in terms of free wreath product motivated fromthe fact H + n ∼ = C ∗ ( Z ) ∗ w C ( S + n ) (see [7]). First of all, we compute the quantum isometry group of n freecopies of Z . Theorem 4.1
Let Γ be Z ∗ Z · · · ∗ Z | {z } n copies , then Q (Γ) will be Q ( Z ) ∗ w C ( S + n ) .Proof: The group is presented as follows: Γ = < a , a , · · a n | o ( a i ) = 4 ∀ i > Now the fundamental unitary is of the form U = A A A A · · · A n − A n ) A ∗ A ∗ A ∗ A ∗ · · · A ∗ n ) A ∗ n − A A A A · · · A n − A n ) A ∗ A ∗ A ∗ A ∗ · · · A ∗ n ) A ∗ n − ... ... A n A n A n A n · · · A n (2 n − A n (2 n ) A ∗ n A ∗ n A ∗ n A ∗ n · · · A ∗ n (2 n ) A ∗ n (2 n − (4)Assuming the unitarity of (4) we have P nj =1 A k (2 j − A k (2 j ) + A k (2 j ) A k (2 j − = 0 ∀ k . Note that the condition α ( λ a k ) = α ( λ a − k ) ∀ k , is equivalent to the following, which are obtained by comparing the coefficients of allterms on both sides A ∗ k (2 j − =( A k (2 j − + A k (2 j ) ) A k (2 j − + A k (2 j ) [ j − X t =1 ( A k (2 t − A k (2 t ) + A k (2 t ) A k (2 t − )+ n X t = j +1 ( A k (2 t − A k (2 t ) + A k (2 t ) A k (2 t − )] ∀ k, j (5)12 ∗ k (2 j ) =( A k (2 j − + A k (2 j ) ) A k (2 j ) + A k (2 j − [ j − X t =1 ( A k (2 t − A k (2 t ) + A k (2 t ) A k (2 t − )+ n X t = j +1 ( A k (2 t − A k (2 t ) + A k (2 t ) A k (2 t − )] ∀ k, j (6)( A k (2 j − + A k (2 j ) ) A k (2 i ) = 0 ∀ i, j, k with i = j (7)( A k (2 j − + A k (2 j ) ) A k (2 i − = 0 ∀ i, j, k with i = j (8) A k (2 i ) ( A k (2 j − + A k (2 j ) ) = 0 ∀ i, j, k with i = j (9) A k (2 i − ( A k (2 j − + A k (2 j ) ) = 0 ∀ i, j, k with i = j (10) A kp A kl A kr = 0 ∀ k, p, l, r (11)except for p = 2 j − , l = 2 j, p = 2 j, l = 2 j − ∀ j,l = 2 j − , r = 2 j, l = 2 j, r = 2 j − ∀ j,p = l, l = r,p = l = r. Moreover, the condition α ( λ a k ) α ( λ a − k ) = α ( λ a − k ) α ( λ a k ) = α ( λ e ) = λ e ⊗ Q is equivalent to the following,which are obtained by comparing the coefficients of all terms on both sides n X j =1 A kj A ∗ kj = 1 , n X j =1 A ∗ kj A kj = 1 ∀ k (12) A k (2 j − A ∗ k (2 i − = A k (2 j − A ∗ k (2 i ) = 0 ∀ i, j, k with i = j (13) A k (2 j ) A ∗ k (2 i − = A k (2 j ) A ∗ k (2 i ) = 0 ∀ i, j, k with i = j (14) A ∗ k (2 j − A k (2 i − = A ∗ k (2 j − A k (2 i ) = 0 ∀ i, j, k with i = j (15) A ∗ k (2 j ) A k (2 i − = A ∗ k (2 j ) A k (2 i ) = 0 ∀ i, j, k with i = j (16) A k (2 j − A ∗ k (2 j ) + A k (2 j ) A ∗ k (2 j − = A ∗ k (2 j − A k (2 j ) + A ∗ k (2 j ) A k (2 j − = 0 ∀ k, j (17)Then, the underlying C ∗ -algebra of Q ( Z ∗ Z · · · ∗ Z ) | {z } n copies is the universal C ∗ -algebra generated by A ij ’s satis-fying the conditions (5) to (17) and U, U t both are unitaries. Now, we will prove the following equations: n X j =1 A ij A ∗ ij = 1 , n X i =1 A ji A ∗ ji = 1 , ∀ i, j (18) A i (2 j − A i (2 k ) = A i (2 j − A i (2 k − = 0 , ∀ i, j, k with j = k (19) A i (2 j ) A i (2 k ) = A i (2 j ) A i (2 k − = 0 , ∀ i, j, k with j = k (20) A ji A ki = 0 , ∀ i, j, k with j = k (21) A ∗ i (2 j − = ( A i (2 j − + A i (2 j ) ) A i (2 j − ∀ i, j (22) A ∗ i (2 j ) = ( A i (2 j − + A i (2 j ) ) A i (2 j ) ∀ i, j (23) A i (2 j − A i (2 j ) + A i (2 j ) A i (2 j − = 0 ∀ i, j (24)Multiplying A ∗ k (2 j ) and A ∗ k (2 j − on the right side of the equations (5) and (6) respectively we can find A ∗ k (2 j − A ∗ k (2 j ) = ( A k (2 j − + A k (2 j ) ) A k (2 j − A ∗ k (2 j ) ∀ k, j (25)13 ∗ k (2 j ) A ∗ k (2 j − = ( A k (2 j − + A k (2 j ) ) A k (2 j ) A ∗ k (2 j − ∀ k, j (26)by using (13) and (14). Now adding the equations (25) and (26) we get A ∗ k (2 j − A ∗ k (2 j ) + A ∗ k (2 j ) A ∗ k (2 j − =( A k (2 j − + A k (2 j ) )( A k (2 j − A ∗ k (2 j ) + A k (2 j ) A ∗ k (2 j − ) = 0 (by using (17)). Taking the adjoint we have A k (2 j − A k (2 j ) + A k (2 j ) A k (2 j − = 0 ∀ k, j , which means (24) is satisfied. Thus, from (5) and (6) we get (22)and (23). From (24), one can easily get A i (2 j − A i (2 j ) = A i (2 j ) A i (2 j − and A i (2 j ) A i (2 j − = A i (2 j − A i (2 j ) .Then, we can conclude that all A ij ’s are normal from the equations (22) and (23). Hence, the equations(19), (20) are obtained from (13) to (16) by Proposition 2.22. Applying antipode on (19) and (20), usingProposition 2.22 we obtain (21). Moreover, (18) follows by (12), unitarity of U t and normality of A ij ’s.Now consider the universal C ∗ -algebra B generated by B ij ’s satisfying the equations (18) to (24) replac-ing A ij ’s by B ij ’s. Thus by universal property of B we always get a surjective C ∗ -morphism from B to theunderlying C ∗ -algebra of Q ( Z ∗ Z · · · ∗ Z ) | {z } n copies sending B ij to A ij .On the other hand, we want a surjective C ∗ -morphism from the associated C ∗ -algebra of Q ( Z ∗ Z · · · ∗ Z ) | {z } n copies to B sending A ij to B ij , which will give an isomorphism between B and the underlying C ∗ -algebra of Q ( Z ∗ Z · · · ∗ Z ) | {z } n copies . Now, we have equations (18) to (24) with A ij ’s replaced by B ij ’s. We input B ij , B ∗ ij inthe matrix (4) instead of A ij , A ∗ ij , call it ˜ U . Observe that each B ij is normal from (22) to (24). Using theequations (18) to (24) and normality of B ij ’s we get the unitarity of ˜ U and ˜ U t , hence (12) holds. Equations(7) to (11), (13) to (16) are obtained from (19) to (21) by Proposition 2.22 and (5), (6), (17) are also satisfiedusing the equations (22), (23) and (24).Then by the universal property of the associated C ∗ -algebra of Q ( Z ∗ Z · · · ∗ Z ) | {z } n copies , we get a surjective C ∗ -morphism from Q ( Z ∗ Z · · · ∗ Z ) | {z } n copies to B sending A ij to B ij , completing the proof of the claim that theunderlying C ∗ -algebra of Q ( Z ∗ Z · · · ∗ Z ) | {z } n copies is the universal C ∗ -algebra generated by A ij ’s satisfying theequations (18) to (24).Now, consider the transpose of the matrix (4). We denote the entries by v ij . From the co-associativitycondition we can easily deduce the co-product given by ∆( v ij ) = Σ nk =1 v ik ⊗ v kj .Recall the quantum group Q ( Z ) from [11]. The underlying C ∗ -algebra associated to Q ( Z ) is the universal C ∗ -algebra generated by two elements u and v satisfying the following relations: uu ∗ + vv ∗ = 1 , uv + vu = 0 ,u ∗ = ( u + v ) u, v ∗ = ( u + v ) v. Moreover, Q ( Z ) ∗ w C ( S + n ) is the universal C ∗ -algebra C ∗ { U i − , U i , t ij | i = 1 , · · n and j = 1 , · · ( n − , n } satisfying the following conditions: U i − U ∗ i − + U i U ∗ i = 1 , U i − U i + U i U i − = 0 , ∀ iU ∗ i − = ( U i − + U i ) U i − , U ∗ i = ( U i − + U i ) U i , ∀ it ij = t ij = t ∗ ij , X i t ij = X j t ji = 1 ,t ij t ik = 0 , t ji t ki = 0 ∀ i, j, k with j = k,U i − t ij = t ij U i − , U i t ij = t ij U i ∀ i, j. Its coproduct is given by ∆ ′ ( U i − ) = U i − ⊗ U i − + U ∗ i ⊗ U i , ′ ( U i ) = U i ⊗ U i − + U ∗ i − ⊗ U i , ∆ ′ ( t ij ) = n X l =1 t il ⊗ t lj . It is clear from the description of Q ( Z ∗ Z · · · ∗ Z ) | {z } n copies as the universal C ∗ -algebra generated by A ij ’s subjectto (18)-(24) that we can define a C ∗ -morphism η from ( C ∗ { A ij | i = 1 , · · n and j = 1 , · · (2 n − , n } , ∆) to( C ∗ { U i − , U i , t ij | i = 1 , · · n and j = 1 , · · ( n − , n } , ∆ ′ ) given by A j (2 i − U i − t ij ,A j (2 i ) U i t ij ∀ i, j. Conversely, we can define a C ∗ -morphism η ′ from ( C ∗ { U i − , U i , t ij | i = 1 , · · n and j = 1 , · · ( n − , n } , ∆ ′ )to ( C ∗ { A ij | i = 1 , · · n and j = 1 , · · (2 n − , n } , ∆) given by U i − Σ nj =1 A j (2 i − ,U i Σ nj =1 A j (2 i ) ,t ij A j (2 i − A ∗ j (2 i − + A j (2 i ) A ∗ j (2 i ) . It is easy to see that η ′ ◦ η = id Q ( Z ∗ Z · · · ∗ Z ) | {z } n copies , η ◦ η ′ = id Q ( Z ) ∗ w C ( S + n ) . In fact, η and η ′ are CQGisomorphisms. This completes the proof. ✷ Remark 4.2
The quantum groups H + s ( n, , K + n can be described in a similar way. For finite s > H + s ( n, ∼ = [ C ∗ ( Z s ) ⊕ C ∗ ( Z s )] ∗ w C ( S + n ) , and K + n ∼ = [ C ∗ ( Z ) ⊕ C ∗ ( Z )] ∗ w C ( S + n ) , where [ C ∗ ( Z s ) ⊕ C ∗ ( Z s )] and [ C ∗ ( Z ) ⊕ C ∗ ( Z )] admit a CQG structure as in [11]. These facts can be provedby essentially the same arguments of Theorem 4.1. Corollary 4.3
Using the Theorem 4.1, Remark 4.2 and the result of [7] we can conclude that for every finites, Q ( Z s ∗ Z s · · · ∗ Z s ) | {z } n copies ∼ = Q ( Z s ) ∗ w C ( S + n ) . Remark 4.4
If we consider
Γ = Z n ∗ Z n where n is finite, then Q (Γ) is doubling of the quantum group Q ( Z n ) ⋆ Q ( Z n ) . In particular for n = 2 , Q (Γ) becomes doubling of the group algebra as Q ( Z ) ∼ = ( C ∗ ( Z ) , ∆ Z ) and C ∗ ( Z ) ⋆ C ∗ ( Z ) ∼ = C ∗ ( Z ∗ Z ) . (Γ , S ) for which Q (Γ) ∼ = D θ ( C ∗ (Γ)) We already mentioned in Subsection 2.4 that, if there exists a non trivial automorphism of order 2 whichpreserves the generating set, then D θ ( C ∗ (Γ)) ([22], [16]) will be always a quantum subgroup of Q (Γ). In[11], [16], [23] the authors could show that Q (Γ) coincides with doubled group algebra for some examples.In Section 5 of [19] together with Goswami we also gave few examples of groups where this happens. Ouraim in this section is to give more examples of such groups.15 .1 Z ⋊ Z The above group has a presentation Γ = < h, g | o ( g ) = 9 , o ( h ) = 3 , h − gh = g > .Using Lemma 5 . A B B ∗ A ∗ G H H ∗ G ∗ . Now the action is defined as, α ( λ h ) = λ h ⊗ A + λ h − ⊗ B,α ( λ h − ) = λ h ⊗ B ∗ + λ h − ⊗ A ∗ ,α ( λ g ) = λ g ⊗ G + λ g − ⊗ H,α ( λ g − ) = λ g ⊗ H ∗ + λ g − ⊗ G ∗ . First we are going to show that B = 0.We have α ( λ gh ) = α ( λ hg ) , α ( λ g ) = λ g ⊗ G + λ h ⊗ H as GH = HG = 0. Equating all the terms of α ( λ gh ) = α ( λ hg ) on both sides we deduce, GA = AG , HA = AH , GB = HB = BG = BH = 0 . Thus, B = ( G ∗ G + H ∗ H ) B = 0 as ( G ∗ G + H ∗ H ) = 1 , GB = HB = 0.This gives the following reduction : A A ∗ G H H ∗ G ∗ . Moreover, using the relations between the generators one can find A ∗ GA = G , A ∗ HA = H , A ∗ G = G A ∗ , A ∗ H = H A ∗ . Now using the above relations we can easily show that G ∗ G, H ∗ H are central projections of the desiredalgebra, hence Q (Γ , S ) is isomorphic to D θ ( C ∗ (Γ)) by Proposition 2.20, with the automorphism g g − , h h . ✷ ( Z ∗ Z ) × Z The group is presented as Γ = < a, b, c | ba = ab, bc = cb, a = b = c = e > .Here S = { a, b, c } . The action is given by, α ( λ a ) = λ a ⊗ A + λ b ⊗ B + λ c ⊗ C,α ( λ b ) = λ a ⊗ D + λ b ⊗ E + λ c ⊗ F,α ( λ c ) = λ a ⊗ G + λ b ⊗ H + λ c ⊗ K. Write the fundamental unitary as A B CD E FG H K . Our aim is to show D = B = F = H = 0. 16pplying α ( λ a ) = λ e ⊗ Q and comparing the coefficients of λ ac , λ ca on both sides we have AC = CA = 0.Using the antipode one can get AG = GA = 0.Applying the same process with b, c we can deduce, DF = F D = BH = HB = 0 ,GK = KG = CK = KC = 0 . Further, using the condition α ( λ ab ) = α ( λ ba ) comparing the coefficients of λ ac , λ ca on both sides we can get AF = DC, CD = F A . Applying κ we have HA = GB, AH = BG . Proceeding the same argument with α ( λ cb ) = α ( λ bc ) one can find, DK = GF, KD = F G,KB = HC, BK = CH.
Again we have, GH + HG = 0 from α ( λ a ) = λ e ⊗ Q comparing the coefficient of λ ab on both sides. Now AHG = BG as we know AH = BG . Further we have − AGH = BG as GH = − HG . Thus we get BG = 0 as AG = 0. Similarly it can be shown that BK = 0, obviously BH = 0 as BH = 0. Hence, B = B ( G + H + K ) = 0 as ( G + H + K ) = 1. This gives D = 0 by using the antipode.Now HA = 0 , HC = 0 as we get before HA = GB, HC = KB . This implies H = H ( A + C ) = 0 , andapplying the antipode F = 0.Thus the fundamental unitary is reduced to the form A C E G K (27)It now follows from Proposition 2.20 that Q (Γ) ∼ = D θ ( C ∗ (Γ)) with respect to the automorphism a c, c a, b b . ✷ Remark 5.1
The above CQG can be identified with Q ( Z ∗ Z ) ˆ ⊗ Q ( Z ) , which is clear from the form offundamental unitary (27) after reduction. The group is presented as Γ = < a, t | a = [ t m at − m , t n at − n ] = e > where m, n ∈ Z .Fundamental unitary is of the form A B CD E FD ∗ F ∗ E ∗ . Now the aim is to show B = C = D = 0.Using the condition α ( λ a ) = α ( λ e ) = λ e ⊗ Q we deduce D = 0, this implies B = C = 0 applying theantipode. Further, we know DD ∗ + EE ∗ + F F ∗ = 1, which gives us DEE ∗ + DF F ∗ = D as D = 0. If wecan show DE = DF = 0 then we will be able to prove our first claim i.e, D = 0.Using group relations we deduce t ( m − n ) at − ( m − n ) a = at ( m − n ) at − ( m − n ) [where m, n ∈ Z ]. In particular, t − ata = atat − , tat − a = atat − , which gives us at = tat − ata . Now using the condition α ( λ at ) = α ( λ tat − ata ) comparing the coefficient of λ t on both sides we have BE = 0 because there are no terms withcoefficient λ t on the right hand side as D = BF = BE ∗ = F B = E ∗ B = 0. Applying the antipode onecan get DE = 0. Similarly, using the relation α ( λ at − ) = α ( λ t − atat − a ) following the same argument wecan deduce BF ∗ = 0 , DF = 0. Hence, we get D = 0. This gives us B = C = 0 using the antipode. Thus,the fundamental unitary is reduced to the form A E F F ∗ E ∗ . AE = EAE ∗ AEA, E ∗ A = AE ∗ AEAE ∗ , AE ∗ = E ∗ AEAE ∗ A. Thus we have,
AEE ∗ = EAE ∗ AEAE ∗ = E ( AE ∗ AEAE ∗ ) = EE ∗ A, hence EE ∗ is a central projection. Similarly, F F ∗ is a central projection. Now we can define the mapfrom C ∗ { A, E, F } to C ∗ (Γ) ⊕ C ∗ (Γ) such as A ( λ a ⊕ λ a ) , E ( λ t ⊕ , F (0 ⊕ λ t − ). This givesthe isomorphism between these two algebras, which is also a CQG isomorphism and by Proposition 2.20corresponding to the automorphism a a, t t − we can conclude that Q (Γ) ∼ = D θ ( C ∗ (Γ)) . ✷ Acknowledgements :
I would like to thank Debashish Goswami and Jyotishman Bhowmick for usefuldiscussions. I would also like to thank the anonymous referee for pointing out some mistakes in the olderversion of the paper.
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