Cocycle twisting of semidirect products and transmutation

Erik Habbestad Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, NO-0316 Oslo, Norway [email protected] and Sergey Neshveyev [email protected]

(Date: April 2, 2023; revised: January 11, 2024)

Abstract.

We apply Majid’s transmutation procedure to Hopf algebra maps $H\to{\mathbb{C}}[T]$ , 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.

Supported by the NFR project 300837 “Quantum Symmetry”.

Introduction

Assume that $H$ and $K$ are Hopf algebras together with Hopf algebra maps

p\colon H\to K,\quad i\colon K\to H,\quad p\circ i=\mathrm{id}_{K}.

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 $A\otimes A$ 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 $\mathcal{C}\to\mathcal{D}$ , where $\mathcal{C}$ is rigid and $\mathcal{D}$ is braided, gives rise to a Hopf algebra object in $\mathcal{D}$ . A particular example of this is when the functor is induced from a map $\pi\colon H\to K$ , where $H$ is a Hopf algebra and $K$ is a coquasitriangular Hopf algebra: the natural monoidal functor between the categories of comodules

\mathcal{F}_{\pi}\colon\mathcal{M}^{H}\to\mathcal{M}^{K},

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 $X_{1}X_{2}-\bar{q}X_{2}X_{1}$ 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 nevertheless allows one to treat an essential part of the theory purely algebraically.

In this paper our starting point is a map $\pi\colon H\to{\mathbb{C}}[T]$ , 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 ${\mathbb{C}}[T/T_{0}]\ltimes H$ . This allows us to treat a number of examples in a unified and efficient way.

The paper is organized as follows. In Section 1 we collect some facts about braided Hopf $*$ -algebras, twisting and transmutation. Section 2 contains our general results. Here we also suggest a definition of braided compact matrix quantum groups, covering in particular transmutations of compact matrix quantum groups, and discuss a connection of transmutation with a recent construction of Bochniak and Sitarz [BS19]. Section 3, which constitutes a large part of this text, consists of computing examples.

1. Generalities

We refer the reader to Majid’s book [Maj95] for more information on the notions in this section. We only consider unital algebras over the complex numbers.

1.1. Categories of comodules

Let $(H,\Delta,S,\varepsilon)$ be a Hopf $*$ -algebra. As is usual in the theory of Hopf algebras, we adopt Sweedler’s sumless notation: we write $\Delta(h)=h_{(1)}\otimes h_{(2)}$ , but remember that the expression represents a sum of simple tensors. We write $\mathcal{M}^{H}$ for the monoidal category of right $H$ -comodules. For an object $M\in\mathcal{M}^{H}$ with corresponding map $\delta_{M}\colon M\to M\otimes H$ , we write $\delta_{M}(m)=m^{(1)}\otimes m^{(2)}$ .

Recall that a coquasitriangular structure on $H$ is a linear map $R\colon H\otimes H\to{\mathbb{C}}$ which is convolution invertible, with convolution inverse $R^{-1}$ , and satisfies

R(xy,z)=R(x,z_{(1)})R(y,z_{(2)}),\quad R(x,yz)=R(x_{(1)},z)R(x_{(2)},y),

y_{(1)}x_{(1)}R(x_{(2)},y_{(2)})=R(x_{(1)},y_{(1)})x_{(2)}y_{(2)}

for all $x,y,z\in H$ . We say that $R$ is unitary if it is unitary as an element of the convolution $*$ -algebra $(H\otimes H)^{*}$ , or equivalently, as $R=R\circ(S\otimes S)$ ,

R^{-1}(x,y)=\overline{R(x^{*},y^{*})}.

For the rest of this subsection we assume that $H$ is given a unitary coquasitriangular structure $R$ . Then $\mathcal{M}^{H}$ has a braiding given by

M\otimes N\to N\otimes M,\quad m\otimes n\mapsto R(m^{(2)},n^{(2)})n^{(1)}\otimes m^{(1)}.

There is also an induced right $H$ -module structure on $M$ given by

m\triangleleft h=R(m^{(2)},h)m^{(1)},\quad h\in H,\ m\in M.

(1.1)

The right comodule and module structures are compatible in the sense that

\delta_{M}(m\triangleleft h)=m^{(1)}\triangleleft h_{(2)}\otimes S(h_{(1)})m^{(2)}h_{(3)},

so $M$ becomes a Yetter–Drinfeld module over $H$ .

We denote by $\mathrm{Alg}^{*}(H)$ the category of right $H$ -comodule $*$ -algebras. In other words, an object in $\mathrm{Alg}^{*}(H)$ is a $*$ -algebra $A$ and a right $H$ -comodule such that the map $\delta_{A}\colon A\to A\otimes H$ is a $*$ -homomorphism. For $A,B\in\mathrm{Alg}^{*}(H)$ we denote by $A\otimes_{R}B$ the $*$ -algebra with underlying vector space $A\otimes B$ equipped with the product

(a\otimes b)\cdot(a^{\prime}\otimes b^{\prime})=R(b^{(2)},a^{\prime(2)})aa^{\prime(1)}\otimes b^{(1)}b^{\prime}

and the $*$ -structure

(a\otimes b)^{*}=(1\otimes b^{*})\cdot(a^{*}\otimes 1)=R(b^{(2)*},a^{(2)*})a^{(1)*}\otimes b^{(1)*}.

(1.2)

The braided tensor product $\otimes_{R}$ turns $\mathrm{Alg}^{*}(H)$ into a monoidal category $\mathrm{Alg}^{*}(H,R)$ 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)):

H\#A=H\otimes_{R}A.

Let $\mathrm{Hopf}^{*}(H,R)$ denote the category of Hopf $*$ -algebras internal to the braided monoidal category $(\mathcal{M}^{H},R)$ . An object $A\in\mathrm{Hopf}^{*}(A,R)$ is thus an $H$ -comodule $*$ -algebra together with $H$ -comodule maps

\Delta_{A}\colon A\to A\otimes_{R}A,\quad S_{A}\colon A\to A,\quad\varepsilon_{A}\colon A\to{\mathbb{C}},

which are required to fit in commutative diagrams analogous to those defining Hopf $*$ -algebras. An object in $\mathrm{Hopf}^{*}(H,R)$ is called a braided Hopf $*$ -algebra, and it is usually not a genuine Hopf $*$ -algebra. It is, however, always closely related to one:

Definition 1.1.

The bosonization of $A\in\mathrm{Hopf}^{*}(H,R)$ is the Hopf $*$ -algebra with underlying $*$ -algebra $H\#A$ , counit $\varepsilon(h\#a)=\varepsilon_{H}(h)\varepsilon_{A}(a)$ , coproduct

\Delta(h\#a)=(h_{(1)}\#{a_{(1)}}^{(1)})\otimes(h_{(2)}{a_{(1)}}^{(2)}\#a_{(2)}),

(1.3)

and antipode

S(h\#a)=(1\#S_{A}(a^{(1)}))(S_{H}(ha^{(2)})\#1).

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 $\mathcal{M}^{H}$ . Thus, the notion of an $A$ -comodule will be reserved for triples $(M,\delta_{M},\gamma_{M})$ , where $\delta_{M}\colon M\to M\otimes H$ is an $H$ -comodule and $\gamma_{M}\colon M\to M\otimes A$ 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:

Proposition 1.2 (cf. [Maj95, Theorem 9.4.12]).

Let $A\in\mathrm{Hopf}^{*}(H,R)$ . Then the category of $A$ -comodules is isomorphic to the category of $(H\#A)$ -comodules through the assignment

(M,\delta_{M},\gamma_{M})\mapsto(M,(\delta_{M}\otimes\iota)\gamma_{M}).

The inverse is given by

(M,\delta)\mapsto(M,(\iota\otimes(\iota\#\varepsilon_{A}))\delta,(\iota\otimes(\varepsilon_{H}\#\iota))\delta).

We will say that a finite dimensional $A$ -comodule is unitary, if the corresponding $(H\#A)$ -comodule is unitary. Recall that a finite dimensional comodule $(M,\delta)$ over a Hopf $*$ -algebra $H^{\prime}$ is called unitary, if $M$ is equipped with a scalar product and

\langle\delta(m),\delta(m)\rangle=(m,m^{\prime})1\ \ \text{for all}\ \ m,m^{\prime}\in M,

where the $H^{\prime}$ -valued sesquilinear form $\langle\cdot,\cdot\rangle$ is defined by $\langle m\otimes a,m^{\prime}\otimes b\rangle=(m,m^{\prime})b^{*}a\in H^{\prime}$ .

1.2. Twisting and transmutation

Assume that $J\colon H\otimes H\to{\mathbb{C}}$ is a Hopf 2-cocycle. This means that $J$ is convolution invertible and satisfies

J(x_{(1)},y_{(1)})J(x_{(2)}y_{(2)},z)=J(y_{(1)},z_{(1)})J(x,y_{(2)}z_{(2)})

for $x,y,z\in H$ . We say that $J$ is unitary if $J^{*}=J^{-1}$ in the convolution $*$ -algebra $(H\otimes H)^{*}$ , that is,

J^{-1}(x,y)=\overline{J(S(x)^{*},S(y)^{*})}.

Given $J$ , we can define a convolution invertible element $u\colon H\to{\mathbb{C}}$ by

u(x)=J(x_{(1)},S(x_{(2)})).

The following is the well-known twisting procedure for Hopf $*$ -algebras, see, e.g., [Maj95, Theorem 2.3.4].

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

x\star_{J}y=J(x_{(1)},y_{(1)})x_{(2)}y_{(2)}J^{-1}(x_{(3)},y_{(3)}),

S^{J}(x)=u(x_{(1)})S(x_{(2)})u^{-1}(x_{(3)}),\quad x^{*_{J}}=u^{-1}(S(x_{(1)})^{*})x^{*}_{(2)}u(S(x_{(3)})^{*})=S^{J}(S^{-1}(x^{*})).

We remark that we are interested in unitary cocycles, while Majid in [Maj95, Section 2.3] considers so-called real ones. That the $*$ -structure on ${}_{J}H_{J^{-1}}$ defined above is the correct one in the unitary case is explained in [NT13, Example 2.3.9] in the context of compact quantum groups. It is not difficult to check directly that the definition works for general Hopf $*$ -algebras.

Next we consider a way to produce braided Hopf algebras, due to Majid [Maj93]. Recall that the adjoint comodule for $H$ is given by

\mathrm{ad}\colon H\to H\otimes H,\quad\mathrm{ad}(x)=x^{(1)}\otimes x^{(2)}=x_{(2)}\otimes S(x_{(1)})x_{(3)}.

Let us for the moment forget about the $*$ -structure on $H$ . The process of transmutation produces a braided Hopf algebra from the $H$ -comodule $(H,\mathrm{ad})$ .

Proposition 1.4 ([Maj93, Theorem 4.1]).

Assume $H$ is a Hopf algebra with a coquasitriangular structure $R\colon H\otimes H\to{\mathbb{C}}$ . Then $H$ with the same coalgebra structure and new product $\cdot_{R}$ and antipode $S_{R}$ , given by

x\cdot_{R}y=x_{(2)}y_{(3)}R(x_{(3)},S(y_{(1)}))R(x_{(1)},y_{(2)}),

S_{R}(x)=S(x_{(2)})R(S^{2}(x_{(3)})S(x_{(1)}),x_{(4)}),

defines a braided Hopf algebra $H_{R}$ over $(H,R)$ , where the comodule structure is given by $\mathrm{ad}$ .

Note that even though the coproduct is not changed, it is now considered as an algebra map $H\to H\otimes_{R}H$ .

More generally, assume that $\pi\colon H\to K$ is a map of Hopf algebras, where $K$ has a coquasitriangular structure $R$ . Although $\pi^{*}R=R\circ(\pi\otimes\pi)$ is not a coquasitriangular structure on $H$ in general, the same formulas as above, with $R$ replaced by $\pi^{*}R$ , define a braided Hopf algebra $H_{R}$ over $(K,R)$ with the restricted coaction

\mathrm{ad}_{\pi}=(\iota\otimes\pi)\mathrm{ad}\colon H\to H\otimes K.

(1.4)

The next result relates the bosonization of $H_{R}$ to a Hopf algebra with the tensor product coalgebra structure.

Theorem 1.5.

Assume that $\pi\colon H\to K$ is a map of Hopf algebras and $R\colon K\otimes K\to{\mathbb{C}}$ is a coquasitriangular structure on $K$ . Let $K\bowtie_{R}H$ denote the Hopf algebra that coincides with $K\otimes H$ as a coalgebra, but has the twisted product

(k\otimes h)\cdot(k^{\prime}\otimes h^{\prime})=R^{-1}(\pi(h_{(1)}),k_{(1)}^{\prime})\,kk_{(2)}^{\prime}\otimes h_{(2)}h^{\prime}\,R(\pi(h_{(3)}),k_{(3)}^{\prime}).

Then the map

K\bowtie_{R}H\to K\#H_{R},\quad k\otimes h\mapsto k\pi(h_{(1)})\#h_{(2)},

defines an isomorphism of Hopf algebras.

Proof.

For $K=H$ and $\pi=\operatorname{id}$ , this is [Maj95, Theorem 7.4.10]. The general case can be proved similarly or by a direct computation. ∎

Note that $K\otimes{\mathbb{C}}1$ is a Hopf subalgebra of $K\bowtie_{R}H$ . Therefore $K\bowtie_{R}H$ is a Hopf algebra with projection $K\bowtie_{R}H\to K$ , $k\otimes h\mapsto k\pi(h)$ , 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 $K\bowtie_{R}H$ is the same as for the cocycle twisting by

J((k\otimes h),(k^{\prime}\otimes h^{\prime}))=\varepsilon_{K}(k)\varepsilon_{H}(h^{\prime})R^{-1}(\pi(h),k^{\prime}),

but the element $J$ is not a Hopf $2$ -cocycle on $K\otimes H$ 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 $K\bowtie_{R}H$ coincides with $K\#H=K\otimes_{R}H$ .

2. Transmutation over abelian groups

Let $(H,\Delta,S,\varepsilon)$ be a Hopf $*$ -algebra and $T$ be a compact abelian group. We write $\hat{T}$ for the dual discrete group and ${\mathbb{C}}\hat{T}$ for the group Hopf $*$ -algebra of $\hat{T}$ . We will frequently identity ${\mathbb{C}}\hat{T}$ with the function algebra ${\mathbb{C}}[T]$ , 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 $\pi\colon H\to{\mathbb{C}}\hat{T}$ .

2.1. Braided Hopf algebras over quotients of $T$

There are canonical commuting left and right coactions

\delta_{L}=(\pi\otimes\iota)\Delta,\quad\delta_{R}=(\iota\otimes\pi)\Delta

by ${\mathbb{C}}\hat{T}$ on $H$ . It follows that $H$ is bi-graded by $\hat{T}$ . More precisely, $H=\bigoplus_{a,b\in\hat{T}}H_{a,b}$ , where

H_{a,b}=\{x\in H\,|\,\delta_{L}(x)=a\otimes x\mbox{ and }\delta_{R}(x)=x\otimes b\}.

A consequence of coassociativity is then that for any $x\in H$ we have

\Delta(x)=x_{(1)}\otimes x_{(2)}\in\bigoplus_{a,b,c}H_{a,b}\otimes H_{b,c}.

(2.1)

We use the shorthand notation $x_{(1)}^{a,b}\otimes x_{(2)}^{b,c}$ to mean the part of the sum $x_{(1)}\otimes x_{(2)}$ which is in $H_{a,b}\otimes H_{b,c}$ . Thus, $\Delta(x)=\sum_{a,b,c}x_{(1)}^{a,b}\otimes x_{(2)}^{b,c}$ .

Consider a closed subgroup $T_{0}\subset T$ , with the corresponding restriction map $q\colon{\mathbb{C}}[T]\to{\mathbb{C}}[T_{0}]$ . We say that $T_{0}$ is $H$ -cocentral, or that the map $q\pi$ is cocentral, if the induced adjoint coaction

\mathrm{ad}_{q\pi}=(\iota\otimes q\pi)\mathrm{ad}\colon H\to H\otimes{\mathbb{C}}[T_{0}]

is trivial. In other words,

H_{a,b}=0\quad\text{whenever}\quad q(a)\neq q(b).

This condition implies that if we view ${\mathbb{C}}[T/T_{0}]$ as a Hopf $*$ -subalgebra of ${\mathbb{C}}[T]$ , then

\mathrm{ad}_{\pi}(H)\subset H\otimes{\mathbb{C}}[T/T_{0}],

(2.2)

so that $(H,\mathrm{ad}_{\pi})$ can be viewed as a ${\mathbb{C}}[T/T_{0}]$ -comodule.

Denote by $\Delta(T_{0})$ the diagonal subgroup in $T_{0}\times T_{0}$ . It is $({\mathbb{C}}[T]\otimes H)$ -cocentral with respect to the composition of $\iota\otimes\pi\colon{\mathbb{C}}[T]\otimes H\to{\mathbb{C}}[T]\otimes{\mathbb{C}}[T]$ with the restriction map ${\mathbb{C}}[T\times T]\to{\mathbb{C}}[\Delta(T_{0})]$ . It follows that the spaces of right and left coinvariants coincide,

({\mathbb{C}}[T]\otimes H)^{\Delta(T_{0})}={}^{\Delta(T_{0})}({\mathbb{C}}[T]\otimes H)=\bigoplus_{\begin{subarray}{c}a,b,c:\\ q(ab)=1\end{subarray}}a\otimes H_{b,c}=\bigoplus_{\begin{subarray}{c}a,b,c:\\ q(ac)=1\end{subarray}}a\otimes H_{b,c},

(2.3)

and define a Hopf $*$ -subalgebra of ${\mathbb{C}}[T]\otimes H$ .

In the present setting we observe that $H$ equipped with $\mathrm{ad}_{\pi}$ is an object in $\mathrm{Hopf}^{*}({\mathbb{C}}[T],\varepsilon\otimes\varepsilon)$ without any modifications. Moreover, by (2.2) it can also be viewed as an object in $\mathrm{Hopf}^{*}({\mathbb{C}}[T/T_{0}],\varepsilon\otimes\varepsilon)$ . The corresponding bosonization ${\mathbb{C}}[T/T_{0}]\ltimes H$ 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 ${\mathbb{C}}[T/T_{0}]$ and $H$ ):

\Delta_{{\mathbb{C}}[T/T_{0}]\ltimes H}(a\otimes x)=\sum_{d}(a\otimes x^{b,d}_{(1)})\otimes(ab^{-1}d\otimes x^{d,c}_{(2)})

for $a\in\widehat{T/T_{0}}\subset\hat{T}$ and $x\in H_{b,c}$ .

Proposition 2.1.

Consider $H$ as a ${\mathbb{C}}[T]$ -comodule Hopf $*$ -algebra under the adjoint coaction $\mathrm{ad}_{\pi}$ . Then

\Theta\colon{\mathbb{C}}[T]\otimes H\to{\mathbb{C}}[T]\ltimes H,\quad\Theta(a\otimes x)=a\pi(x_{(1)})\otimes x_{(2)},

is an isomorphism of Hopf $*$ -algebras. Moreover, $\Theta$ restricts to an isomorphism of Hopf $*$ -algebras

({\mathbb{C}}[T]\otimes H)^{\Delta(T_{0})}\cong{\mathbb{C}}[T/T_{0}]\ltimes H

for any $H$ -cocentral closed subgroup $T_{0}\subset T$ .

Proof.

That $\Theta$ is an isomorphism is easily verified, but except for the $*$ -structure it is also a special case of Theorem 1.5. As $\Theta(a\otimes H_{b,c})=ab\otimes H_{b,c}$ and ${\mathbb{C}}[T/T_{0}]$ is spanned by $a\in\hat{T}$ 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 $T\cap Z(G)$ , where $Z(G)$ is the center of $G$ . As $T_{0}$ acts trivially under the conjugation action $(t,g)\mapsto tgt^{-1}$ of $T$ on $G$ , we have an induced action on $G$ by the quotient group $T/T_{0}$ . Then the map

(T/T_{0})\ltimes G\to(T\times G)/\Delta(T_{0}),\quad([t],g)\mapsto[(t,tg)],

is a group isomorphism. The above result can be seen as a generalization of this. $\diamond$

Next, fix a bicharacter $\beta\colon\hat{T}\times\hat{T}\to{\mathbb{T}}$ . This simply means that, for all $a,b,c$ in $\hat{T}$ ,

\beta(ab,c)=\beta(a,c)\beta(b,c)\quad\mbox{ and }\quad\beta(a,bc)=\beta(a,b)\beta(a,c).

We will write $\beta$ also for the extension of the bicharacter to a linear map ${\mathbb{C}}\hat{T}\otimes{\mathbb{C}}\hat{T}\to{\mathbb{C}}$ . This defines a unitary coquasitriangular structure on ${\mathbb{C}}[T]={\mathbb{C}}\hat{T}$ . We want to understand the corresponding transmutation $H_{\beta}\in\mathrm{Hopf}^{*}({\mathbb{C}}[T],\beta)$ .

By definition, the product on $H_{\beta}$ is determined by

x\cdot_{\beta}y=\beta(a^{-1}b,c^{-1})xy,\quad x\in H_{a,b},\ y\in H_{c,d},

(2.4)

the coproduct and counit remain unchanged, while the antipode is determined by

S_{\beta}(x)=\beta(a^{-1}b,b)S(x),\quad x\in H_{a,b}.

(2.5)

In the present setting we may also introduce a $*$ -structure on $H_{\beta}$ :

Lemma 2.3.

The formula

x^{*_{\beta}}=\beta(a^{-1}b,a^{-1})x^{*},\quad x\in H_{a,b},

(2.6)

turns $H_{\beta}$ into a braided Hopf $*$ -algebra.

Proof.

Using that $x^{*}\in H_{a^{-1},b^{-1}}$ for $x\in H_{a,b}$ , it is straightforward to check that $*_{\beta}$ is involutive. Using formula (2.4), it is also easy to see that $(x\cdot_{\beta}y)^{*_{\beta}}=y^{*_{\beta}}\cdot_{\beta}x^{*_{\beta}}$ . To show that $\Delta\colon H_{\beta}\to H_{\beta}\otimes_{\beta}H_{\beta}$ is $*$ -preserving, notice that if $x\in H_{a,b}$ and $y\in H_{b,c}$ , then by (1.2) we have

	$\displaystyle(x\otimes y)^{_{\beta}\,\otimes_{\beta}\,_{\beta}}$	$\displaystyle=\beta(a^{-1}b,a^{-1})\beta(b^{-1}c,b^{-1})\beta(b^{-1}c,a^{-1}b)\,x^{}\otimes y^{}$
		$\displaystyle=\beta(a^{-1}c,a^{-1})\,x^{}\otimes y^{}.$

That $\Delta$ 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 $T_{0}\subset T$ , the transmutation $H_{\beta}$ can be viewed as an object in $\mathrm{Hopf}^{*}({\mathbb{C}}[T/T_{0}],i^{*}\beta)$ , where $i\colon{\mathbb{C}}[T/T_{0}]\to{\mathbb{C}}[T]$ is the embedding map.

The following is the main result of this section.

Theorem 2.5.

Given a bicharacter $\beta\colon\hat{T}\times\hat{T}\to{\mathbb{T}}$ , let $J_{1},J_{2}\colon(\hat{T}\times\hat{T})\times(\hat{T}\times\hat{T})\to{\mathbb{T}}$ be the $2$ -cocycles defined by

J_{1}((a,b),(c,d))=\overline{\beta(b,c)},\quad J_{2}((a,b),(c,d))=\overline{\beta(b,cd^{-1})}.

Then, for any $H$ -cocentral closed subgroup $T_{0}\subset T$ , we have Hopf $*$ -algebra isomorphisms

	$\displaystyle{}_{J_{1}}(({\mathbb{C}}[T]\otimes H)^{\Delta(T_{0})})_{J_{1}{}^{-1}}$	$\displaystyle\cong{\mathbb{C}}[T/T_{0}]\#H_{\beta},\quad a\otimes x\mapsto a\pi(x_{(1)})\#x_{(2)},$
	$\displaystyle{}_{J_{2}}({\mathbb{C}}[T/T_{0}]\ltimes H)_{J_{2}{}^{-1}}$	$\displaystyle\cong{\mathbb{C}}[T/T_{0}]\#H_{\beta},\quad a\otimes x\mapsto a\#x.$

More pedantically, by restriction $J_{1}$ defines a $2$ -cocycle on $\Delta(T_{0})^{\perp}\subset\hat{T}\times\hat{T}$ , hence a Hopf $2$ -cocycle on ${\mathbb{C}}[(T\times T)/\Delta(T_{0})]$ . The latter gives rise to a Hopf $2$ -cocycle on $({\mathbb{C}}[T]\otimes H)^{\Delta(T_{0})}$ using the map $\iota\otimes\pi\colon({\mathbb{C}}[T]\otimes H)^{\Delta(T_{0})}\to{\mathbb{C}}[(T\times T)/\Delta(T_{0})]$ . This is the cocycle we use to define ${}_{J_{1}}(({\mathbb{C}}[T]\otimes H)^{\Delta(T_{0})})_{J_{1}{}^{-1}}$ . Similarly, $J_{2}$ defines a Hopf $2$ -cocycle on ${\mathbb{C}}[T/T_{0}]\ltimes H$ .

Proof of Theorem 2.5..

We note that ${}_{J_{1}}({\mathbb{C}}[T]\otimes H)_{{J_{1}}{}^{-1}}={\mathbb{C}}[T]\bowtie_{\beta}H$ , where the latter Hopf algebra is defined as in Theorem 1.5. By that theorem, 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 ${\mathbb{C}}[T]\#H_{\beta}$ is given by

(a\#x)^{*}=\beta(b^{-1}c,a)\,a^{-1}\#x^{*_{\beta}}=\beta(b^{-1}c,ab^{-1})\,a^{-1}\#x^{*},\quad a\in\hat{T},\ x\in H_{b,c},

while by Proposition 1.3 the involution on ${}_{J_{1}}({\mathbb{C}}[T]\otimes H)_{J_{1}{}^{-1}}$ is given by

(a\otimes x)^{*_{J_{1}}}=\beta(b^{-1}c,a)\,a^{-1}\otimes x^{*},\quad a\in\hat{T},\ x\in H_{b,c}.

For the same reason as in Proposition 2.1, for every $H$ -cocentral closed subgroup $T_{0}\subset T$ , the isomorphism ${}_{J_{1}}({\mathbb{C}}[T]\otimes H)_{J_{1}{}^{-1}}\cong{\mathbb{C}}[T]\#H_{\beta}$ 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:

Corollary 2.6.

For any $H$ -cocentral closed subgroup $T_{0}\subset T$ , the category $\mathcal{M}^{{\mathbb{C}}[T/T_{0}]\#H_{\beta}}$ is monoidally equivalent to $\mathcal{M}^{{\mathbb{C}}[T/T_{0}]\ltimes H}$ .

Remark 2.7.

The cocentral homomorphism $q\pi\colon H\to{\mathbb{C}}[T_{0}]$ defines a $\hat{T}_{0}$ -grading on the category $\mathcal{M}^{H}_{f}$ of finite dimensional comodules. Then $\mathcal{M}^{{\mathbb{C}}[T]\otimes H}_{f}\cong\operatorname{Vect}_{f}^{\hat{T}}\boxtimes\mathcal{M}^{H}_{f}$ , where $\operatorname{Vect}_{f}^{\hat{T}}$ is the category of finite dimensional $\hat{T}$ -graded vector spaces, is bi-graded by $\hat{T}\times\hat{T}_{0}$ . We can conclude that $\mathcal{M}^{{\mathbb{C}}[T/T_{0}]\#H_{\beta}}_{f}$ is monoidally equivalent to the subcategory of $\operatorname{Vect}_{f}^{\hat{T}}\boxtimes\mathcal{M}^{H}_{f}$ generated by the homogeneous components of bi-degree $(a,b)$ such that $q(a)b=1$ . $\diamond$

From Theorems 1.5 or 2.5 we see that we have a Hopf $*$ -algebra inclusion

\phi\colon H\to{\mathbb{C}}[T]\#H_{\beta},\quad\phi(x)=\pi(x_{(1)})\#x_{(2)}.

(2.7)

This map induces a monoidal functor from the category $H$ -comodules to the category of $H_{\beta}$ -comodules. It will be convenient to have the following description of this functor.

Lemma 2.8.

Let $\delta\colon M\to M\otimes H$ be an $H$ -comodule. Then $(M,(\iota\otimes\pi)\delta,\delta)$ is an $H_{\beta}$ -comodule.

Proof.

Write $\delta^{\prime}=(\iota\otimes\phi)\delta$ . Then

(\iota\otimes(\iota\#\varepsilon))\delta^{\prime}=(\iota\otimes(\pi\otimes\varepsilon)\Delta)\delta=(\iota\otimes\pi)\delta,

and

(\iota\otimes(\varepsilon\#\iota))\delta^{\prime}=(\iota\otimes(\varepsilon\pi\otimes\iota)\Delta)\delta=\delta.

Hence, by Proposition 1.2, the claim follows. ∎

2.2. Another view on $H_{\beta}$

Motivated by the recent of work of Bochniak and Sitarz [BS19], we now give another interpretation of the structure maps for $H_{\beta}$ .

Using the left and right coactions of ${\mathbb{C}}[T]$ on $H$ , we can view $H$ as a ${\mathbb{C}}[T\times T]$ -comodule algebra. Then the new product $\cdot_{\beta}$ on $H_{\beta}$ is obtained by cocycle twisting (see [Maj95, Section 2.3]) the original product by the $2$ -cocycle

(\hat{T}\times\hat{T})\times(\hat{T}\times\hat{T})\to{\mathbb{T}},\quad((a,b),(c,d))\mapsto\beta(a^{-1}b,c^{-1}).

On the other hand, $H$ is also a ${\mathbb{C}}[T]$ -comodule coalgebra, so its coalgebra structure can be twisted by a $2$ -cocycle $\omega\in Z^{2}(\hat{T};{\mathbb{T}})$ :

\Delta_{\omega}(x)=\sum_{c}\omega(a^{-1}c,c^{-1}b)x^{a,c}_{(1)}\otimes x^{c,b}_{(2)},\quad x\in H_{a,b}.

As a consequence of the following lemma, this always gives an isomorphic comodule coalgebra.

Lemma 2.9.

Assume $\omega\in Z^{2}(\hat{T};{\mathbb{T}})$ is a normalized cocycle (so $\omega(1,1)=1$ ) and $\gamma\colon\hat{T}\times\hat{T}\to{\mathbb{T}}$ is a function. Then the identity

\omega(a^{-1}c,c^{-1}b)=\gamma(a,c)\gamma(c,b)\gamma(a,b)^{-1}

(2.8)

holds for all $a,b,c\in\hat{T}$ if and only if

\gamma(a,b)=\omega(a^{-1},b)^{-1}f(a)g(b)

(2.9)

for all $a,b\in{\mathbb{T}}$ , where $f,g\colon\hat{T}\to{\mathbb{T}}$ are arbitrary functions such that

f(a)g(a)=\omega(a,a^{-1}),\quad a\in\hat{T}.

Proof.

Assume (2.8) holds. Letting $c=1$ we get

\omega(a^{-1},b)=\gamma(a,1)\gamma(1,b)\gamma(a,b)^{-1}.

Therefore (2.9) holds with $f(a)=\gamma(a,1)$ and $g(b)=\gamma(1,b)$ . The identity $f(a)g(a)=\omega(a,a^{-1})$ is satisfied by (2.9), since by letting $a=b=c$ in (2.8) we see that $\gamma(a,a)=1$ (recall also that $\omega(a,a^{-1})=\omega(a^{-1},a)$ by the cocycle identity).

Conversely, assume (2.9) holds for some functions $f,g\colon\hat{T}\to{\mathbb{T}}$ . Then

\gamma(a,c)\gamma(c,b)\gamma(a,b)^{-1}=\omega(a^{-1},c)^{-1}\omega(c^{-1},b)^{-1}\omega(a^{-1},b)f(c)g(c).

Therefore (2.8) holds if and only if

\omega(a^{-1},c)\omega(a^{-1}c,c^{-1}b)\omega(c^{-1},b)=\omega(a^{-1},b)f(c)g(c).

Using the cocycle identity twice, the left hand side equals

\omega(a^{-1},b)\omega(c,c^{-1}b)\omega(c^{-1},b)=\omega(a^{-1},b)\omega(c,c^{-1})\omega(1,b)=\omega(a^{-1},b)\omega(c,c^{-1}),

where the last identity holds, since $\omega$ is normalized. Thus, identity (2.8) holds if and only if $f(c)g(c)=\omega(c,c^{-1})$ . ∎

From this we see that up to an isomorphism $H_{\beta}$ can be obtained in many different ways by simultaneously twisting the product and coproduct on $H$ :

Proposition 2.10.

Given a bicharacter $\beta\colon\hat{T}\times\hat{T}\to{\mathbb{T}}$ and a normalized $2$ -cocycle $\omega\in Z^{2}(\hat{T};{\mathbb{T}})$ , choose a function $\gamma\colon\hat{T}\times\hat{T}\to{\mathbb{T}}$ satisfying (2.8) and define a $2$ -cocycle $\Omega\in Z^{2}(\hat{T}\times\hat{T};{\mathbb{T}})$ by

\Omega((a,b),(c,d))=\beta(a^{-1}b,c^{-1})\gamma(a,b)^{-1}\gamma(c,d)^{-1}\gamma(ac,bd).

Then we can define new product, coproduct and involution on the ${\mathbb{C}}[T]$ -comodule $H$ by

x\cdot_{\Omega}y=\Omega((a,b),(c,d))xy,\quad x\in H_{a,b},\ y\in H_{c,d},

\Delta_{\omega}(x)=\sum_{c}\omega(a^{-1}c,c^{-1}b)x^{a,c}_{(1)}\otimes x^{c,b}_{(2)},\quad x\in H_{a,b},

x^{\star}=\overline{\Omega((a,b),(a,b)^{-1})}x^{*}=\beta(a^{-1}b,a^{-1})\gamma(a,b)\gamma(a^{-1},b^{-1})x^{*},\quad x\in H_{a,b},

to get a braided Hopf $*$ -algebra $H_{\Omega,\omega}\in\mathrm{Hopf}^{*}({\mathbb{C}}[T],\beta)$ . We have an isomorphism

H_{\beta}\cong H_{\Omega,\omega},\quad H_{a,b}\ni x\mapsto\gamma(a,b)x.

Example 2.11.

Consider $T={\mathbb{T}}$ and, identifying $\hat{\mathbb{T}}$ with ${\mathbb{Z}}$ , let

\beta(m,n)=e^{2i\phi mn},\qquad\omega(m,n)=e^{i\phi mn}

for some $\phi\in{\mathbb{R}}$ . By taking $\gamma(m,n)=\omega(-m,n)^{-1}e^{-i\phi m^{2}}=e^{i\phi(mn-m^{2})}$ , we get

\Omega((k,l),(m,n))=e^{i\phi(kn-lm)}.

Then $H_{\Omega,\omega}$ coincides with the braided Hopf algebra defined in [BS19].

2.3. 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 ${\mathbb{C}}[G]$ that is spanned (equivalently, generated as an 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 ${\mathbb{C}}[G]$ -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 $U=(u_{ij})_{i,j}\in\operatorname{Mat}_{m}(H)$ such that

\Delta(u_{ij})=\sum_{k}u_{ik}\otimes u_{kj},\quad\varepsilon(u_{ij})=\delta_{ij}.

(2.10)

Conversely any such matrix defines an $m$ -dimensional $H$ -comodule $\delta_{U}\colon M\to M\otimes H$ , by setting

\delta_{U}(e_{j})=\sum_{i}e_{i}\otimes u_{ij}

(2.11)

for a fixed vector space $M$ with basis $(e_{i})_{i}$ . If $U$ is unitary, then the conjugate corepresentation matrix is $\bar{U}=(u_{ij}^{*})_{i,j}$ .

Definition 2.12 ([Wor87, Wor91]).

A compact matrix quantum group $G$ is a Hopf $*$ -algebra ${\mathbb{C}}[G]$ with generators $u_{ij}$ , $1\leq i,j\leq m$ , such that

(i)

$U=(u_{ij})_{i,j}$ is a unitary corepresentation matrix;
(ii)

$\bar{U}=(u_{ij}^{*})_{i,j}$ is equivalent to a unitary corepresentation matrix.

The coproduct $\Delta$ , counit $\varepsilon$ and antipode $S$ are then given by

\Delta(u_{ij})=\sum_{k}u_{ik}\otimes u_{kj},\quad\varepsilon(u_{ij})=\delta_{ij},\quad S(u_{ij})=u_{ji}^{*}.

The matrix $U$ is called the fundamental unitary for the compact matrix quantum group.

We remark that this is not the original definition of Woronowicz, but it is equivalent to that by a result of Dijkhuizen and Koornwinder [DK94], see also [NT13, Section 1.6].

Next, we want to introduce a braided analogue of this definition, but first we need some preparation. Suppose that $A\in\mathrm{Hopf}^{*}(K,R)$ for a Hopf $*$ -algebra $K$ with a unitary coquasitriangular structure $R$ . Suppose $U=(u_{ij})_{i,j}\in\operatorname{Mat}_{m}(A)$ satisfies $\Delta_{A}(u_{ij})=\sum_{k}u_{ik}\otimes u_{kj}$ , $\varepsilon_{A}(u_{ij})=\delta_{ij}$ . We can still define $\delta_{U}\colon M\to M\otimes A$ as in (2.11) and this gives a comodule for the coalgebra $A$ . By definition, if $Z=(z_{ij})_{i,j}\in\operatorname{Mat}_{m}(K)$ is a corepresentation matrix, the triple $(M,\delta_{Z},\delta_{U})$ defines an $A$ -comodule if and only if

(\delta_{U}\otimes\iota)\circ\delta_{Z}=\delta_{M\otimes A}\circ\delta_{U},

where $\delta_{M\otimes A}$ denotes the tensor product comodule in $\mathcal{M}^{K}$ . We have the following characterization:

Lemma 2.13.

In the above setting, the pair $(Z,U)$ defines an $A$ -comodule if and only if

\delta(u_{ij})=\sum_{s,t}u_{st}\otimes S(z_{is})z_{tj},

(2.12)

where $\delta\colon A\to A\otimes K$ 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 $(\bar{Z},\bar{U}_{Z})$ , where

\bar{U}_{Z}=(\bar{u}_{ij}^{Z})_{i,j},\quad\bar{u}_{ij}^{Z}=\sum_{s,l,t}R(z_{tj}^{*}z_{sl},z_{il}^{*})u_{st}^{*},

(2.13)

while the antipode on $A$ satisfies $S_{A}(u_{ij})=u^{*}_{ji}$ .

Here by the conjugate $A$ -comodule we mean the comodule obtained by taking the conjugate ( $K\#A)$ -comodule.

Proof.

The condition $(\delta_{U}\otimes\iota)\circ\delta_{Z}=\delta_{M\otimes A}\circ\delta_{U}$ is satisfied if and only if

\sum_{t}u_{st}\otimes z_{tj}=\sum_{k}u_{kj}^{(1)}\otimes z_{sk}u_{kj}^{(2)}

for $1\leq s,j\leq m$ , where we write $\delta(a)=a^{(1)}\otimes a^{(2)}$ , $a\in A$ . Multiplying both sides by $S(z_{is})$ and summing over $s$ yields

\sum_{s,t}u_{st}\otimes S(z_{is})z_{tj}=\sum_{k,s}u_{kj}^{(1)}\otimes S(z_{is})z_{sk}u_{kj}^{(2)}=u_{ij}^{(1)}\otimes u_{ij}^{(2)}.

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

W=(w_{ij})_{i,j}=Z\#U,\quad\text{so}\quad w_{ij}=\sum_{k}z_{ik}\#u_{kj}.

Using that $\iota\#\varepsilon_{A}\colon K\#A\to K$ 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 $\bar{U}_{Z}=(\varepsilon_{K}\#\iota)(\bar{W})$ . Alternatively, we can use the formula for the antipode in Definition 1.1 to get

\bar{u}^{Z}_{ij}=(\varepsilon_{K}\#\iota)S(w_{ji})=\sum_{r}R(S_{A}(u_{jr})^{(2)},S_{K}(z_{ri}))S_{A}(u_{jr})^{(1)}.

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 $\bar{U}_{Z}$ above. However, we stick to the notation $\bar{U}_{Z}$ for the rest of the paper, as $R$ will always be given by a fixed bicharacter $\beta$ .

Definition 2.14.

Let $T$ be a compact abelian group with a fixed unitary corepresentation matrix $Z\in\operatorname{Mat}_{m}({\mathbb{C}}[T])$ and a bicharacter $\beta$ on $\hat{T}$ . A braided compact matrix quantum group over the triple $(T,Z,\beta)$ is an object $A\in\mathrm{Hopf}^{*}({\mathbb{C}}[T],\beta)$ generated as a $*$ -algebra by elements $u_{ij}$ , $1\leq i,j\leq m$ , such that, for $U=(u_{ij})_{i,j}$ ,

(i)

$(Z,U)$ defines a unitary $A$ -comodule;
(ii)

$(\bar{Z},\bar{U}_{Z})$ 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 $U\in\operatorname{Mat}_{m}(A)$ is unitary, there is $F\in\operatorname{GL}_{m}({\mathbb{C}})$ such that both $F\bar{Z}F^{-1}$ and $F\bar{U}_{Z}F^{-1}$ are unitary, and the structure maps for $A$ satisfy the following properties: the coaction of ${\mathbb{C}}[T]$ on $A$ is given by (2.12), and

\Delta_{A}(u_{ij})=\sum_{k}u_{ik}\otimes u_{kj},\qquad\varepsilon_{A}(u_{ij})=\delta_{ij},\qquad S_{A}(u_{ij})=u_{ji}^{*}.

We remark that in view of Lemma 2.13 we can similarly define a braided compact matrix quantum group over $(K,Z,R)$ for any Hopf $*$ -algebra $K$ with a unitary coquasitriangular structure $R$ and a unitary corepresentation matrix $Z=(z_{ij})_{i,j}\in\operatorname{Mat}_{m}(K)$ , but the adjective “compact” in this generality might be somewhat misleading.

Proposition 2.15.

Given a compact abelian group $T$ , a unitary corepresentation matrix $Z\in\operatorname{Mat}_{m}({\mathbb{C}}[T])$ and a bicharacter $\beta$ on $\hat{T}$ , the bosonization of any braided compact matrix quantum group $A$ over $(T,Z,\beta)$ is a compact quantum group.

Proof.

By working in an orthonormal basis where $Z$ is diagonal, we see that the $*$ -algebra ${\mathbb{C}}[T]\#A$ is generated by the matrix coefficients of the fundamental unitary representation and the characters of $T$ . ∎

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 $\pi\colon{\mathbb{C}}[G]\to{\mathbb{C}}[T]$ , and let $\beta$ be a bicharacter on $\hat{T}$ . Then the transmutation ${\mathbb{C}}[G]_{\beta}$ is a braided compact matrix quantum group over $(T,\pi(U),\beta)$ with fundamental unitary $U$ .

Proof.

Put $Z=\pi(U)$ . Recall that by (2.7) we have a Hopf $*$ -algebra map

\phi\colon{\mathbb{C}}[G]\to{\mathbb{C}}[T]\#{\mathbb{C}}[G]_{\beta},\quad\phi(x)=\pi(x_{(1)})\#x_{(2)}.

By Lemma 2.8 this implies that the pair $(Z,U)$ defines a unitary ${\mathbb{C}}[G]_{\beta}$ –comodule, with the corresponding $({\mathbb{C}}[T]\#{\mathbb{C}}[G]_{\beta})$ -comodule given by the unitary $\phi(U)=Z\#U$ . The conjugate $({\mathbb{C}}[T]\#{\mathbb{C}}[G]_{\beta})$ -comodule is given by $\phi(\bar{U})$ . As $\bar{U}\in\operatorname{Mat}_{m}({\mathbb{C}}[G])$ 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 ${\mathbb{C}}[G]_{\beta}$ 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 ${\mathbb{C}}[G]$ and ${\mathbb{C}}[G]_{\beta}$ coincide up to phase factors. ∎

Remark 2.17.

Even though ${\mathbb{C}}[G]_{\beta}$ has fundamental representation $(\pi(U),U)$ , condition (2.12) can be satisfied for another pair $(Z^{\prime},U)$ , which is then also a fundamental representation. A particularly interesting situation is when $Z^{\prime}\in\operatorname{Mat}_{m}({\mathbb{C}}[T/T_{0}])$ for a ${\mathbb{C}}[G]$ -cocentral subgroup $T_{0}\subset T$ . In this case we can view ${\mathbb{C}}[G]_{\beta}$ as a braided compact matrix quantum group over $(T/T_{0},Z^{\prime},i^{*}\beta)$ . $\diamond$

We record a useful lemma related to the above remark.

Lemma 2.18.

Assume $A\in\mathrm{Hopf}^{*}({\mathbb{C}}[T],\beta)$ and take $w\in\hat{T}$ . Then $(Z,U)$ defines an $A$ -comodule if and only if $(wZ,U)$ defines an $A$ -comodule. If in addition $Z$ and $U$ are unitary, then we have the relation

\bar{U}_{wZ}=D\bar{U}_{Z}D^{-1},\quad D=(\beta(z_{ij}^{*},w))_{i,j}.

Proof.

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 $W=(\sum_{k}z_{ik}\#u_{kj})_{i,j}$ for ${\mathbb{C}}[T]\#A$ we have

(w\#1)W=XW(w\#1)X^{-1},

where $X=(\beta(z_{ij},w))_{i,j}$ , which is a matrix commuting with $Z$ . This implies that $\bar{U}_{wZ}=\bar{X}\bar{U}_{Z}\bar{X}^{-1}$ . It remains to observe that $\overline{\beta(x,w)}=\beta(x^{*},w)$ for all $x\in{\mathbb{C}}[T]$ to see that $\bar{X}=D$ . ∎

3. Examples: transmuting matrix quantum groups

Before we embark on the examples, we remark that in a number of recent papers (see, e.g., [Kas+16, MR22, BJR22, Ans+22]) 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 be 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 $H:={\mathbb{C}}[SU_{q}(2)]$ is the universal unital $*$ -algebra with generators $\alpha$ and $\gamma$ subject to the relations

\alpha\gamma=q\gamma\alpha,\quad\alpha\gamma^{*}=q\gamma^{*}\alpha,\quad\gamma^{*}\gamma=\gamma\gamma^{*},

\quad\alpha^{*}\alpha+\gamma^{*}\gamma=1,\quad\alpha\alpha^{*}+q^{2}\gamma\gamma^{*}=1.

It is a Hopf $*$ -algebra with coproduct

\Delta(\alpha)=\alpha\otimes\alpha-q\gamma^{*}\otimes\gamma,\quad\Delta(\gamma)=\gamma\otimes\alpha+\alpha^{*}\otimes\gamma.

Consider the map

\pi\colon H\to{\mathbb{C}}[{\mathbb{T}}]={\mathbb{C}}[z,z^{-1}],\quad\pi(\alpha)=z,\quad\pi(\gamma)=0.

Under the identification ${\mathbb{Z}}=\hat{\mathbb{T}}$ , we have

\alpha\in H_{1,1},\qquad\gamma\in H_{{-1},1},

and the restricted right adjoint coaction $\mathrm{ad}_{\pi}$ is determined by

\mathrm{ad}_{\pi}(\alpha)=\alpha\otimes 1,\quad\mathrm{ad}_{\pi}(\gamma)=\gamma\otimes z^{2}.

For $\lambda\in{\mathbb{T}}$ , define a bicharacter on ${\mathbb{Z}}=\hat{\mathbb{T}}$ by $\beta_{\lambda}(m,n)=\lambda^{-mn}$ . To find relations in the transmutation $H_{\lambda}=H_{\beta_{\lambda}}$ we write $a\cdot b=a\cdot_{\beta_{\lambda}}b$ and $a^{*_{\lambda}}=a^{*_{\beta_{\lambda}}}$ . Then, by (2.4) and (2.6),

\alpha^{*_{\lambda}}=\beta_{\lambda}(0,{-1})\alpha^{*}=\alpha^{*},\qquad\gamma^{*_{\lambda}}=\beta_{\lambda}(2,1)\gamma^{*}=\lambda^{-2}\gamma^{*},

and

\alpha\cdot\gamma=\beta_{\lambda}(0,1)\alpha\gamma=\alpha\gamma=q\gamma\alpha=q\beta_{\lambda}(2,{-1})^{-1}\gamma\cdot\alpha=q\lambda^{-2}\gamma\cdot\alpha,

\alpha\cdot\gamma^{*_{\lambda}}=\beta_{\lambda}(0,{-1})\alpha\gamma^{*_{\lambda}}=\alpha\gamma^{*_{\lambda}}=q\gamma^{*_{\lambda}}\alpha=q\beta_{\lambda}({-2},{-1})^{-1}\gamma^{*_{\lambda}}\cdot\alpha=q\lambda^{2}\gamma^{*_{\lambda}}\cdot\alpha,

\gamma^{*_{\lambda}}\cdot\gamma=\beta_{\lambda}({-2},1)\gamma^{*_{\lambda}}\gamma=\gamma^{*}\gamma=\gamma\gamma^{*}=\beta_{\lambda}(2,{-1})\gamma\gamma^{*_{\lambda}}=\gamma\cdot\gamma^{*_{\lambda}},

\alpha\cdot\alpha^{*_{\lambda}}=\alpha\alpha^{*},\qquad\alpha^{*}\alpha=\alpha\cdot\alpha^{*_{\lambda}}.

Defining $q^{\prime}=q\lambda^{2}$ we get the following relations in $H_{\lambda}$ :

\alpha\cdot\gamma=\bar{q}\,^{\prime}\gamma\cdot\alpha,\quad\alpha\cdot\gamma^{*_{\lambda}}=q^{\prime}\gamma^{*_{\lambda}}\cdot\alpha,\quad\gamma^{*_{\lambda}}\cdot\gamma=\gamma\cdot\gamma^{*_{\lambda}},

\quad\alpha^{*_{\lambda}}\cdot\alpha+\gamma^{*_{\lambda}}\cdot\gamma=1,\quad\alpha\cdot\alpha^{*_{\lambda}}+|q^{\prime}|^{2}\gamma\cdot\gamma^{*_{\lambda}}=1.

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

\Delta(\alpha)=\alpha\otimes\alpha-q^{\prime}\gamma^{*_{\lambda}}\otimes\gamma,\qquad\Delta(\gamma)=\gamma\otimes\alpha+\alpha^{*_{\lambda}}\otimes\gamma.

These formulas are the same as for the braided quantum group $SU_{q^{\prime}}(2)$ constructed in [Kas+16], modulo a small but important nuance. By Proposition 2.4, $H_{\lambda}$ can be viewed as a braided quantum group over different tori. Namely, we see that $H_{\lambda}$ can be viewed as a braided compact matrix quantum group over both triples

({\mathbb{T}},\,\begin{pmatrix}z&0\\ 0&z^{-1}\end{pmatrix},\,\beta_{\lambda})\quad\mbox{ and }\quad({\mathbb{T}}/T_{0},\,\begin{pmatrix}z^{2}&0\\ 0&1\end{pmatrix},\,i^{*}\beta_{\lambda}),

where $T_{0}=\{-1,1\}\subset{\mathbb{T}}$ , see Remark 2.17. As ${\mathbb{C}}[{\mathbb{T}}/T_{0}]$ is generated by $z^{2}$ we have the isomorphism

f\colon{\mathbb{C}}[\mathbb{T}/T_{0}]\to{\mathbb{C}}[w,w^{-1}],\quad f(z^{2})=w.

Moreover, $(i\circ f^{-1})^{*}\beta_{\lambda}=\beta_{\zeta}$ , where $\zeta=\lambda^{4}=q^{\prime}/\bar{q}\,^{\prime}$ . Therefore we can consider $H_{\lambda}$ as a braided compact matrix quantum group over the triple

({\mathbb{T}},\,\begin{pmatrix}w&0\\ 0&1\end{pmatrix},\,\beta_{\zeta}).

This is the braided quantum group ${\mathbb{C}}[SU_{q^{\prime}}(2)]\in\mathrm{Hopf}^{*}({\mathbb{C}}[{\mathbb{T}}],\beta_{\zeta})$ considered in [Kas+16].

Finally, let us consider the bosonizations. By Theorem 2.5 we have

{\mathbb{C}}[z,z^{-1}]\#H_{\lambda}\cong{}_{J}({\mathbb{C}}[z,z^{-1}]\otimes{\mathbb{C}}[SU_{q}(2)])_{J^{-1}},

where $J((m,n),({m^{\prime}},{n^{\prime}}))=\lambda^{nm^{\prime}}$ . In other words, the bosonization is a cocycle twist of the compact quantum group ${\mathbb{T}}\times SU_{q}(2)$ .

On the other hand, by the same theorem, the bosonization ${\mathbb{C}}[w,w^{-1}]\,\#\,H_{\lambda}$ is a cocycle twist of $({\mathbb{T}}\times SU_{q}(2))/\Delta(T_{0})$ . It is easy to see that the latter quantum group is isomorphic to $U_{q}(2)$ , similarly to the classical isomorphism

({\mathbb{T}}\times SU(2))/{\Delta(T_{0})}\cong U(2),\quad[(z,U)]\mapsto\begin{pmatrix}z&0\\ 0&z\end{pmatrix}U,

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 $m\geq 2$ be a natural number, $F\in\mathrm{GL}_{m}({\mathbb{C}})$ and assume that $F\bar{F}=\pm 1$ . Let ${\mathbb{C}}[O_{F}^{+}]$ be the universal unital $*$ -algebra generated by elements $u_{ij}$ , $1\leq i,j\leq m$ , subject to the relations

U=(u_{ij})_{i,j}\quad\mbox{is unitary and}\quad U=F\bar{U}F^{-1}.

The Hopf $*$ -algebra structure on ${\mathbb{C}}[O_{F}^{+}]$ 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 $Z\in\operatorname{Mat}_{m}({\mathbb{C}}[T])$ be a unitary corepresentation matrix satisfying $F\bar{Z}F^{-1}=Z$ . By the universality of ${\mathbb{C}}[O_{F}^{+}]$ there is a Hopf $*$ -algebra map $\pi\colon{\mathbb{C}}[O_{F}^{+}]\to{\mathbb{C}}[T]$ such that $\pi(u_{ij})=z_{ij}$ . Fix a bicharacter $\beta\colon\hat{T}\times\hat{T}\to{\mathbb{T}}$ and consider the transmutation ${\mathbb{C}}[O_{F}^{+}]_{\beta}$ . It is natural to call it a braided free orthogonal quantum group.

Proposition 3.1.

The braided Hopf $*$ -algebra ${\mathbb{C}}[O_{F}^{+}]_{\beta}$ is a braided compact matrix quantum group over $(T,Z,\beta)$ 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=(u_{ij})_{i,j}\quad\mbox{is unitary and}\quad U=F\bar{U}_{Z}F^{-1},

(3.1)

where $\bar{U}_{Z}=(\bar{u}_{ij}^{Z})_{i,j}$ and $\bar{u}_{ij}^{Z}=\sum_{s,l,t}\beta(z_{tj}^{*}z_{sl},z_{il}^{*})u_{st}^{*}$ .

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 ${\mathbb{C}}[O_{F}^{+}]_{\beta}$ . 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 $\phi\colon{\mathbb{C}}[O^{+}_{F}]\to{\mathbb{C}}[T]\#A$ , $\phi(x)=\pi(x_{(1)})\#x_{(2)}$ , and using that $\phi(V)=Z\#U$ , $\phi(\bar{V})=\bar{Z}\#\bar{U}_{Z}$ .

It remains to show that as a $*$ -algebra $A$ is completely described by relations (3.1). Consider a universal unital $*$ -algebra $\tilde{A}$ with generators $\tilde{u}_{ij}$ satisfying these relations, and let $\rho\colon\tilde{A}\to A$ be the $*$ -homomorphism such that $\rho(\tilde{u}_{ij})=u_{ij}$ .

Working in a basis where $Z$ is diagonal, it is not difficult to check that $\tilde{A}$ is a ${\mathbb{C}}[T]$ -comodule $*$ -algebra, with the coaction of ${\mathbb{C}}[T]$ given by

\tilde{\delta}(\tilde{u}_{ij})=\sum_{s,t}\tilde{u}_{st}\otimes z_{si}^{*}z_{tj},\quad\text{or}\quad(\iota\otimes\tilde{\delta})(U)=Z^{*}_{13}\tilde{U}_{12}Z_{13}.

Consider the smash product ${\mathbb{C}}[T]\#\tilde{A}={\mathbb{C}}[T]\otimes_{\beta}\tilde{A}$ . It is again not difficult to check that we have a $*$ -homomorphism

\tilde{\phi}\colon{\mathbb{C}}[O^{+}_{F}]\to{\mathbb{C}}[T]\#\tilde{A},\quad\tilde{\phi}(v_{ij})=\sum_{k}z_{ik}\#\tilde{u}_{kj}.

Define linear maps

	$\displaystyle\tilde{\psi}\colon{\mathbb{C}}[T]\otimes{\mathbb{C}}[O^{+}_{F}]\to{\mathbb{C}}[T]\#\tilde{A},\qquad\tilde{\psi}(x\otimes a)=x\tilde{\phi}(a),$
	$\displaystyle\psi\colon{\mathbb{C}}[T]\otimes{\mathbb{C}}[O^{+}_{F}]\to{\mathbb{C}}[T]\#A,\qquad\psi(x\otimes a)=x\pi(a_{(1)})\#a_{(2)}.$

Then $\psi=(\iota\#\rho)\tilde{\psi}$ . The map $\psi$ is a linear isomorphism, e.g., by Theorem 2.5. On the other hand, the map $\tilde{\psi}$ is surjective, which becomes particularly clear if we work in a basis where $Z$ is diagonal and therefore $\tilde{\phi}(v_{ij})=z_{ii}\#\tilde{u}_{ij}$ . (Alternatively, we can observe that $\tilde{\psi}$ defines a homomorphism ${\mathbb{C}}[T]\#{\mathbb{C}}[O^{+}_{F}]\to{\mathbb{C}}[T]\#\tilde{A}$ , cf. Remark 1.6, and its image contains the elements $1\#\tilde{u}_{ij}$ .) It follows that $\tilde{\psi}$ is a linear isomorphism and hence $\rho$ 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 $A\in\mathrm{GL}_{m}({\mathbb{C}})$ ( $m\geq 2$ ) be a matrix such that $A\bar{A}$ is unitary, and choose a sign $\tau=\pm 1$ , with $\tau=1$ if $m$ is odd. Then there are a compact abelian group $T$ , a unitary corepresentation matrix $X=(x_{ij})_{i,j}\in\operatorname{Mat}_{m}({\mathbb{C}}[T])$ , a character $w\in\hat{T}$ and a bicharacter $\beta\colon\hat{T}\times\hat{T}\to{\mathbb{T}}$ such that

A(w^{2}\bar{X})A^{-1}=X\quad\mbox{in}\quad\operatorname{Mat}_{m}({\mathbb{C}}[T])

(3.2)

and $AC\overline{AC}=\tau 1$ , where $C=(\beta(x_{ij}^{*},w))_{i,j}$ . For every such quadruple $(T,X,w,\beta)$ , consider a universal unital $*$ -algebra ${\mathbb{C}}[O_{A}^{X,\beta}]$ with generators $u_{ij}$ satisfying the relations

U=(u_{ij})_{i,j}\quad\mbox{is unitary and}\quad U=A\bar{U}_{X}A^{-1},

where $\bar{U}_{X}=(\bar{u}_{ij}^{X})_{i,j}$ and $\bar{u}_{ij}^{X}=\sum_{s,l,t}\beta(x_{tj}^{*}x_{sl},x_{il}^{*})u_{st}^{*}$ . Then ${\mathbb{C}}[O_{A}^{X,\beta}]$ , equipped with the coaction

\delta(u_{ij})=\sum_{s,t}u_{st}\otimes x_{si}^{*}x_{tj},

is a braided compact matrix quantum group over $(T,X,\beta)$ with fundamental unitary $U$ .

Proof.

Assume first that a quadruple $(T,X,w,\beta)$ as in the formulation indeed exists. Define $F=AC$ . By our assumptions this matrix satisfies $F\bar{F}=\tau 1$ and, as $C$ commutes with $\bar{X}$ , we have

F(w\bar{X})F^{-1}=w^{-1}X.

By universality there is a Hopf $*$ -homomorphism $\pi\colon{\mathbb{C}}[O_{F}^{+}]\to{\mathbb{C}}[T]$ sending the fundamental representation to $Z=w^{-1}X$ . We claim that the corresponding transmutation ${\mathbb{C}}[O_{F}^{+}]_{\beta}$ satisfies all the required properties of ${\mathbb{C}}[O_{A}^{X,\beta}]$ .

Indeed, by Lemma 2.18, in ${\mathbb{C}}[O_{F}^{+}]_{\beta}$ we have

\bar{U}_{X}=\bar{U}_{wZ}=D\bar{U}_{Z}D^{-1},

where $D=(\beta(z^{*}_{ij},w))_{i,j}=(\beta(wx^{*}_{ij},w))_{i,j}=\beta(w,w)C$ . Then $A=\beta(w,w)FD^{-1}$ , and the claim follows from Proposition 3.1.

Next we explain the existence of $(T,X,w,\beta)$ . By [HN21, Proposition 1.5], we can find a unitary $v$ such that $vAv^{t}$ has the form

\begin{pmatrix}0&&a_{m}\\ &\iddots&\\ a_{1}&&0\end{pmatrix},\quad a_{i}\bar{a}_{m-i+1}=\lambda_{i}\in{\mathbb{T}}.

(3.3)

If we can find a quadruple $(T,X,w,\beta)$ for this matrix, then $(T,v^{*}X(\cdot)v,w,\beta)$ 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 $X(t)=\operatorname{diag}(x_{1}(t),\dots,x_{m}(t))$ for some characters $x_{i}$ . The conditions (3.2) and $AC\overline{AC}=\tau 1$ for $C=\operatorname{diag}(\beta(x_{1}^{-1},w),\dots,\beta(x_{m}^{-1},w))$ mean then that

x_{i}x_{m-i+1}=w^{2}\qquad\text{and}\qquad\beta(x_{i}^{-1}x_{m-i+1},w)=\tau\lambda_{i}.

(3.4)

If $m=2k$ , these conditions can be easily satisfied for the dual $T$ of a free abelian group with independent generators $x_{1},\dots,x_{k},w$ by letting $x_{m-i+1}=w^{2}x_{i}^{-1}$ for $1\leq i\leq k$ . If $m=2k+1$ , then $\tau=1$ , $\lambda_{k+1}=1$ and the conditions can be satisfied for the dual $T$ of a free abelian group with independent generators $x_{1},\dots,x_{k+1}$ by letting $w=x_{k+1}$ and $x_{m-i+1}=w^{2}x_{i}^{-1}$ for $1\leq i\leq k$ . ∎

As is clear from the proof of this proposition, the braided quantum groups $O^{X,\beta}_{A}$ lie within the class of braided free orthogonal quantum groups that we defined by transmutation. Namely, we have the following:

Corollary 3.3.

The braided Hopf $*$ -algebra ${\mathbb{C}}[O_{A}^{X,\beta}]\in\mathrm{Hopf}^{*}({\mathbb{C}}[T],\beta)$ is isomorphic to the transmutation ${\mathbb{C}}[O_{F}^{+}]_{\beta}$ with respect to the map ${\mathbb{C}}[O_{F}^{+}]\to{\mathbb{C}}[T]$ , $U\mapsto w^{-1}X$ , 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 $w=x_{1}^{j}$ for any $j\neq 0,1$ . Similarly, for $m=2k+1$ we could take $x_{k+1}=w=x_{1}^{j}$ for any $j\neq 0,1$ . In both cases we cannot choose groups of a smaller rank in general, since the numbers $\tau\lambda_{i}$ generate a group of rank up to $k=[m/2]$ .

Remark 3.5.

Once (3.2) is satisfied, condition $AC\overline{AC}=\tau 1$ can be formulated as follows. Let $t_{w}\in T$ be the element such that $x(t_{w})=\beta(x,w)$ for all $x\in\hat{T}$ , so that $C=\overline{X(t_{w})}$ . Then the requirement is

A\bar{A}=\tau\beta(w,w)^{2}X(t_{w})^{-2}.

(3.5)

As a prerequisite for constructing $O^{X,\beta}_{A}$ this condition can be written as

A\bar{A}=c\,X(t_{w})^{-2}\quad\text{for some}\quad c\in{\mathbb{T}}.

(3.6)

Indeed, assume (3.2) and (3.6) are satisfied. Then applying complex conjugation and conjugation by $A$ to the last identity we get

A\bar{A}=\bar{c}\,\beta(w,w)^{4}X(t_{w})^{-2}.

Hence $c=\bar{c}\,\beta(w,w)^{4}$ and therefore $\tau:=c\,\beta(w,w)^{-2}=\pm 1$ , so that (3.5) is satisfied for this $\tau$ . Note that the sign must be $+1$ for odd $m$ , which becomes obvious if we choose a unitary $v$ such that $vA\,\overline{X(t_{w})}v^{t}$ is of the form (3.3). Thus, the braided compact matrix quantum groups $O^{X,\beta}_{A}$ are defined under assumptions (3.2) and (3.6). $\diamond$

In the setting of Proposition 3.1, assume now that $T_{0}\subset T$ is a closed ${\mathbb{C}}[O_{F}^{+}]$ -cocentral subgroup. Since the elements $u_{ij}\in{\mathbb{C}}[O^{+}_{F}]$ are linearly independent, this means that the matrices $Z(t)$ , $t\in T_{0}$ , are scalar. Then the condition $Z=F\bar{Z}F^{-1}$ implies that $Z(t)=\pm 1$ , so we get a character $\chi\colon T_{0}\to\{\pm 1\}$ . When it is nontrivial, it defines the standard $({\mathbb{Z}}/2{\mathbb{Z}})$ -grading on $\operatorname{Rep}O^{+}_{F}$ .

It is known that the quantum group $O^{+}_{F}$ is monoidally equivalent to $SU_{q}(2)$ for an appropriate $q$ (see [NT13, Theorem 2.5.11]), and this equivalence respects the $({\mathbb{Z}}/2{\mathbb{Z}})$ -gradings. Combining this with Remark 2.7, one can conclude that the bosonization of the braided Hopf $*$ -algebra ${\mathbb{C}}[O^{+}_{F}]_{\beta}\in\mathrm{Hopf}^{*}({\mathbb{C}}[T/T_{0}],i^{*}\beta)$ is monoidally equivalent to

(T\times SU_{q}(2))/{(\operatorname{id}\times\chi)\Delta(T_{0})}.

Together with Corollary 3.3 this leads to the following conclusion.

Proposition 3.6.

In the setting of Proposition 3.2, let $q\in[-1,1]\setminus\{0\}$ be such that $\operatorname{sgn}q=-\tau$ and $|q+q^{-1}|=\operatorname{Tr}(A^{*}A)$ . Assume $T_{0}\subset T$ is a closed subgroup satisfying $X(t)=\pm w(t)1$ for all $t\in T_{0}$ , and let $\chi\colon T_{0}\to\{\pm 1\}$ be the character such that $X=\chi w1$ on $T_{0}$ . Then the bosonization of ${\mathbb{C}}[O^{X,\beta}_{A}]\in\mathrm{Hopf}^{*}({\mathbb{C}}[T/T_{0}],i^{*}\beta)$ is a compact quantum group monoidally equivalent to

(T\times SU_{q}(2))/{(\operatorname{id}\times\chi)\Delta(T_{0})}.

(3.7)

We remark that the sign of $q$ is not uniquely determined by a monoidal equivalence between (3.7) and the bosonization of $O^{X,\beta}_{A}$ in general, since $U_{q}(2)$ and $U_{-q}(2)$ are cocycle twists of each other.

Next, it is possible to obtain a classification of braided free orthogonal quantum groups up to isomorphism similar to the known classification of ordinary free orthogonal quantum groups, see again [NT13, Theorem 2.5.11].

Proposition 3.7.

Consider a compact abelian group $T$ and a bicharacter $\beta\colon\hat{T}\times\hat{T}\to{\mathbb{T}}$ . Assume $A\in\mathrm{GL}_{m}({\mathbb{C}})$ ( $m\geq 2$ ) is a matrix such that $A\bar{A}$ is unitary and $X=(x_{ij})_{i,j}\in\operatorname{Mat}_{m}({\mathbb{C}}[T])$ is a unitary corepresentation matrix such that conditions (3.2) and (3.6) are satisfied for some $w\in\hat{T}$ . Assume $A^{\prime}\in\mathrm{GL}_{m^{\prime}}({\mathbb{C}})$ ( $m^{\prime}\geq 2$ ) and $X^{\prime}=(x^{\prime}_{ij})_{i,j}\in\operatorname{Mat}_{m^{\prime}}({\mathbb{C}}[T])$ is another such pair, with (3.2) and (3.6) satisfied for some $w^{\prime}\in\hat{T}$ . Then the braided Hopf $*$ -algebras ${\mathbb{C}}[O^{X,\beta}_{A}]$ and ${\mathbb{C}}[O^{X^{\prime},\beta}_{A^{\prime}}]$ over $({\mathbb{C}}[T],\beta)$ are isomorphic if and only if $m=m^{\prime}$ and there exist a unitary matrix $v\in\operatorname{U}(m)$ and a character $\chi\in\hat{T}$ such that

vXv^{*}=\chi X^{\prime}\quad\text{and}\quad vADv^{t}=A^{\prime},

(3.8)

where $D=(\beta(x^{*}_{ij},\chi))_{i,j}=\overline{X(t_{\chi})}$ .

Proof.

Denote by $U$ and $U^{\prime}$ the fundamental unitaries of ${\mathbb{C}}[O^{X,\beta}_{A}]$ and ${\mathbb{C}}[O^{X^{\prime},\beta}_{A^{\prime}}]$ , resp. If conditions (3.8) are satisfied, then $\bar{U}_{\chi^{-1}X}=D^{-1}\bar{U}_{X}D$ by Lemma 2.18, hence $U=AD\bar{U}_{\chi^{-1}X}(AD)^{-1}$ , and it is easy to check that we get an isomorphism ${\mathbb{C}}[O^{X,\beta}_{A}]\cong{\mathbb{C}}[O^{X^{\prime},\beta}_{A^{\prime}}]$ of braided Hopf $*$ -algebras such that $U\mapsto v^{*}U^{\prime}v$ .

Conversely, assume we have an isomorphism $\rho\colon{\mathbb{C}}[O^{X,\beta}_{A}]\to{\mathbb{C}}[O^{X^{\prime},\beta}_{A^{\prime}}]$ . We then get an isomorphism $\iota\#\rho\colon{\mathbb{C}}[T]\#{\mathbb{C}}[O^{X,\beta}_{A}]\to{\mathbb{C}}[T]\#{\mathbb{C}}[O^{X^{\prime},\beta}_{A^{\prime}}]$ of the bosonizations, which in turn defines a monoidal equivalence of the corresponding C^∗-tensor categories of finite dimensional unitary comodules. For ${\mathbb{C}}[T]\#{\mathbb{C}}[O^{X,\beta}_{A}]$ , as we discussed before Proposition 3.6, this C^∗-tensor category is $\operatorname{Rep}T\boxtimes\operatorname{Rep}SU_{q}(2)$ for suitable $q\in[-1,1]\setminus\{0\}$ . The simple noninvertible objects of the smallest intrinsic dimension (equal to $|q+q^{-1}|$ ) in this category are the tensor products of characters $\chi\in\hat{T}$ with the fundamental representation of $SU_{q}(2)$ . At the level of ${\mathbb{C}}[T]\#{\mathbb{C}}[O^{X,\beta}_{A}]$ , these objects are defined by the unitary corepresentation matrices $\chi X\#U$ . For the same reason, the simple noninvertible objects of the smallest intrinsic dimension in the category of finite dimensional unitary comodules of ${\mathbb{C}}[T]\#{\mathbb{C}}[O^{X^{\prime},\beta}_{A^{\prime}}]$ are defined by the unitary corepresentation matrices $\chi X^{\prime}\#U^{\prime}$ . It follows that there exist $\chi\in\hat{T}$ and a unitary $v\colon{\mathbb{C}}^{m}\to{\mathbb{C}}^{m^{\prime}}$ such that

v(\iota\#\rho)(X\#U)v^{*}=\chi X^{\prime}\#U^{\prime}.

In particular, we must have $m=m^{\prime}$ . As $(\iota\#\rho)(X\#U)=X\#\rho(U)$ , by applying $\iota\#\varepsilon_{{\mathbb{C}}[O^{X^{\prime},\beta}_{A^{\prime}}]}$ we first conclude that $vXv^{*}=\chi X^{\prime}$ and then that $v\rho(U)v^{*}=U^{\prime}$ .

As $v\chi^{-1}Xv^{*}=X^{\prime}$ and $v\rho(U)v^{*}=U^{\prime}$ , we have $\bar{v}\rho(\bar{U}_{\chi^{-1}X})\bar{v}^{*}=\bar{U}^{\prime}_{X^{\prime}}$ . Recall also that $\bar{U}_{\chi^{-1}X}=D^{-1}\bar{U}_{X}D$ for $D=(\beta(x^{*}_{ij},\chi))_{i,j}$ . By the relations in ${\mathbb{C}}[O^{X,\beta}_{A}]$ we then get

U^{\prime}=v\rho(U)v^{*}=v\rho(AD\bar{U}_{\chi^{-1}X}(AD)^{-1})v^{*}=vADv^{t}\bar{U}^{\prime}_{X^{\prime}}\bar{v}(AD)^{-1}v^{*}.

On the other hand, $U^{\prime}=A^{\prime}\bar{U}^{\prime}_{X^{\prime}}{A^{\prime}}^{-1}$ by the relations in ${\mathbb{C}}[O^{X^{\prime},\beta}_{A^{\prime}}]$ . Since the matrix coefficients of $U^{\prime}$ are linearly independent, it follows that

vADv^{t}=\lambda A^{\prime}\quad\text{for some}\quad\lambda\in{\mathbb{C}}^{\times}.

As both $A^{\prime}\bar{A^{\prime}}$ and $AD\overline{AD}$ are unitary (recall that $D=\overline{X(t_{\chi})}$ is unitary and $AD$ coincides with $\bar{D}A$ up to a phase factor), we must have $\lambda\in{\mathbb{T}}$ . Hence, by multiplying $v$ by a phase factor we can achieve that $vADv^{t}=A^{\prime}$ , while the equality $vXv^{*}=\chi X^{\prime}$ is still satisfied. ∎

Note that, for fixed $(A,X)$ and $(A^{\prime},X^{\prime})$ , there can only be finitely many $\chi$ such $vXv^{*}=\chi X^{\prime}$ for some $v$ . Furthermore, if $\hat{T}$ is torsion-free (equivalently, $T$ is connected), then the only possible candidate for such $\chi$ is $w{w^{\prime}}^{-1}$ , since a nontrivial translation of a finite symmetric subset of $\hat{T}$ is never symmetric. Once $\chi$ is fixed, the question whether there is $v$ satisfying both conditions in (3.8) can be solved by writing $(AD,\chi^{-1}X)$ and $(A^{\prime},X^{\prime})$ in a standard form in the following sense.

Lemma 3.8.

Assume $A\in\mathrm{GL}_{m}({\mathbb{C}})$ ( $m\geq 2$ ) is a matrix such that $A\bar{A}$ is unitary and $X=(x_{ij})_{i,j}\in\operatorname{Mat}_{m}({\mathbb{C}}[T])$ is a unitary corepresentation matrix such that $A(w_{0}\bar{X})A^{-1}=X$ for some $w_{0}\in\hat{T}$ . Then there is $v\in\operatorname{U}(m)$ such that $vAv^{t}$ and $vXv^{*}$ are block-diagonal matrices $\operatorname{diag}(A_{1},\dots,A_{k})$ and $\operatorname{diag}(X_{1},\dots,X_{k})$ , where the blocks are

–

either $2\times 2$ matrices

A_{i}=\begin{pmatrix}0&\theta_{i}\lambda_{i}^{-1}\\ \lambda_{i}&0\end{pmatrix},\qquad X_{i}=\begin{pmatrix}\chi_{i}&0\\ 0&w_{0}\bar{\chi}_{i}\end{pmatrix},

with $0<\lambda_{i}\leq 1$ , $\theta_{i}\in{\mathbb{T}}$ and $\chi_{i}\in\hat{T}$ such that if $\lambda_{i}=1$ , then either $0<\arg\theta_{i}\leq\pi$ or ( $\theta_{i}=1$ and $\chi^{2}_{i}\neq w_{0}$ );

–

or $1\times 1$ matrices $A_{i}=1$ , $X_{i}=\chi_{i}$ , with $\chi_{i}\in\hat{T}$ such that $\chi^{2}_{i}=w_{0}$ .

Proof.

The lemma can be viewed as a refinement of [HN21, Proposition 1.5], but the proof is almost the same, so we will be brief.

First one observes that classification of the pairs $(A,X)$ up to the transformations $A\mapsto vAv^{t}$ and $X\mapsto vXv^{*}$ is the same as classification of the pairs $(AJ,X)$ up to unitary conjugacy, where $J\colon{\mathbb{C}}^{m}\to{\mathbb{C}}^{m}$ is the complex conjugation. Consider the polar decomposition $AJ=u|AJ|$ , so $|AJ|$ is positive and $u$ is anti-unitary. Then the joint spectrum, considered together with multiplicities, of the commuting operators $|AJ|$ , $u^{2}$ and $X(t)$ ( $t\in T$ ) is a complete invariant of the unitary conjugacy class, and a standard form of $(A,X)$ as in the formulation of the lemma is obtained from an appropriate orthonormal basis diagonalizing these operators, as follows.

For $\lambda>0$ , $\theta\in{\mathbb{T}}$ and $\chi\in\hat{T}$ , consider the space

H_{\lambda,\theta,\chi}=\{\xi\in{\mathbb{C}}^{m}:|AJ|\xi=\lambda\xi,\ u^{2}\xi=\theta\xi,\ X(t)\xi=\chi(t)\xi\ \text{for all}\ t\in T\}.

As $u|AJ|=|AJ|^{-1}u$ and $w_{0}uX=u\bar{w}_{0}X=Xu$ , we have $uH_{\lambda,\theta,\chi}=H_{\lambda^{-1},\bar{\theta},w_{0}\bar{\chi}}$ . In particular, the spaces $H_{\lambda,\theta,\chi}+H_{\lambda^{-1},\bar{\theta},w_{0}\bar{\chi}}$ are invariant under $AJ$ and $X$ . For every pair of triples $(\lambda,\theta,\chi)$ and $(\lambda^{-1},\bar{\theta},w_{0}\bar{\chi})$ such that $H_{\lambda,\theta,\chi}\neq 0$ , pick a representative and consider three cases.

1) Assume $(\lambda,\theta,\chi)\neq(\lambda^{-1},\bar{\theta},w_{0}\bar{\chi})$ . Then, by changing the representative if necessary, we may also assume that either $\lambda<1$ or ( $\lambda=1$ and $0\leq\arg\theta\leq\pi$ ). Choose an orthonormal basis $(\xi_{j})_{j}$ in $H_{\lambda,\theta,\chi}$ . Then $(u\xi_{j})_{j}$ is an orthonormal basis in $H_{\lambda^{-1},\bar{\theta},w_{0}\bar{\chi}}$ , and the restrictions of $AJ$ and $X$ to every $2$ -dimensional space with basis $\{\xi_{j},u\xi_{j}\}$ have the required form.

2) Assume $\lambda=1$ , $\theta=-1$ , $\chi^{2}=w_{0}$ . In this case $u\xi\perp\xi$ for every $\xi\in H_{1,-1,\chi}$ . Hence we can find an orthonormal system $(\xi_{j})_{j}$ in $H_{1,-1,\chi}$ such that $(\xi_{j})_{j}\cup(u\xi_{j})_{j}$ is an orthonormal basis in $H_{1,-1,\chi}$ . The restrictions of $AJ$ and $X$ to every $2$ -dimensional space with basis $\{\xi_{j},u\xi_{j}\}$ have the required form.

3) Finally, assume $\lambda=1$ , $\theta=1$ , $\chi^{2}=w_{0}$ . Then $\{\xi\in H_{1,1,\chi}\mid u\xi=\xi\}$ is a real form of $H_{1,1,\chi}$ . By choosing an orthonormal basis in this Euclidean space we get a decomposition of $H_{1,1,\chi}$ into a direct sum of one-dimensional spaces that are invariant under $AJ$ and $X$ . ∎

From the proof one can see that the only source of nonuniqueness of the standard form of $(A,X)$ , apart from the order of the blocks, is the choice of a representative from each pair of the triples $(1,\pm 1,\chi)$ and $(1,\pm 1,w_{0}\bar{\chi})$ with $\chi^{2}\neq w_{0}$ .

Remark 3.9.

If $\hat{T}$ has zero $2$ -torsion, then there is at most one $\chi$ such that $\chi^{2}=w_{0}$ . Then, as in [HN21], we can choose an orthonormal basis $(\xi_{j})^{l}_{j=1}$ in $H_{1,1,\chi}$ such that $u\xi_{j}=\xi_{l-j+1}$ . It follows that instead of making $vAv^{t}$ block-diagonal, we can make it to be of the form (3.3), with $0<a_{i}\leq 1$ for $i\leq[\frac{m+1}{2}]$ and $0\leq\arg a_{i}\leq\pi$ for $i>[\frac{m+1}{2}]$ whenever $|a_{i}|=1$ . $\diamond$

We can now parameterize the isomorphism classes of braided free orthogonal quantum groups $O^{X,\beta}_{A}$ , at least when $\hat{T}$ is torsion-free. We already know that it suffices to consider only the transmutations of free orthogonal quantum groups, that is, we may assume that $A\bar{A}=\pm 1$ and $w=1$ (the trivial character), and that then the only choice for $\chi$ in (3.8) is $\chi=1$ . Hence we get the following result.

Corollary 3.10.

Consider a compact connected abelian group $T$ and a bicharacter $\beta\colon\hat{T}\times\hat{T}\to{\mathbb{T}}$ . Then representatives ${\mathbb{C}}[O^{X,\beta}_{A}]\in\mathrm{Hopf}^{*}({\mathbb{C}}[T],\beta)$ of all isomorphism classes of braided free orthogonal quantum groups are obtained by considering the pairs $(A,X)$ such that

–

the matrix $A$ has the form (3.3), with $0<a_{i}\leq 1$ for $i\leq[\frac{m+1}{2}]$ and $a_{i}a_{m-i+1}=\tau$ for all $i$ , where $\tau=\pm 1$ (hence $a_{\frac{m+1}{2}}=1$ and $\tau=1$ if $m$ is odd);
–

the corepresentation matrix $X$ is diagonal, $X=\operatorname{diag}(\chi_{1},\dots,\chi_{m})$ , with $\chi_{i}\chi_{m-i+1}=1$ for all $i$ (hence $\chi_{\frac{m+1}{2}}=1$ if $m$ is odd).

Two such pairs $(A,X)$ and $(A^{\prime},X^{\prime})$ define isomorphic braided Hopf $*$ -algebras over $({\mathbb{C}}[T],\beta)$ if and only if $(A^{\prime},X^{\prime})$ can be obtained from $(A,X)$ by

–

permuting the pairs $(a_{i},a_{m-i+1})$ and $(\chi_{i},\chi_{m-i+1})$ ( $i\leq[\frac{m}{2}]$ );
–

replacing $\chi_{i}$ by $\chi_{m-i+1}$ and $\chi_{m-i+1}$ by $\chi_{i}$ for some $i\leq[\frac{m}{2}]$ such that $a_{i}=1$ .

Now, observe that the quantum group (3.7) is isomorphic to $U_{q}(2)$ when $\chi$ is nontrivial and $T/\ker\chi\cong{\mathbb{T}}$ . All compact quantum groups monoidally equivalent to $U_{q}(2)$ are known by the work of Mrozinski [Mro14]. Our next goal is to describe the subclass of these quantum groups that can be obtained as the bosonizations of ${\mathbb{C}}[O^{X,\beta}_{A}]\in\mathrm{Hopf}^{*}({\mathbb{C}}[T/T_{0}],i^{*}\beta)$ for suitable $T_{0}$ .

Assume that $B\in\mathrm{GL}_{m}({\mathbb{C}})$ is such that $B\bar{B}$ is unitary. Following the conventions of [HN22], consider the quantum group $\tilde{O}_{B}^{+}$ of unitary transformations leaving the noncommutative polynomial $P=\sum_{i,j}B_{ji}X_{i}X_{j}$ invariant up to a phase factor: ${\mathbb{C}}[\tilde{O}_{B}^{+}]$ is the universal unital $*$ -algebra generated by a unitary $d$ and elements $w_{ij}$ , $1\leq i,j\leq m$ , such that

W=(w_{ij})_{i,j}\mbox{ is unitary and }W=B\bar{W}B^{-1}d,

(3.9)

and the coproduct is given by $\Delta(w_{ij})=\sum_{k}w_{ik}\otimes w_{kj}$ , $\Delta(d)=d\otimes d$ . We remark that in [Mro14] the Hopf $*$ -algebra ${\mathbb{C}}[\tilde{O}_{B}^{+}]$ is denoted by $A_{\tilde{o}}(\bar{B}^{t})$ .

Proposition 3.11.

Assume that $B\in\mathrm{GL}_{m}({\mathbb{C}})$ ( $m\geq 2$ ) is such that $B\bar{B}$ is unitary. Then the following conditions are equivalent:

1)

${\mathbb{C}}[\tilde{O}_{B}^{+}]$ is the bosonization of ${\mathbb{C}}[O^{X,\beta}_{B}]\in\mathrm{Hopf}^{*}({\mathbb{C}}[{\mathbb{T}}/\{\pm 1\}],i^{*}\beta)$ for some unitary corepresentation matrix $X\in\operatorname{Mat}_{m}({\mathbb{C}}[{\mathbb{T}}])$ and a bicharacter $\beta$ on ${\mathbb{Z}}=\hat{\mathbb{T}}$ as in Proposition 3.2 with $X(-1)=1$ and $w=z^{-1}$ ;
2)

${\mathbb{C}}[\tilde{O}_{B}^{+}]$ is the bosonization of a braided Hopf $*$ -algebra over $({\mathbb{C}}[{\mathbb{T}}],\beta)$ for some bicharacter $\beta$ on ${\mathbb{Z}}=\hat{\mathbb{T}}$ ;
3)

$m$ is even and the spectrum of $B\bar{B}$ consists of odd powers of a single number $\lambda\in{\mathbb{T}}$ .

Proof.

1) $\Rightarrow$ 2) is obvious.

2) $\Rightarrow$ 3): As $\tilde{O}_{B}^{+}$ has fusion rules of $U(2)$ , the group-like elements of ${\mathbb{C}}[\tilde{O}_{B}^{+}]$ are powers of $d$ . It follows that if we can identify ${\mathbb{C}}[\tilde{O}_{B}^{+}]$ with ${\mathbb{C}}[{\mathbb{T}}]\#A$ for some $A\in\mathrm{Hopf}^{*}({\mathbb{C}}[{\mathbb{T}}],\beta)$ , then $z\#1=d$ or $z\#1=d^{-1}$ . By replacing $B$ by $\bar{B}^{-1}$ if necessary, we may assume that $z\#1=d$ , since there is an isomorphism $\tilde{O}_{B}^{+}\cong\tilde{O}_{\bar{B}^{-1}}^{+}$ mapping $d$ into $d^{-1}$ , see [Mro14, Proposition 3.4].

Consider the Hopf $*$ -algebra map $\iota\#\varepsilon_{A}\colon{\mathbb{C}}[{\mathbb{T}}]\#A\to{\mathbb{C}}[{\mathbb{T}}]$ . Put $X=(\iota\#\varepsilon_{A})(W)$ . Then from the defining relations for $\tilde{O}_{B}^{+}$ we get that $X$ is a unitary corepresentation matrix and $X=B\bar{X}B^{-1}z$ . By Remark 3.9 we may assume that

B=\begin{pmatrix}0&&b_{m}\\ &\iddots&\\ b_{1}&&0\end{pmatrix},\quad b_{i}\bar{b}_{m-i+1}=\lambda_{i}\in{\mathbb{T}},

(3.10)

and $X=\mathrm{diag}(z^{k_{1}},z^{k_{2}},...,z^{k_{m}})$ for some $k_{i}\in{\mathbb{Z}}$ , $1\leq i\leq m$ . The identity $X=B\bar{X}B^{-1}z$ means then that $k_{i}+k_{m-i+1}=1$ for all $i$ . This implies that $m$ must be even.

Next, consider $U=(\varepsilon_{{\mathbb{C}}[{\mathbb{T}}]}\#\iota)(W)\in\operatorname{Mat}_{m}(A)$ . On the one hand, by (2.12), we have

\delta_{A}(u_{ij})=u_{ij}\otimes z^{k_{j}-k_{i}}.

As $W=X\#U=(z^{k_{i}}\#u_{ij})_{i,j}$ and $z\#1=d$ , by the definition of the smash product ${\mathbb{C}}[{\mathbb{T}}]\#A$ it follows that

dWd^{*}=(\beta(k_{i}-k_{j},1)w_{ij})_{i,j}.

On the other hand, the relation $W=B\bar{W}B^{-1}d$ implies $\bar{W}=d^{*}\bar{B}W\bar{B}^{-1}$ , hence

dWd^{*}=B\bar{B}W(B\bar{B})^{-1}=(\lambda_{i}\bar{\lambda}_{j}w_{ij})_{i,j}.

We therefore have that $\beta(k_{i}-k_{j},1)=\lambda_{i}\bar{\lambda}_{j}$ for all $i,j$ , whence $\lambda_{i}=\lambda\zeta^{k_{i}}$ for some $\lambda\in{\mathbb{T}}$ , where $\zeta=\beta(1,1)$ . As $\bar{\lambda}_{i}=\lambda_{m-i+1}$ and $k_{i}+k_{m-i+1}=1$ , we must have $\lambda^{-2}=\zeta$ and thus $\lambda_{i}=\lambda^{-2k_{i}+1}$ .

3) $\Rightarrow$ 1): We may assume that $B$ is of the form (3.10). By the assumption on the spectrum there exist $l_{i}\in{\mathbb{Z}}$ for $i=1,\dots,m/2$ such that $\lambda_{i}=\lambda^{-2l_{i}-1}$ . Let $l_{i}=-1-l_{m-i+1}$ for $i=m/2+1,\dots,m$ . Then $\lambda_{i}=\lambda^{-2l_{i}-1}$ for all $i$ .

Put $X=\mathrm{diag}(z^{l_{1}},z^{l_{2}},...,z^{l_{m}})$ . Define a bicharacter $\beta$ on $\hat{\mathbb{T}}$ by $\beta(k,l)=\lambda^{-2kl}$ . We then have $X=B\bar{X}B^{-1}z^{-1}$ and $B\bar{B}=\lambda^{-1}X(\lambda^{-2})$ . Consider the double cover $p\colon{\mathbb{T}}\to{\mathbb{T}}$ , $t\mapsto t^{2}$ . Denote by $\beta^{1/4}$ the bicharacter on $\hat{\mathbb{T}}$ defined in the same way as $\beta$ , but with $\lambda^{2}$ replaced by one of its fourth roots. Therefore if $i\colon{\mathbb{C}}[{\mathbb{T}}]\to{\mathbb{C}}[{\mathbb{T}}]$ is defined by $i(z)=z^{2}$ , we get $i^{*}\beta^{1/4}=\beta$ . By Remark 3.5 the braided quantum group $O_{B}^{Xp,\beta^{1/4}}$ is well-defined. As $(Xp)(-1)=1$ , we can view ${\mathbb{C}}[O_{B}^{Xp,\beta^{1/4}}]$ as an object $A$ in $\mathrm{Hopf}^{*}({\mathbb{C}}[{\mathbb{T}}/\{\pm 1\}],i^{*}\beta^{1/4})=\mathrm{Hopf}^{*}({\mathbb{C}}[{\mathbb{T}}],\beta)$ .

The defining relations in $A$ say that $B$ is an intertwiner of the $A$ -comodules defined by $(z^{-1}\bar{X},\bar{U}_{X})$ and $(X,U)$ . In other words, for $W^{\prime}=X\#U=(z^{l_{i}}\#u_{ij})_{i,j}$ we have

B(z^{-1}\#1)\overline{W^{\prime}}=W^{\prime}B.

The relation $W=B\bar{W}B^{-1}d$ in ${\mathbb{C}}[\tilde{O}^{+}_{B}]$ can be written as

d^{*}W=Bd^{*}\overline{d^{*}W}B^{-1}.

It follows that we have a well-defined Hopf $*$ -algebra map ${\mathbb{C}}[\tilde{O}^{+}_{B}]\to{\mathbb{C}}[{\mathbb{T}}]\#A$ such that $d\mapsto z\#1$ and $W=(w_{ij})_{i,j}\mapsto zX\#U=(z^{l_{i}+1}\#u_{ij})_{i,j}$ . Using the relations in both algebras it is also easy to construct the inverse map. ∎

Remark 3.12.

The proof of the proposition implies that the procedure described in the proof of the implication 3) $\Rightarrow$ 1) is the only way of getting decompositions ${\mathbb{C}}[\tilde{O}^{+}_{B}]={\mathbb{C}}[{\mathbb{T}}]\#A$ such that $A\in\mathrm{Hopf}^{*}({\mathbb{C}}[{\mathbb{T}}],\beta)$ and $d=z\#1$ .

Remark 3.13.

As in the proof of the proposition, it is not difficult to see that given $B\in\mathrm{GL}_{m}({\mathbb{C}})$ such that $B\bar{B}$ is unitary, a unitary corepresentation matrix $X\in\mathrm{Mat}_{m}({\mathbb{C}}[{\mathbb{T}}])$ such that $X=B\bar{X}B^{-1}z$ exists if and only if $m$ is even. Therefore, for even $m$ , we always get a Hopf $*$ -algebra map $p:{\mathbb{C}}[\tilde{O}_{B}^{+}]\to{\mathbb{C}}[{\mathbb{T}}]$ such that $p(d)=z$ with a right inverse $z\mapsto d$ . By Radford’s theorem [Rad85] we then get a Hopf $*$ -algebra object in the braided category $\mathcal{YD}({\mathbb{T}})$ of ${\mathbb{T}}$ -Yetter–Drinfeld modules. From this perspective the above proposition characterizes when this object lies in the subcategory $(\mathcal{M}^{{\mathbb{C}}[{\mathbb{T}}]},\beta)\subset\mathcal{YD}({\mathbb{T}})$ for some $\beta$ . $\diamond$

Finally, let us compare the transmutations of ${\mathbb{C}}[O_{F}^{+}]$ to the braided quantum groups constructed in [MR22]. Fix numbers $d_{1},d_{2},...,d_{m},d\in{\mathbb{Z}}$ and consider the representation

X\colon{\mathbb{T}}\to\operatorname{Mat}_{m}({\mathbb{C}}),\quad X(t)=\begin{pmatrix}t^{d_{1}}&&0\\ &\ddots&\\ 0&&t^{d_{m}}\end{pmatrix}.

Take $\Omega=(\omega_{ij})_{i,j}\in\mathrm{GL}_{m}({\mathbb{C}})$ and assume

\omega_{ij}\neq 0\,\Rightarrow\,d_{i}+d_{j}=d,\quad\text{or equivalently},\quad\Omega^{t}(z^{d}\bar{X})(\Omega^{t})^{-1}=X,

where $z\in{\mathbb{C}}[{\mathbb{T}}]$ is the generator. Assume further that there is $\zeta\in{\mathbb{T}}$ such that

\bar{\Omega}\Omega=c\,X(\zeta^{d})\quad\text{for some}\quad c\in{\mathbb{T}}.

Define a bicharacter on $\hat{{\mathbb{T}}}={\mathbb{Z}}$ by $\beta(m,n)=\zeta^{mn}$ .

In [MR22], Meyer and Roy construct an object $A_{o}(\Omega,X)\in\mathrm{Hopf}^{*}({\mathbb{C}}[{\mathbb{T}}],\beta)$ from this data. By [MR22, Theorem 2.6] and [BJR22, Remark 2.20(2)], in our terminology, $A_{o}(\Omega,X)$ is defined as a universal braided compact matrix quantum group over $({\mathbb{T}},X,\beta)$ with fundamental unitary $U=(u_{ij})_{i,j}$ subject to the relations

U=A\bar{U}_{X}A^{-1},

where $A=(\omega_{ji}\zeta^{dd_{j}})_{i,j}=\Omega^{t}X(\zeta^{d})=\zeta^{d^{2}}X(\zeta^{-d})\Omega^{t}$ . Note that we have $\bar{U}_{X}=(\zeta^{d_{i}(d_{j}-d_{i})}u_{ij}^{*})_{i,j}$ and $A\bar{A}=c\,\zeta^{d^{2}}X(\zeta^{-d})$ .

Assume that $d=2n$ . Put $w(t)=t^{n}$ . Then we see from the description above and Remark 3.5 that $A_{o}(\Omega,X)\cong{\mathbb{C}}[O_{A}^{X,\beta}]$ in $\mathrm{Hopf}^{*}({\mathbb{C}}[{\mathbb{T}}],\beta)$ .

Next, assume that $d$ is odd. Then, as in the proof of Proposition 3.11, $m$ is even. Similarly to that proof, consider the double cover $p\colon{\mathbb{T}}\to{\mathbb{T}}$ , $t\mapsto t^{2}$ , to write the character $z^{d}$ as a square. Write $\beta^{1/4}$ for the bicharacter on $\hat{\mathbb{T}}$ defined in the same way as $\beta$ , but with $\zeta$ replaced by a fourth root $\zeta^{1/4}$ . Define a character $w$ on the double cover by $w(t)=t^{d}$ . Then

A_{o}(\Omega,X)\cong{\mathbb{C}}[O_{A}^{Xp,\beta^{1/4}}],

where we now view the latter braided Hopf $*$ -algebra as an object in $\mathrm{Hopf}^{*}({\mathbb{C}}[{\mathbb{T}}/\{\pm 1\}],i^{*}\beta^{1/4})=\mathrm{Hopf}^{*}({\mathbb{C}}[{\mathbb{T}}],\beta)$ .

3.3. Braided free unitary quantum groups

Let $m\geq 2$ be a natural number. We recall the definition of the free unitary quantum group $U_{F}^{+}$ . Let $F=(f_{ij})_{i,j}\in\mathrm{GL}_{m}({\mathbb{C}})$ be such that $\operatorname{Tr}(F^{*}F)=\operatorname{Tr}((F^{*}F)^{-1})$ . Then ${\mathbb{C}}[U_{F}^{+}]$ denotes the universal $*$ -algebra with generators $u_{ij}$ , $1\leq i,j\leq m$ , and relations determined by

U=(u_{ij})_{i,j}\mbox{ and }F\bar{U}F^{-1}\mbox{ are unitaries in }\operatorname{Mat}_{m}({\mathbb{C}}[U_{F}^{+}]).

The Hopf $*$ -algebra structure on ${\mathbb{C}}[U_{F}^{+}]$ 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 $Z\in\operatorname{Mat}_{m}({\mathbb{C}}[T])$ such that $F\bar{Z}F^{-1}$ is unitary, equivalently, $\bar{Z}$ commutes with $|F|$ . Then, by the universality of ${\mathbb{C}}[U_{F}^{+}]$ , there is a Hopf $*$ -algebra map $\pi\colon{\mathbb{C}}[U_{F}^{+}]\to{\mathbb{C}}[T]$ mapping $U$ to $Z$ . Let $\beta\colon\hat{T}\times\hat{T}\to{\mathbb{T}}$ be a bicharacter, and consider the transmutation ${\mathbb{C}}[U^{+}_{F}]_{\beta}$ . Then, similarly to Proposition 3.1, we get the following result.

Proposition 3.14.

The braided Hopf $*$ -algebra ${\mathbb{C}}[U_{F}^{+}]_{\beta}$ is a braided compact matrix quantum group over $(T,Z,\beta)$ 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\quad\mbox{and}\quad F\bar{U}_{Z}F^{-1}\quad\text{are unitaries},

where $\bar{U}_{Z}=(\bar{u}_{ij}^{Z})_{i,j}$ and $\bar{u}_{ij}^{Z}=\sum_{s,l,t}\beta(z_{tj}^{*}z_{sl},z_{il}^{*})u_{st}^{*}$ .

By Theorem 2.5, the bosonization of ${\mathbb{C}}[U_{F}^{+}]_{\beta}$ is a compact quantum group that is a cocycle twist of $T\times U^{+}_{F}$ .

When $T={\mathbb{T}}$ and $Z$ is such that both $Z$ and $F\bar{Z}F^{-1}$ 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 $F\bar{Z}F^{-1}$ is diagonal by choosing an appropriate orthonormal basis, but it is usually impossible to make both of them diagonal simultaneously.

3.4. Anyonic quantum permutation groups

Let $N\geq 2$ be a natural number. The quantum symmetric group $S_{N}^{+}$ is the universal compact matrix quantum group with fundamental unitary representation $U=(u_{ij})_{i,j=0}^{N-1}$ subject to the relations

\sum_{i}u_{ij}=1=\sum_{j}u_{ij}\quad\mbox{ and }\quad u_{ij}^{*}=u_{ij}^{2}=u_{ij}.

We can view the cyclic group ${\mathbb{Z}}/N{\mathbb{Z}}$ as a subgroup of $S_{N}\subset S_{N}^{+}$ , so we get a Hopf $*$ -algebra map

\pi\colon{\mathbb{C}}[S_{N}^{+}]\to{\mathbb{C}}[{\mathbb{Z}}/N{\mathbb{Z}}],\quad\pi(u_{ij})=\delta_{j-i},

where $\delta_{k}\in{\mathbb{C}}[{\mathbb{Z}}/N{\mathbb{Z}}]$ are the usual delta-functions.

To describe the transmutation we want to express the relations in ${\mathbb{C}}[S_{N}^{+}]$ in terms of homogeneous elements with respect to the bi-grading by ${\mathbb{Z}}/N{\mathbb{Z}}$ . Fix a primitive $N$ -th root of unity $\omega$ . By Lemma 4.9 in [Ans+22] such generators can be obtained by considering the elements $a_{ij}$ defined by

(a_{ij})_{i,j}=\Omega^{-1}U\Omega,\quad\Omega=\Big{(}\dfrac{1}{N}\omega^{-ij}\Big{)}_{i,j=0}^{N-1}.

The elements $a_{ij}$ are then bi-graded by

(\pi\otimes\iota)\Delta(a_{ij})=z^{i}\otimes a_{ij},\quad(\iota\otimes\pi)\Delta(u_{ij})=a_{ij}\otimes z^{j},

where $z\in{\mathbb{C}}[{\mathbb{Z}}/N{\mathbb{Z}}]$ is the function $z(k)=\omega^{k}$ . In terms of the new generators the relations in ${\mathbb{C}}[S_{N}^{+}]$ become

a_{0i}=a_{i0}=\delta_{i,0},\quad a_{ij}^{*}=a_{-i,-j},

a_{k,i+j}=\sum_{l}a_{k-l,i}a_{lj},\quad a_{i+j,k}=\sum_{l}a_{jl}a_{i,k-l}.

Define a bicharacter by $\beta(z^{i},z^{j})=\omega^{-ij}$ . It is then readily verified, by using formulas (2.4) and (2.6), that the transmutation ${\mathbb{C}}[S_{N}^{+}]_{\beta}$ is described by the relations

a_{0i}=a_{i0}=\delta_{i,0},\quad a_{ij}^{*}=\omega^{i(j-i)}a_{-i,-j},

a_{k,i+j}=\sum_{l}\omega^{-l(i-k+l)}a_{k-l,i}a_{lj},\quad a_{i+j,k}=\sum_{l}\omega^{-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 ${\mathbb{C}}[S_{N}^{+}]_{\beta}$ is a cocycle twist of the quantum group $({\mathbb{Z}}/N{\mathbb{Z}})\times S_{N}^{+}$ .

References

[Ans+22] Anshu, Suvrajit Bhattacharjee, Atibur Rahaman and Sutanu Roy “Anyonic quantum symmetries of finite spaces” arXiv, 2022 DOI: 10.48550/ARXIV.2207.08153
[BJR22] Suvrajit Bhattacharjee, Soumalya Joardar and Sutanu Roy “Braided quantum symmetries of graph $\mathrm{C}^{*}$ -algebras” arXiv, 2022 DOI: 10.48550/ARXIV.2201.09885
[BS19] Arkadiusz Bochniak and Andrzej Sitarz “Braided Hopf algebras from twisting” In J. Algebra Appl. 18.9, 2019, pp. 1950178\bibrangessep18 DOI: 10.1142/S0219498819501780
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[HN21] Erik Habbestad and Sergey Neshveyev “Subproduct systems with quantum group symmetry” Preprint, 2021 DOI: 10.48550/ARXIV.2111.10911
[HN22] Erik Habbestad and Sergey Neshveyev “Subproduct systems with quantum group symmetry. II” Preprint, 2022 URL: https://arxiv.org/abs/2212.08512
[Kas+16] Paweł Kasprzak, Ralf Meyer, Sutanu Roy and Stanisław Lech Woronowicz “Braided quantum $\rm SU(2)$ groups” In J. Noncommut. Geom. 10.4, 2016, pp. 1611–1625 DOI: 10.4171/JNCG/268
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[Maj95] Shahn Majid “Foundations of quantum group theory” Cambridge University Press, Cambridge, 1995, pp. x+607 DOI: 10.1017/CBO9780511613104
[MR22] Ralf Meyer and Sutanu Roy “Braided free orthogonal quantum groups” In Int. Math. Res. Not. IMRN, 2022, pp. 8890–8915 DOI: 10.1093/imrn/rnaa379
[Mro14] Colin Mrozinski “Quantum groups of $\rm GL(2)$ representation type” In J. Noncommut. Geom. 8.1, 2014, pp. 107–140 DOI: 10.4171/JNCG/150
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Cocycle twisting of semidirect products and transmutation

Abstract.

Introduction

1. Generalities

1.1. Categories of comodules

Definition 1.1.

Proposition 1.2 (cf. [Maj95, Theorem 9.4.12]).

1.2. Twisting and transmutation

Proposition 1.3.

Proposition 1.4 ([Maj93, Theorem 4.1]).

Theorem 1.5.

Proof.

Remark 1.6.

2. Transmutation over abelian groups

2.1. Braided Hopf algebras over quotients of TT

Proposition 2.1.

Proof.

Remark 2.2.

Lemma 2.3.

Proof.

Proposition 2.4.

Theorem 2.5.

Proof of Theorem 2.5..

Corollary 2.6.

Remark 2.7.

Lemma 2.8.

Proof.

2.2. Another view on HβH_{\beta}

Lemma 2.9.

Proof.

Proposition 2.10.

Example 2.11.

2.3. Braided compact matrix quantum groups

Definition 2.12 ([Wor87, Wor91]).

Lemma 2.13.

Proof.

Definition 2.14.

Proposition 2.15.

Proof.

Proposition 2.16.

Proof.

Remark 2.17.

Lemma 2.18.

Proof.

3. Examples: transmuting matrix quantum groups

3.1. Braided S​Uq​(2)SU_{q}(2)

3.2. Braided free orthogonal quantum groups

Proposition 3.1.

Proof.

Proposition 3.2.

Proof.

Corollary 3.3.

Remark 3.4.

Remark 3.5.

Proposition 3.6.

Proposition 3.7.

Proof.

Lemma 3.8.

Proof.

Remark 3.9.

Corollary 3.10.

Proposition 3.11.

Proof.

Remark 3.12.

Remark 3.13.

3.3. Braided free unitary quantum groups

Proposition 3.14.

3.4. Anyonic quantum permutation groups

References

2.1. Braided Hopf algebras over quotients of $T$

2.2. Another view on $H_{\beta}$

3.1. Braided $SU_{q}(2)$