Cocycle twisting of semidirect products and transmutation

. We apply Majid’s transmutation procedure to Hopf algebra maps H → C [ T ], where T is a compact abelian group, and explain how this construction gives rise to braided Hopf algebras over quotients of T by subgroups that are cocentral in H . This allows us to unify and generalize a number of recent constructions of braided compact quantum groups, starting from the braided SU q (2) quantum group, and describe their bosonizations.


Introduction
Assume that H and K are Hopf algebras together with Hopf algebra maps This setting was considered in 1985 by Radford in his paper The structure of Hopf algebras with a projection, [Rad85].He explained that this data is equivalent to having an object A in the category of Yetter-Drinfeld modules over K satisfying certain conditions reminiscent of a Hopf algebra.This object is often not a genuine Hopf algebra, because the algebra structure considered on A ⊗ A involves the braiding on the Yetter-Drinfeld modules.Today, following Majid, it is common to call A a braided Hopf algebra.It is an example of a Hopf algebra object in a braided monoidal category.
As for genuine Hopf algebras, there is a Tannaka-Krein type reconstruction theorem for braided Hopf algebras [Maj95,Chapter 9].That is, under certain representability conditions, a monoidal functor C → D, where C is rigid and D is braided, gives rise to a Hopf algebra object.A particular example of this is when the functor is induced from a map π : H → K, where H is a Hopf algebra and K is a coquasitriangular Hopf algebra: the natural monoidal functor between the categories of comodules gives rise to a braided Hopf algebra, which Majid calls a transmutation of H, [Maj93].
More recently, there have been a number of constructions of braided compact quantum groups, [Ans+22], [BJR22], [Kas+16], [MR22].For the authors of the present paper they showed up "in nature" through a connection with C * -algebras arising from certain subproduct systems, [HN22].In particular, a Cuntz-Pimsner type algebra associated to the noncommutative polynomial X 1 X 2 − qX 2 X 1 turned out to be the C * -algebra of continuous functions on the braided SU q (2) quantum group, constructed in [Kas+16].Although the analytic/C * -algebraic aspects are therefore important to us, compactness nevetherless allows one to treat an essential part of the theory purely algebraically.
In this paper our starting point is a map π : where H is a Hopf * -algebra and T is a compact abelian group.We observe that the resulting transmutation may be viewed as a braided Hopf * -algebra over the quotients of T by subgroups T 0 that are cocentral in H. Using a theorem due to Majid we describe the corresponding Hopf algebra with projection, called bosonization, in terms of 2-cocycle twists of C[T /T 0 ] ⋉ H.This allows us to treat a number of examples in a unified and efficient way.
For the rest of this subsection we assume that H is given a unitary coquasitriangular structure R. Then M H has a braiding given by M ⊗ N → N ⊗ M, m ⊗ n → R(m (2) , n (2) )n (1) ⊗ m (1) .
There is also an induced right H-module structure on M given by m ⊳ h = R(m (2) , h)m (1) , h ∈ H, m ∈ M. (1.1) We denote by Alg * (H) the category of right H-comodule * -algebras.In other words, an object in Alg * (H) is a * -algebra A and a right H-comodule such that the map δ A : A → A ⊗ H is a * -homomorphism.For A, B ∈ Alg * (H) we denote by A ⊗ R B the * -algebra with underlying vector space A ⊗ B equipped with the product and the * -structure (1.2) The braided tensor product ⊗ R turns Alg * (H) into a monoidal category Alg * (H, R) with equivariant * -homomorphisms as morphisms.
Considering H as an H-comodule * -algebra with the coaction given by the coproduct, we recover the smash product (with respect to the right action (1.1)): Let Hopf * (H, R) denote the category of Hopf * -algebras internal to the braided monoidal category (M H , R).An object A ∈ Hopf * (A, R) is thus an H-comodule * -algebra together with H-comodule maps ∆ A : A → A ⊗ R A, S A : A → A, ε A : A → C, which are required to fit in commutative diagrams analogous to those defining Hopf * -algebras.
An object in Hopf * (H, R) is called a braided Hopf * -algebra, and it is usually not a genuine Hopf * -algebra.It is, however, always closely related to one: That H#A is a Hopf algebra, is a special case of results of Radford [Rad85], who proved that a Hopf algebra object in the category of Yetter-Drinfeld-modules is equivalent to a Hopf algebra with projection.
Let A be a braided Hopf * -algebra.We will only consider A-comodules internal to M H . Thus the notion of an A-comodule will be reserved for triples (M, δ M , γ M ), where δ M : M → M ⊗ H is an H-comodule and γ M : M → M ⊗ A is a morphism of H-comodules that defines a comodule for the coalgebra A in the usual sense.We record the following well-known result: Then the category of A-comodules is isomorphic to the category of (H#A)-comodules through the assignment

The inverse is given by
We will say that an A-comodule is unitary, if the corresponding (H#A)-comodule is unitary.
Assume that π : H → K is a map of Hopf * -algebras, where K has coquasitriangular structure R. Then there is an induced coquasitriangular structure on H given by π * R := R • (π ⊗ π).Then the strict monoidal functor 1.2.Twisting and transmutation.Assume that J : H ⊗ H → C is a Hopf 2-cocycle.This means that J is convolution invertible and satisfies Given J we can define a convolution invertible element u : The following is the well-known twisting procedure for Hopf * -algebras.
Proposition 1.3.Let J be a unitary Hopf 2-cocycle on H.There is a Hopf * -algebra J H J −1 having the same coalgebra structure as H and new product, antipode and involution defined by Next, we consider a way to produce braided Hopf algebras, due to Majid [Maj93].Recall that the adjoint comodule for H is given by ad : Let us for the moment forget about the * -structure on H.The process of transmutation produces a braided Hopf algebra from the H-comodule (H, ad).
Proposition 1.4 ([Maj93, Theorem 4.1]).Assume H is a Hopf algebra with a coquasitriangular structure R : H ⊗ H → C. Then H with the same coalgebra structure and new product • R and antipode S R , given by defines a braided Hopf algebra over H, where the comodule structure is given by ad.
Note that even though the coproduct is not changed, it is now considered as an algebra map Assume now that π : H → K is a map of Hopf algebras, where K has a coquasitriangular structure R. Then the transmutation H π * R constructed above is considered as a braided Hopf algebra over K with the restricted coaction (1.4) We will simply write H R := H π * R if the map π is clear from the context.The next result relates the bosonization of H R to a Hopf algebra with tensor product coalgebra structure.
Theorem 1.5 ( [Maj95]).Assume that π : H → K is a map of Hopf algebras and R : K ⊗K → C is a coquasitriangular structure on K. Let K ⊲⊳ R H denote the Hopf algebra that coincides with K ⊗ H as a coalgebra, but has the twisted product , defines an isomorphism of Hopf algebras.
Proof.The result follows from [Maj95,Theorem 7.4.10]by applying π to appropriate factors.
Note that K ⊗C1 is a Hopf subalgebra of K ⊲⊳ R H. Therefore K ⊲⊳ R H is a Hopf algebra with projection K ⊲⊳ R H → K, k ⊗ h → kπ(h), and then the transmuted braided Hopf algebra H R can be viewed as a particular case of the construction of Radford [Rad85].
Remark 1.6.The formula for the product on K ⊲⊳ R H is the same as for the cocycle twisting by but the element J is not a Hopf 2-cocycle on K ⊗ H in general.It becomes a 2-cocycle, when K is cocommutative.Note also that if K is both commutative and cocommutative, then H is a K-comodule algebra with respect to the adjoint coaction and then as an algebra K ⊲⊳ R H coincides with K#H = K ⊗ R H.

Transmutation over abelian groups
Let (H, ∆, S, ε) be a Hopf * -algebra and T be a compact abelian group.We write T for the dual discrete group and C T for the group Hopf * -algebra of T .We will frequently identity C T with the function algebra C[T ], and will use the latter notation when it is natural to focus on the compact group T .Throughout this section we assume that we are given a Hopf * -algebra map π : H → C T .

Braided Hopf algebras over quotients of
A consequence of coassociativity is then that for any x ∈ H we have We use the shorthand notation x a,b (1) ⊗ x b,c (2) to mean the part of the sum . Consider a closed subgroup T 0 ⊂ T , with the corresponding restriction map q : We say that T 0 is H-cocentral, or that the map qπ is cocentral, if the induced adjoint coaction is trivial.In other words, H a,b = 0 whenever q(a) = q(b).This condition implies that if we view C[T /T 0 ] as a Hopf * -subalgebra of C[T ], then so that (H, ad π ) can be viewed as a C[T /T 0 ]-comodule.Denote by ∆(T 0 ) the diagonal subgroup in It follows that the spaces of right and left coinvariants coincide, and define a Hopf * -subalgebra of C[T ] ⊗ H.
In the present setting we observe that H equipped with ad π is an object in Hopf * (C[T ], ε ⊗ ε) without any modifications.Moreover, by (2.2) it can also be viewed as an object in Hopf * (C[T /T 0 ], ε ⊗ ε).The corresponding bosonization C[T /T 0 ] ⋉ H is just the tensor product * -algebra with the coproduct defined by (1.3) (in other words, as a coalgebra, it is the smash coproduct of C[T /T 0 ] and H): for a ∈ T /T 0 ⊂ T and x ∈ H b,c .
Proposition 2.1.Consider H as a C[T ]-comodule Hopf * -algebra under the adjoint coaction ad π .Then is an isomorphism of Hopf * -algebras.Moreover, Θ restricts to an isomorphism of Hopf * - Proof.That Θ is an isomorphism is easily verified, but except for the * -structure it is also a special case of Theorem 1.5.As Θ(a ⊗ H b,c ) = ab ⊗ H b,c and C[T /T 0 ] is spanned by a ∈ T such that q(a) = 1, the second part of the proposition follows from (2.3).
Remark 2.2.Let G be a compact group with a closed abelian subgroup T .Assume that T 0 is a closed subgroup of T ∩ Z(G).As T 0 acts trivially under the conjugation action (t, g) → tgt −1 , we have an induced action on G by the quotient T /T 0 .Then the map is a group isomorphism.The above result can be seen as a generalization of this.♦ Next, fix a bicharacter β : T × T → T. This simply means that, for all a, b in T , We will write β also for the extension of the bicharacter to a linear map C T ⊗ C T → C.This defines a unitary coquasitriangular structure on C[T ] = C T , and thus also on H through the map π.We want to understand the corresponding transmutation The product on H β is determined by the coproduct and counit remain unchanged, while the antipode is determined by (2.5) In the present setting we may also introduce a * -structure on H β : Lemma 2.3.The formula turns H β into a braided Hopf * -algebra.
Using formula (2.4), it is also easy to see that (x and y ∈ H b,c , then by (1.2) we have That ∆ preserves the new * -structure is thus a consequence of (2.1).

By (2.2) we thus get the following:
Proposition 2.4.For any H-cocentral closed subgroup T 0 ⊂ T , the transmutation H β can be viewed as an object in The following is the main result of this section.
Theorem 2.5.Given a bicharacter Then, for any H-cocentral closed subgroup T 0 ⊂ T , we have Hopf * -algebra isomorphisms More pedantically, by restriction . This is the cocycle we use to define where the latter Hopf algebra is defined as in Theorem 1.5.This immediately gives the first Hopf algebra isomorphism for trivial T 0 .The isomorphism is readily verified to be * -preserving, using that by (1.2) and Definition 1.1 the involution on C[T ]#H β is given by For the same reason as in Proposition 2.1, for every defines by restriction the first isomorphism in the formulation of the theorem.The second isomorphism follows from this and again Proposition 2.1.
It is well-known that 2-cocycle twisting preserves the monoidal categories of comodules.We thus get the following: is monoidally equivalent to the subcategory of Vect T f ⊠M H f generated by the homogeneous components of bi-degree (a, b) such that q(a)b = 1.♦ From Theorems 1.5 or 2.5 we see that we have a Hopf * -algebra inclusion (2.7) This map induces a monoidal functor from the category H-comodules to the category of H βcomodules.It will be convenient to have the following description of this functor.
On the other hand, H is also a C[T ]-comodule coalgebra, so its coalgebra structure can be twisted by a 2-cocycle ω ∈ Z 2 ( T ; T): As a consequence of the following lemma, this always gives an isomorphic comodule coalgebra.
Lemma 2.9.Assume ω ∈ Z 2 ( T ; T) is a normalized cocycle (so ω(1, 1) = 1) and γ : T × T → T is a function.Then the identity holds for all a, b, c ∈ T if and only if for all a, b ∈ T, where f, g : T → T are arbitrary functions such that f (a)g(a) = ω(a, a −1 ), a ∈ T .
Proof.Assume (2.8) holds.Letting c = 1 we get Therefore (2.9) holds with f (a) = γ(a, 1) and g(b is satisfied by (2.9), since by letting a = b = c in (2.8) we see that γ(a, a) = 1 (recall also that ω(a, a −1 ) = ω(a −1 , a) by the cocycle identity).Conversely, assume (2.9) holds for some functions f, g : T → T. Then Therefore (2.8) holds if and only if Using the cocycle identity twice, the left hand side equals where the last identity holds, since ω is normalized.Thus, identity (2.8) holds if and only if f (c)g(c) = ω(c, c −1 ).
From this we see that up to an isomorphism H β can be obtained in many different ways by simultaneously twisting the product and coproduct on H: Proposition 2.10.Given a bicharacter β : T × T → T and a normalized 2-cocycle ω ∈ Z 2 ( T ; T), choose a function γ : T × T → T satisfying (2.8) and define a 2-cocycle Ω ∈ Z 2 ( T × T ; T) by
Then H Ω,ω coincides with the braided Hopf algebra defined in [BS19].

Braided compact matrix quantum groups.
In our examples we will mainly be interested in transmutations of compact quantum groups.A compact quantum group G is a Hopf * -algebra C[G] that is spanned (equivalently, generated as a * -algebra) by matrix coefficients of finite dimensional unitary comodules.We refer the reader to [NT13] for an introduction to the subject and we will often use the terminology there.For instance, a C[G]-comodule will sometimes be called a representation of G.
Recall that fixing a basis in the underlying vector space of an m-dimensional H-comodule defines a corepresentation matrix for H.This is a matrix (2.10) Conversely any such matrix defines an m-dimensional H-comodule δ U : M → M ⊗ H, by setting for a fixed vector space M with basis (e i ) i .If U is unitary, then the conjugate corepresentation matrix is Ū = (u * ij ) i,j .Definition 2.12.
is equivalent to a unitary corepresentation matrix.The coproduct ∆, counit ε and antipode S are then given by The matrix U is called the fundamental unitary for the compact matrix quantum group.
Next we want to introduce a braided analogue of this definition, but first we need some preparation.Suppose that A ∈ Hopf * (K, R) for a Hopf * -algebra K with a unitary coquasitriangular structure R.
We can still define δ U : M → M ⊗ A as in (2.11) and this gives a comodule for the coalgebra A. By definition, if Z ∈ Mat m (K) is a corepresentation matrix, the triple (M, δ Z , δ U ) defines an A-comodule if and only if , where δ M ⊗A denotes the tensor product comodule in M K .We have the following characterization: Lemma 2.13.In the above setting, the pair (Z, U ) defines an A-comodule if and only if (2.12) where δ : A → A ⊗ K is the coaction by K on A. Furthermore, the A-comodule we thus get is unitary (that is, the corresponding (K#A)-comodule is unitary) if and only if U and Z are unitary, and then the conjugate comodule is given by the pair ( Z, ŪZ ), where (2) kj for 1 ≤ s, j ≤ m, where we write δ(a) = a (1) ⊗ a (2) , a ∈ A. Multiplying both sides by S(z is ) and summing over s yields This implies the first statement.
For the second one, note that the A-comodule defined by (Z, U ) corresponds to the (K#A)comodule given by the matrix Using that ι#ε A : K#A → K is a * -homomorphism, it is easy to see that W is unitary if and only Z and U are unitary.
When U is unitary, the equality S A (u ij ) = u * ji holds by the antipode identity.The claim about the conjugate comodule follows from the * -structure on K#A and the fact that ŪZ = (ε K #ι)( W ). Alternatively, we can use the formula for the antipode in Definition 1.1 to get As S A (u jr ) = u * rj , we recover (2.13).
We remark that it is important to keep track of both R and Z in the definition of ŪZ above.However, we stick to the notation ŪZ for the rest of the paper, as R will always be given by a fixed bicharacter β.
Definition 2.14.Let T be a compact abelian group with a fixed unitary corepresentation matrix Z ∈ Mat m (C[T ]) and a bicharacter β on T .A braided compact matrix quantum group over the triple (T, Z, β) is an object A ∈ Hopf * (C[T ], β) generated as a * -algebra by elements u ij , 1 ≤ i, j ≤ m, such that, for U = (u ij ) i,j , (i) (Z, U ) defines a unitary A-comodule; (ii) ( Z, ŪZ ) defines a unitarizable A-comodule.
We say that U is the fundamental unitary for A, while the pair (Z, U ) is the fundamental unitary representation.
More explicitly, by Lemma 2.13, conditions (i) and (ii) mean that U ∈ Mat m (A) is unitary, there is F ∈ GL m (C) such that both F ZF −1 and F ŪZ F −1 are unitary, and the structure maps for A satisfy the following properties: the coaction of C[T ] on A is given by (2.12), and Proposition By Lemma 2.8 this implies that the pair (Z, U ) defines a unitary C[G] β -comodule, with the corresponding (C[T ]#C[G] β )-comodule given by the unitary φ(U ) is unitarizable, by Lemma 2.13 we see that both conditions (i) and (ii) in Definition 2.14 are satisfied.
It remains to check that C[G] β is generated by the matrix coefficients of U as a * -algebra.This becomes clear if we work in an orthonormal basis where Z is diagonal, as then the products of the elements u ij and their adjoints in C[G] and C[G] β coincide up to phase factors.
Remark 2.17.Even though C[G] β has fundamental representation (π(U ), U ), condition (2.12) can be satisfied for another pair (Z ′ , U ), which is then also a fundamental representation.A particularly interesting situation is when The first claim is obvious from condition (2.12).The second claim is easy to check in an orthonormal basis where Z is diagonal, in which case it follows immediately from (2.13).More conceptually, one can check that for the corepresentation matrix W = ( k z ik #u kj ) i,j for C[T ]#A we have (w#1)W = XW (w#1)X −1 , where X = (β(z ij , w)) i,j , which is a matrix commuting with Z.This implies that ŪwZ = X ŪZ X−1 .It remains to observe that β(x, w) = β(x * , w) for all x ∈ C[T ] to see that X = D.

Examples: transmuting matrix quantum groups
Before we embark on the examples, we remark that in a number of recent papers (see, e.g., [Ans+22; BJR22; Kas+16; MR22]) braided quantum groups are constructed in a C * -algebraic setting.However, the corresponding bosonizations are C * -algebraic compact quantum groups, and these always have dense * -subalgebras of matrix coefficients, which leads to purely algebraic results.Conversely, in our examples the bosonizations will be compact quantum groups by Theorem 2.5 (as unitary cocycle twisting preserves compactness) or Proposition 2.15, and hence they can completed to C * -algebraic compact quantum groups.We can therefore go back and forth between the * -algebraic and C * -algebraic settings.Below we will not dwell on the specific details of this but rather stick to the algebraic picture.
3.1.Braided SU q (2).Fix q > 0 and recall that H := C[SU q (2)] is the universal * -algebra with generators α and γ subject to the relations Then H is a Hopf * -algebra with coproduct

Consider the map
Under the identification Z = T, we have and the restricted right adjoint coaction ad π is determined by For λ ∈ T define a bicharacter on Z = T by β λ (m, n) = λ −mn .To find relations in the transmutation H λ = H β λ we write a • b = a • β λ b and a * λ = a * β λ .Then, by (2.4) and (2.6), Defining q ′ = qλ 2 we get the following relations in H λ : It is not difficult to see that these relations completely describe the transmuted algebra, in the next subsection we will prove a more general result.The coproduct remains unchanged, so we have These formulas are the same as for the braided quantum group SU q ′ (2) constructed in [Kas+16], modulo a small but important nuance.By Proposition 2.4, H λ can be viewed as a braided quantum group over different tori.Namely, we see that H λ can be viewed as a braided compact matrix quantum group over both triples (T, z 0 0 z −1 , β λ ) and (T/T 0 , where Therefore we can consider H λ as a braided compact matrix quantum group over the triple This is the braided quantum group Finally, let us consider the bosonizations.By Theorem 2.5 we have where J((m, n), (m ′ , n ′ )) = λ nm ′ .In other words, the bosonization is a cocycle twist of the compact quantum group T × SU q (2).
On the other hand, by the same theorem, the bosonization C[w, w −1 ] # H λ is a cocycle twist of (T × SU q (2))/∆(T 0 ).It is easy to see that the latter quantum group is isomorphic to U q (2), similarly to the classical isomorphism and therefore its cocycle twist must be one of the quantum deformations of U (2) studied in [ZZ05], cf.[Kas+16].
3.2.Braided free orthogonal quantum groups.Let m ≥ 2 be a natural number, F ∈ GL m (C) and assume that F F = ±1.Let C[O + F ] be the universal * -algebra generated by elements u ij , 1 ≤ i, j ≤ m, subject to the relations The Hopf * -algebra structure on C[O + F ] is defined as in Definition 2.12, and O + F is called a free orthogonal quantum group.
Let T be a compact abelian group and Z ∈ Mat m (C[T ]) be a unitary corepresentation matrix satisfying F ] β is a braided compact matrix quantum group over (T, Z, β) with fundamental unitary U = (u ij ) i,j .As a * -algebra, it is a universal unital * -algebra with generators u ij satisfying the relations where ŪZ = (ū Z ij ) i,j and ūZ ij = s,l,t β(z * tj z sl , z * il )u * st .
Proof.For the purpose of this proof let us denote the fundamental unitary of O + F by V = (v ij ) i,j and write A for C[O + F ] β .The first claim follows from Proposition 2.16.Relations (3.1) are obtained by considering, as in the proof of that proposition, the Hopf * -algebra map φ : It remains to show that as a * -algebra A is completely described by relations (3.1).Consider a universal unital * -algebra Ã with generators ũij satisfying these relations, and let ρ : Ã → A be the * -homomorphism such that ρ(ũ ij ) = u ij .
Working in a basis where Consider the smash product Define linear maps ψ : Then ψ = (ι#ρ) ψ.The map ψ is a linear isomorphism, e.g., by Theorem 2.5.On the other hand, the map ψ is surjective, which becomes particularly clear if we work in a basis where Z is diagonal and therefore ψ(v ij ) = z ii #ũ ij .(Alternatively, we can observe that ψ defines a 6, and its image contains the elements 1#ũ ij .)It follows that ψ is a linear isomorphism and hence ρ is an isomorphism as well.
Next, we want to change the perspective on the braided free orthogonal quantum groups and show how they can be associated with a larger class of matrices than F as above.
Proposition 3.2.Let A ∈ GL m (C) be a matrix such that A Ā is unitary, and choose a sign τ = ±1, with τ = 1 if m is odd.Then there are a compact abelian group T , a unitary corepresentation matrix and ACAC = τ 1, where C = (β(x * ij , w)) i,j .For every such quadruple (T, X, w, β), consider a universal unital * -algebra is a braided compact matrix quantum group over (T, X, β) with fundamental unitary U .
Proof.Assume first that a quadruple (T, X, w, β) as in the formulation indeed exists.Define F = AC.By our assumptions this matrix satisfies F F = τ 1 and, as C commutes with X, we have By universality there is a Hopf * -homomorphism π : Next we explain the existence of (T, X, w, β).By [HN21, Proposition 1.5], we can find a unitary v such that vAv t has the form If we can find a quadruple (T, X, w, β) for this matrix, then (T, v * X(•)v, w, β) is a quadruple for A. Thus, we may assume that A has the above form.We will construct T and X such that X is diagonal, so X(t) = diag(x 1 (t), . . ., x m (t)) for some characters x i .The conditions (3.2) and ACAC = τ 1 for C = diag(β(x −1 1 , w), . . ., β(x −1 m , w)) mean then that If m = 2k, these conditions can be easily satisfied for the dual T of a free abelian group with independent generators x 1 , . . ., x k , w by letting x m−i+1 = w 2 x −1 i for 1 ≤ i ≤ k.If m = 2k + 1, then τ = 1, λ k+1 = 1 and the conditions can be satisfied for the dual T of a free abelian group with independent generators x 1 , . . ., x k+1 by letting w = x k+1 and As is clear from the proof of this proposition, the braided quantum groups O X,β A lie within the class of braided free orthogonal quantum groups that we defined by transmutation.Namely, we have the following: , where F = AC.Remark 3.4.A moment's reflection shows that in the proof of Proposition 3.2 we could take a slightly smaller group T and arrange X to be faithful.Namely, if m = 2k, instead of taking w as a separate independent generator, we could let w = x j 1 for any j = 0, 1.Similarly, for m = 2k +1 we could take x k+1 = w = x j 1 for any j = 0, 1.In both cases we cannot choose groups of a smaller rank in general, since the numbers τ λ i generate a group of rank up to k = [m/2].
We remark also that if A Ā is unitary and ) is a unitary corepresentation matrix such that A(χ X)A −1 = X for a character χ, then by following the proof of [HN21, Proposition 1.5] one can show that there is a unitary v such that vAv t has form (3.3) and vXv * is diagonal.Therefore there are no other ways of constructing (T, X, w, β) than solving equations (3.4).
Remark 3.5.Once (3.2) is satisfied, condition ACAC = τ 1 can be formulated as follows.Let t w ∈ T be the element such that x(t w ) = β(x, w) for all x ∈ T , so that C = X(t w ).Then the requirement is As a prerequisite for constructing O X,β A this condition can be written as Indeed, assume (3.2) and (3.6) are satisfied.Then applying complex conjugation and conjugation by A to the last identity we get Hence c = c β(w, w) 4 and therefore τ := c β(w, w) −2 = ±1, so that (3.5) is satisfied for this τ .Note that the sign must be +1 for odd m, which becomes obvious if we choose a unitary v such that vA X(t w )v t has form (3.3).Thus, the braided compact matrix quantum groups O X,β A are defined under assumptions (3.2) and (3.6).♦ In the setting of Proposition 3.1, assume now that F ] are linearly independent, this means that the matrices Z(t), t ∈ T 0 , are scalar.Then the condition Z = F ZF −1 implies that Z(t) = ±1, so we get a character χ : T 0 → {±1}.When it is nontrivial, it defines the standard (Z/2Z)-grading on Rep O + F .It is known that the quantum group O + F is monoidally equivalent to SU q (2) for an appropriate q, and this equivalence respects the (Z/2Z)-gradings.Combining this with Remark 2.7, one can conclude that the bosonization of C Together with Corollary 3.3 this leads to the following conclusion.Proposition 3.6.In the setting of Proposition 3.2, let q ∈ [−1, 1] \ {0} be such that sgn q = −τ and |q + q −1 | = Tr(A * A).Assume T 0 ⊂ T is a closed subgroup satisfying X(t) = ±w(t)1 for all t ∈ T 0 , and let χ : T 0 → {±1} be the character such that X = χw1 on T 0 .Then the bosonization ) is a compact quantum group monoidally equivalent to (T × SU q (2))/(id ×χ)∆(T 0 ). (3.7) 3) ⇒ 1): We may assume that B is of the form (3.9).By the assumption on the spectrum there exist l i ∈ Z for i = 1, . . ., m/2 such that λ Put X = diag(z l 1 , z l 2 , ..., z lm ).Define a bicharacter β on T by β(k, l) = λ −2kl .We then have X = B XB −1 z −1 and B B = λ −1 X(λ −2 ).Consider the double cover p : T → T, t → t 2 .Denote by β 1/4 the bicharacter on T defined in the same way as β, but with λ 2 replaced by one of its fourth roots.Therefore if i : The defining relations in A say that B is an intertwiner of the A-comodules defined by (z −1 X, ŪX ) and (X, U ).In other words, for It follows that we have a well-defined Hopf * -algebra map . Using the relations in both algebras it is also easy to construct the inverse map.
Remark 3.8.The proof of the proposition implies that the procedure described in the proof of the implication 3) ⇒ 1) is the only way of getting decompositions Remark 3.9.As in the proof of the proposition, it is not difficult to see that given Take Ω = (ω ij ) i,j ∈ GL m (C) and assume , β) from this data.By [MR22, Theorem 2.6] and [BJR22, Remark 2.20(2)], in our terminology, A o (Ω, X) is defined as a universal braided compact matrix quantum group over (T, X, β) with fundamental unitary U = (u ij ) i,j subject to the relations Assume that d = 2n.Put w(t) = t n .Then we see from the description above and Remark 3.5 A ] in Hopf * (C[T], β).Next, assume that d is odd.Then, as in the proof of Proposition 3.7, m is even.Similarly to that proof, consider the double cover p : T → T, t → t 2 , to write the character z d as a square.Write β 1/4 for the bicharacter on T defined in the same way as β, but with ζ replaced by a fourth root ζ 1/4 .Define a character w on the double cover by w(t) = t d .Then where we now view the latter braided Hopf * -algebra as an object in Hopf * (C[T/{±1}], i * β 1/4 ) = Hopf * (C[T], β).By Theorem 2.5, the bosonization of C[U + F ] β is a compact quantum group that is a cocycle twist of T × U + F .When T = T and Z is such that both Z and F ZF −1 are diagonal matrices, we recover the braided free unitary quantum groups defined in [BJR22].We remark that we can of course always assume that one of the matrices Z or F ZF −1 is diagonal by choosing an appropriate orthonormal basis, but it is usually impossible to make both of them diagonal simultaneously.To describe the transmutation we want to express the relations in C[S + N ] in terms of homogeneous elements with respect to the bi-grading by Z/N Z. Fix a primitive N -th root of unity ω.By Lemma 4.9 in [Ans+22] such generators can be obtained by considering the elements a ij defined by .
The elements a ij are then bi-graded by Define a bicharacter by β(z i , z j ) = ω −ij .It is then readily verified, by using formulas (2.4) and (2.6), that the transmutation C[S + N ] β is described by the relations a 0i = a i0 = δ i,0 , a * ij = ω i(j−i) a −i,−j , a k,i+j = l ω −l(i−k+l) a k−l,i a lj , a i+j,k = l ω −i(l−j) a jl a i,k−l .
These are exactly the relations in [Ans+22, Definition 2.7].Finally, by Theorem 2.5, the bosonization of C[S + N ] β is a cocycle twist of the quantum group (Z/N Z) × S + N .
Fix a bicharacter β : T × T → T and consider the transmutation C[O + F ] β .It is natural to call it a braided free orthogonal quantum group.Proposition 3.1.The braided Hopf * -algebra C[O + sending the fundamental representation to Z = w −1 X.We claim that the corresponding transmutation C[O + F ] β satisfies all the required properties of C[O X,β A ]. Indeed, by Lemma 2.18, in C[O + F ] β we have ŪX = ŪwZ = D ŪZ D −1 , where D = (β(z * ij , w)) i,j = (β(wx * ij , w)) i,j = β(w, w)C.Then A = β(w, w)F D −1 , and the claim follows from Proposition 3.1.
B ∈ GL m (C) such that B B is unitary, a unitary corepresentation matrix X ∈ Mat m (C[T]) such that X = B XB −1 z exists if and only if m is even.Therefore, for even m, we always get a Hopf *algebra map p : C[ Õ+ B ] → C[T] such that p(d) = z with a right inverse z → d.By Radford's theorem [Rad85] we then get a Hopf * -algebra object in the braided category YD(T) of T-Yetter-Drinfeld modules.From this perspective the above proposition characterizes when this object lies in the subcategory (M C[T] , β) ⊂ YD(T) for some β.♦ Finally, let us compare the transmutations of C[O + F ] to the braided quantum groups constructed in [MR22].Fix numbers d 1 , d 2 , ..., d m , d ∈ Z and consider the representation where z ∈ C[T] is the generator.Assume further that there is ζ ∈ T such that ΩΩ = c X(ζ d ) for some c ∈ T. Define a bicharacter on T = Z by β(m, n) = ζ mn .In [MR22], Meyer and Roy construct an object A o (Ω, X) ∈ Hopf * (C[T]

3. 3 .
Braided free unitary quantum groups.Let m ≥ 2 be a natural number.We recall the definition of the free unitary quantum groupU + F .Let F = (f ij ) i,j ∈ GL m (C) be such that Tr(F * F ) = Tr((F * F ) −1 ).Then C[U +F ] denotes the universal * -algebra with generators u ij , 1 ≤ i, j ≤ m, and relations determined byU = (u ij ) i,j and F Ū F −1 are unitaries in Mat m (C[U + F ]).The Hopf * -algebra structure on C[U +F ] is defined so that U + F is a compact matrix quantum group as in Definition 2.12.Similarly to the previous example, we fix a compact abelian group T together with a unitary corepresentation matrix Z ∈ Mat m (C[T ]) such that F ZF −1 is unitary, equivalently, Z commutes with |F |.Then, by the universality of C[U + F ], there is a Hopf * -algebra map π :C[U + F ] → C[T ] mapping U to Z.Let β : T × T → T be a bicharacter, and consider the transmutation C[U + F ] β .Then, similarly to Proposition 3.1, we get the following result.Proposition 3.10.The braided Hopf * -algebra C[U + F ] β is a braided compact matrix quantum group over (T, Z, β) with fundamental unitary U = (u ij ) i,j .As a * -algebra, it is a universal unital * -algebra with generators u ij satisfying the relations U and F ŪZ F −1 are unitaries, where ŪZ = (ū Z ij ) i,j and ūZ ij = s,l,t β(z * tj z sl , z * il )u * st .

3. 4 .
Anyonic quantum permutation groups.Let N ≥ 2 be a natural number.The quantum symmetric group S + N is the universal compact matrix quantum group with fundamental unitary representation U = (u ij ) N −1 i,j=0 subject to the relationsi u ij = 1 = j u ij and u * ij = u 2 ij = u ij .We can view the cyclic group Z/N Z as a subgroup of S N ⊂ S + N , so we get a Hopf * -algebra map π :C[S + N ] → C[Z/N Z], π(u ij ) = δ j−i ,where δ k ∈ C[Z/N Z] are the usual delta-functions.
(π ⊗ ι)∆(a ij ) = z i ⊗ a ij , (ι ⊗ π)∆(u ij ) = a ij ⊗ z j , where z ∈ C[Z/N Z] is the function z(k) = ω k .In terms of the new generators the relations in C[S + N ] become a 0i = a i0 = δ i,0 , a * ij = a −i,−j , a k,i+j = l a k−l,i a lj , a i+j,k = l a jl a i,k−l .
T .There are canonical commuting left and right coactions δ L = (π ⊗ ι)∆, δ R = (ι ⊗ π)∆ by C T on H.It follows that H is bi-graded by T .More precisely, H = a,b∈ T H a,b , where by Proposition 1.2, the claim follows.2.2.Another view onH β .Motivated by the recent of work of Bochniak and Sitarz [BS19] we now give another interpretation of the structure maps for H β .Using the left and right coactions of C[T ] on H, we can view H as a C[T × T ]-comodule algebra.Then the new product • β on H β is obtained by cocycle twisting the original product by the 2-cocycle Put Z = π(U ).Recall that by (2.7) we have a Hopf * -algebra map φ 2.15.Given a compact abelian group T , a unitary corepresentation matrix Z ∈ Mat m (C[T ]) and a bicharacter β on T , the bosonization of any braided compact matrix quantum group A over (T, Z, β) is a compact quantum group.Proposition 2.16.Let G be a compact matrix quantum group with fundamental unitary U = (u ij ) m i,j=1 .Assume that T is a compact abelian group with a Hopf * -algebra map π : C[G] → C[T ], and let β be a bicharacter on T .Then the transmutation C[G] β is a braided compact matrix quantum group over (T, π(U ), β) with fundamental unitary U .Proof.
In this case we can view C[G] β as a braided compact matrix quantum group over (T /T 0 , Z ′ , i * β).♦We record a useful lemma related to the above remark.Lemma 2.18.Assume A ∈ Hopf * (C[T ], β) and take w ∈ T .Then (Z, U ) defines an Acomodule if and only if (wZ, U ) defines an A-comodule.If in addition Z and U are unitary, then we have the relation ŪwZ