---
title: Quasitriangular Comodule Algebras
url: https://www.emergentmind.com/topics/quasitriangular-comodule-algebras
type: topic
---

# Quasitriangular Comodule Algebras

Searching arXiv for recent and foundational papers on quasitriangular comodule algebras and related \(K\)-matrix/module-category formulations.
Quasitriangular comodule algebras are comodule-algebra analogues of quasitriangular Hopf algebras, formulated so that a coaction by a quasitriangular Hopf algebra \((H,R)\) is supplemented by an invertible element \(K\in H\otimes A\) satisfying boundary quantum Yang–Baxter identities or, equivalently, a reflection equation. In the Hopf-theoretic formulation, a left \(H\)-comodule algebra \(A\) with coaction \(\delta(a)=a_{[-1]}\otimes a_{[0]}\) is quasitriangular if there exists an invertible \(K=\sum_j g_j\otimes p_j\in H\otimes A\) satisfying three coherence equations: \((\Delta\otimes\mathrm{Id}_A)K = K_{23}R_{21}K_{13}R_{21}^{-1}\), \((\mathrm{Id}_H\otimes\delta)K = R_{21}K_{13}R_{12}\), and \(K\,\delta(a)=\delta(a)\,K\) for all \(a\in A\) [2307.14764]. These data provide the boundary counterpart of universal \(R\)-matrices, induce braided module-category structures on representation categories, and connect coideal subalgebras of quantum groups, generalized Satake diagrams, reflective centers, factorizability, and Morita invariants [1807.02388] [2307.14764] [2411.18453] [2508.19845].

## 1. Foundational definition and reflection-equation formalism

Let \((H,R)\) be a quasitriangular Hopf algebra over a field \(\Bbbk\), with
\[
R=\sum_i s_i\otimes t_i\in H\otimes H,\qquad R^{-1}=\sum_i s^i\otimes t^i.
\]
A left \(H\)-comodule algebra is a \(\Bbbk\)-algebra \(A\) equipped with a coaction
\[
\delta\colon A\to H\otimes A,\qquad \delta(a)=a_{[-1]}\otimes a_{[0]},
\]
such that \(\delta\) is an algebra map [2307.14764]. In the parallel notation of later work, one also writes \(\delta(a)=a_{(-1)}\otimes a_{(0)}\) [2508.19845] [2411.18453].

A quasitriangular structure on \((A,\delta)\) is an invertible element
\[
K=\sum_j K_j\otimes K^j\in H\otimes A
\]
satisfying the three boundary quantum–Yang–Baxter identities
\[
(\Delta\otimes\mathrm{id})(K)=K_{23}R_{21}K_{13}R_{21}^{-1},
\]
\[
(\mathrm{id}\otimes\delta)(K)=R_{21}K_{13}R_{12},
\]
\[
K\bigl(a_{(-1)}\otimes a_{(0)}\bigr)=\bigl(a_{(-1)}\otimes a_{(0)}\bigr)K,\qquad \forall a\in A
\]
[2508.19845]. The same three axioms appear in Kolb’s formulation, where they are denoted \((\mathrm{QT1})\), \((\mathrm{QT2})\), and \((\mathrm{QT3})\) [2411.18453].

An equivalent encoding is the reflection equation
\[
K_{12}R_{21}K_{21}R_{12}=R_{21}K_{21}R_{12}K_{12}
\]
in \(H\otimes H\otimes A\) [2508.19845]. In the ribbon-Hopf setting, one may twist \(K\) by the ribbon element to recover Kolb’s version of the quantum-reflection equations [2307.14764]. This places quasitriangular comodule algebras in direct analogy with quasitriangular Hopf algebras: the universal \(R\)-matrix governs bulk braiding, while the universal \(K\)-matrix governs boundary braiding.

A standard misconception is that a quasitriangular comodule algebra is merely a comodule algebra over a quasitriangular Hopf algebra. The cited works make clear that the essential additional datum is the \(K\)-matrix; without it one has only an \(H\)-module-category structure, not a braided \(H\)-module-category structure [2508.19845] [2411.18453].

## 2. Coideal subalgebras of quantum groups and universal \(K\)-matrices

For Drinfeld–Jimbo quantum groups \(U_q(\mathfrak g)\), quasitriangular comodule-algebra phenomena are realized through coideal subalgebras. A unital subalgebra \(B\subset U_q(\mathfrak g)\) is a right coideal subalgebra if
\[
\Delta(B)\subset B\otimes U_q(\mathfrak g).
\]
Then \(B\) becomes a right \(U_q(\mathfrak g)\)-comodule algebra via the restricted coproduct
\[
\delta:=\Delta|_B\colon B\to B\otimes U_q(\mathfrak g),
\]
which satisfies coassociativity, counitality, multiplicativity, and \(\delta(1_B)=1_B\otimes 1\) [1807.02388]. The standard Borel subalgebra \(U_q(\mathfrak b^+)\) is an example of a coideal subalgebra [1807.02388].

The quantum-pair construction associated to generalized Satake data starts from a compatible decoration \((X,\tau)\), where \(X\subset I\) is a union of connected components of the Dynkin diagram and \(\tau\colon I\to I\) is a diagram involution preserving \(X\) [1807.02388]. For \((X,\tau)\in\mathrm{GSat}(A)\) and parameters
\[
\gamma=(\gamma_i)_{i\in I\setminus X}\in (\Bbbk^\times)^{I\setminus X},\qquad
\sigma=(\sigma_i)_{i\in I\setminus X}\in \Bbbk^{I\setminus X},
\]
the quantum pair coideal subalgebra
\[
B=B_{\gamma,\sigma}(X,\tau)\subset U_q(\mathfrak g)
\]
is generated by the Hopf subalgebra \(U_q(\mathfrak g_X)U_q(\mathfrak h)\) together with elements
\[
B_i=F_i+\gamma_i\,\theta_q(F_i)K_i^{-1}+\sigma_iK_i^{-1},\qquad i\in I\setminus X,
\]
where \(\theta_q=T_{w_X}\circ\tau\circ\omega_q\) is the quantum analogue of \(\theta=-w_X\tau\) [1807.02388]. These are explicitly described as Sklyanin-type elements. Since \(\Delta(B_i)\in B\otimes U_q(\mathfrak g)\), the algebra \(B\) is a right coideal subalgebra [1807.02388].

The central result is the existence criterion for a universal \(K\)-matrix. If \((X,\tau)\in\mathrm{GSat}(A)\) and the parameters satisfy the bar-involution constraints
\[
\gamma_i\gamma_{\tau(i)}=q_i^{(\theta(\alpha_i)-2\rho_X)(h_i)},\qquad
\sigma_i=\sigma_{\tau(i)},
\]
with \(\sigma_i=0\) unless \(i\) lies in a certain fixed set \(I_{\mathrm{ns}}\), then there exists an invertible universal \(K\)-matrix
\[
\mathcal K\in \widehat{U_q(\mathfrak g)}\,\widehat\otimes\,B
\]
such that
\[
\mathcal K\,\delta(b)=\bigl(\mathrm{id}\otimes\Delta\bigr)(\mathcal K)(1\otimes b),\qquad \forall b\in B,
\]
and
\[
R_{21}(1\otimes\mathcal K)R(1\otimes\mathcal K)
=
(1\otimes\mathcal K)R_{21}(1\otimes\mathcal K)R
\]
inside the completed tensor product [1807.02388]. Moreover, \(\mathcal K\) is uniquely determined up to multiplication by a central grouplike element of \(B\) [1807.02388].

The proof proceeds by constructing a quasi-\(K\)-matrix \(\mathfrak X\in\widehat{U_q(\mathfrak n^+)}\), then showing that
\[
\mathcal K=\mathfrak X\,\xi\,T_{w_X}^{-1}
\]
satisfies the intertwining property, and finally deriving the reflection equation from this intertwiner relation and the RTT-relations for the \(R\)-matrix [1807.02388]. This extends the quantum symmetric-pair framework of Letzter, Kolb, and Balagović–Kolb from ordinary Satake diagrams to generalized Satake diagrams [1807.02388].

## 3. Generalized Satake diagrams and the classification problem

Generalized Satake diagrams provide the combinatorial input controlling a broad family of quasitriangular coideal subalgebras of \(U_q(\mathfrak g)\). The set
\[
\mathrm{GSat}(A)
=
\left\{
(X,\tau)\ \text{compatible}\ \middle|\ 
\forall i\in I\setminus X:\ X(i)\cup\{i,\tau(i)\}\neq\emptyset
\right\}
\]
is defined using the Dynkin-node set \(I\), the Cartan matrix \(A=(a_{ij})\), and the compatibility conditions on \((X,\tau)\) [1807.02388]. If \((X,\tau)\in\mathrm{GSat}(A)\), then the restricted root system of the involution \(\theta=-w_X\tau\) is again of Coxeter type [1807.02388].

In the classical theory of quantum symmetric pairs, the relevant pairs \((X,\tau)\) belong to the usual Satake class \(\mathrm{Sat}(A)\). The generalized condition of Heck weakens the Satake compatibility and thereby produces new coideal subalgebras that still admit universal \(K\)-matrices [1807.02388]. The resulting subalgebras satisfy
\[
\mathfrak k\cap\mathfrak h=\mathfrak h^\theta,
\]
but \(\mathfrak k\) need not be the fixed-point subalgebra of \(\theta\) [1807.02388]. This is one of the main structural distinctions between generalized quantum pair algebras and the older quantum symmetric-pair setting.

The article on generalized Satake diagrams states two closely related classification claims. First, it states that Regelskis–Vlaar show that every right coideal subalgebra \(B\subset U_q(\mathfrak g)\) admitting an invertible solution of the reflection equation must arise from a unique generalized Satake diagram \((X,\tau)\) and parameters \((\gamma,\sigma)\) [1807.02388]. Second, it formulates the broader statement as a conjecture:
\[
\text{Every quasitriangular right coideal subalgebra of }U_q(\mathfrak g)
\text{ is, up to Hopf-algebra automorphism and isomorphism, of the form }
B_{\gamma,\sigma}(X,\tau).
\]
This is recorded as Conjecture 5.1 [1807.02388]. Taken together, these statements indicate that the classification program is presented partly as established structure theory and partly as a conjectural global description. A cautious reading therefore distinguishes the proved existence theorem for universal \(K\)-matrices from the strongest universal classification claim.

This framework unifies and extends the Letzter–Kolb classification of quantum symmetric pairs to a strictly larger family [1807.02388]. The larger family still supports a bar involution on \(B\), a factorized quasi-\(K\)-matrix with braid-group symmetries of the restricted Weyl group, and a low-dimensional classification of matrix reflection-equation solutions matching the generalized Satake list [1807.02388]. This suggests that generalized quantum symmetric pairs are the natural boundary counterparts of the usual Satake-based quantum symmetric pairs.

## 4. Braided module categories, reflective centers, and reflective algebras

Quasitriangular comodule algebras have an intrinsic categorical meaning: they are algebraic realizations of braided module categories. Given \((H,R)\) and a quasitriangular comodule algebra \((A,K)\), the category \(\mathsf{Rep}(A)\) of left \(A\)-modules becomes a braided module category over \(\mathsf{Rep}(H)\), with module braiding
\[
e_{X,M}(x\otimes m)=\sum_j (K_j\cdot x)\otimes (K^j\ast m)
\]
for \(X\in\mathsf{Rep}(H)\), \(M\in\mathsf{Rep}(A)\) [2508.19845]. The boundary Yang–Baxter axioms for \(K\) are exactly the conditions required for the braided module-category axioms [2508.19845]. The same construction is stated in finite-dimensional form in terms of \(B\)-\(\mathsf{FdMod}\) over \(H\)-\(\mathsf{FdMod}\) [2411.18453].

A categorical enlargement of this picture is given by the reflective center \(\mathcal E_{\mathcal C}(\mathcal M)\), introduced for a braided monoidal category \(\mathcal C\) and a \(\mathcal C\)-module category \(\mathcal M\) [2307.14764]. It is described as an analogue of the Drinfeld center adapted to module categories and is a canonical braided module category attached to \(\mathcal M\) [2307.14764]. In the Hopf setting with \(\mathcal C=H\text{-mod}\) and \(\mathcal M=A\text{-mod}\), the reflective center is equivalent to a category of modules over an explicit algebra \(R_H(A)\), called the reflective algebra [2307.14764].

The reflective algebra is constructed using Majid’s transmuted coalgebra \(\widehat H\), defined on the vector space \(H\) by
\[
\widehat\Delta(h)=\sum_{i,j} t_jh_{(1)}t_i\otimes h_{(2)}s_iS^{-1}(s_j),
\qquad \varepsilon_{\widehat H}=\varepsilon_H,
\]
and with the adjoint-twisted left \(H\)-action \(\ell\rightharpoonup h=\ell_{(2)}hS^{-1}(\ell_{(1)})\) [2307.14764]. Then \(\widehat H^*\) is a right \(H\)-module algebra, and one forms the crossed product
\[
R_H(A)=A\rtimes_H (\widehat H^*)^{\mathrm{op}}
\]
with underlying vector space \(A\otimes (\widehat H^*)^{\mathrm{op}}\) and relations
\[
\xi\,a=a_{[0]}\,(\xi\!\leftharpoonup a_{[-1]}),\qquad \xi\in\widehat H^*,\ a\in A
\]
[2307.14764].

The reflective algebra is itself quasitriangular. Under the identification above, the element
\[
K_{\mathrm{ref}}=\sum_d h_d\otimes \xi_d\in H\otimes (\widehat H^*)^{\mathrm{op}}\subset H\otimes R_H(A)
\]
is a quantum \(K\)-matrix for the \(H\)-comodule algebra \(R_H(A)\) [2307.14764]. Thus reflective centers provide a canonical method for producing quasitriangular comodule algebras from arbitrary comodule algebras.

A further universal property sharpens this construction. When \(A=\Bbbk\) is the trivial comodule algebra,
\[
R_H(\Bbbk)\cong (\widehat H^*)^{\mathrm{op}}\cong H^*
\]
as an \(H\)-comodule algebra, with its canonical \(K\)-matrix [2307.14764]. For any quasitriangular left \(H\)-comodule algebra \((Q,K)\), there is a unique comodule-algebra map
\[
\kappa\colon R_H(\Bbbk)\to Q
\]
sending the canonical \(K\)-matrix of \(R_H(\Bbbk)\) to the prescribed \(K\in H\otimes Q\), explicitly
\[
\kappa(\xi)=\sum_i \langle \xi,g_i\rangle p_i
\qquad\text{if } K=\sum_i g_i\otimes p_i
\]
[2307.14764]. Therefore \(R_H(\Bbbk)\) is an initial object in the category of quasitriangular \(H\)-comodule algebras [2307.14764].

This categorical lifting is explicitly compared with the Drinfeld-center and Drinfeld-double story: \(R_H(A)\) plays for \(K\)-matrices the role that \(\mathrm{Drin}(H)\) plays for \(R\)-matrices [2307.14764].

## 5. Factorizability and nondegeneracy

The theory of quasitriangular comodule algebras also admits a factorization theory parallel to that of quasitriangular Hopf algebras. For a finite-dimensional quasitriangular Hopf algebra \((H,R)\) and a finite-dimensional quasitriangular left \(H\)-comodule algebra \((B,K)\), one defines
\[
E(H,B)
=
\left\{
\xi\colon H\to B\ \middle|\ 
\xi(b_{[-1]}h)b_{[0]}=b\,\xi(h)\ \forall b\in B,\ h\in H
\right\},
\]
described as the end of the internal Hom-functor for the module category \(B\)-\(\mathsf{FdMod}\) [2411.18453]. In analogy with the Drinfeld map, one then defines the canonical factorization map
\[
\theta_B\colon H^*\to E(H,B),\qquad
\bigl(\theta_B(f)\bigr)(h)
=
\bigl\langle f,\ S(h_{(1)})K_i h_{(2)}\bigr\rangle K^i.
\]
The comodule algebra \((B,K)\) is called factorizable precisely when \(\theta_B\) is an isomorphism of vector spaces [2411.18453].

Walton–Yadav prove that this algebraic condition is exactly the categorical nondegeneracy condition for the associated braided module category. Specifically, if
\[
M=B\text{-}\mathsf{FdMod},
\]
then \(M\) is nondegenerate, in the sense that its universal factorization map \(\theta_M\) is an isomorphism, if and only if \((B,K)\) is factorizable in the sense above [2411.18453]. Equivalently,
\[
B\text{-}\mathsf{FdMod}\ \text{is nondegenerate}
\quad\Longleftrightarrow\quad
\theta_B\colon H^*\xrightarrow{\cong}E(H,B)\ \text{is an isomorphism}
\]
[2411.18453].

The proof uses a monadicity theorem for module categories, an equivalence involving the Deligne product \(M\boxtimes \Fun_{C\!|\!}(M,M)\), and an identification of the reflective center \(E_C(M)\) with a coend-based endomorphism category [2411.18453]. In the Hopf specialization \(C=H\text{-}\mathsf{FdMod}\), \(M=B\text{-}\mathsf{FdMod}\), the universal factorization map \(\theta_M\) identifies with the concrete map \(\theta_B\), up to the antipode of \(H^*\) [2411.18453].

Several examples clarify the concept. When \(B=H\) with \(\delta=\Delta\), one may take
\[
K=R_{21}R\in H\otimes H,
\]
and \(\theta_H\) becomes the usual Drinfeld map of \((H,R)\); thus \((H,R_{21}R)\) is factorizable precisely when \((H,R)\) is factorizable [2411.18453]. In the triangular case, where \(R_{21}=R^{-1}\), one may take \(K=1\otimes 1\), but then the factorization map collapses to a rank-one map, so \((B,K)\) fails to be factorizable except in the trivial case \(H=\Bbbk\) [2411.18453]. For reflective algebras \(R_H(A)\), Laugwitz–Walton–Yakimov show that when \(R_H(A)\) is \(H\)-simple, it is factorizable [2411.18453]. This yields a broad source of nondegenerate braided module categories.

A plausible implication is that factorization for quasitriangular comodule algebras plays the same structural role for braided module categories that Drinfeld factorizability plays for braided tensor categories.

## 6. Invariants, Morita theory, and representative examples

Braided Morita theory provides a mechanism for comparing quasitriangular comodule algebras through their representation categories. Two quasitriangular left \(H\)-comodule algebras \((A,K)\) and \((A',K')\) are braided Morita equivalent if there is an equivalence
\[
F\colon \mathsf{Rep}(A)\xrightarrow{\simeq}\mathsf{Rep}(A')
\]
that is a linear, strong \(H\)-module functor and is compatible with the braidings \(e\) and \(e'\) [2508.19845]. When \(K=1\otimes 1_A\), \(\mathsf{Rep}(A)\) is a plain \(H\)-module category with trivial module braiding; more generally \((A,K)\) is sometimes called a braided comodule algebra or reflection algebra [2508.19845].

The main invariants arise from braid-group representations of Coxeter types \(\mathrm{BC}\) and \(\mathrm{D}\). For a braided monoidal category \((\mathcal C,c)\) and a braided \(\mathcal C\)-module category \((\mathcal M,e)\), any \(X\in\mathcal C\), \(M\in\mathcal M\) give a representation
\[
\rho_n^{X,M}\colon \mathsf{Br}_n^{\mathrm{BC}}\to \mathrm{Aut}(X^{\otimes n}\otimes M)
\]
defined on the standard braid generators by the braiding \(c_{X,X}\) and on the boundary generator \(t\) by \(e_{X,M}\) [2508.19845]. If \((\mathcal M,e)\) is symmetric as a module category, meaning \(e=e^{-1}\), this extends to a representation of \(\mathsf{Br}_n^{\mathrm D}\) [2508.19845].

For finite-dimensional \((H,R)\), if \((A,K)\) and \((A',K')\) are both augmented and \(H\)-simple, then braided Morita equivalence implies that, for every \(n\ge 2\),
\[
\rho_n^{H_{\mathrm{reg}},A_{\mathrm{reg}}}\simeq \rho_n^{H_{\mathrm{reg}},A'_{\mathrm{reg}}}
\]
as representations of \(\mathsf{Br}_n^{\mathrm{BC}}\); under triangularity, the analogous statement holds for type \(\mathrm D\) [2508.19845]. In particular, the characters \(\mathrm{tr}\,\rho_n^{H,A}(1)\) and the spectra of the images of the generators \(\sigma_i\) and \(t\) are braided Morita invariants [2508.19845].

Several concrete examples are classified.

| Setting | Quasitriangular data | Classification statement |
|---|---|---|
| Group algebra \(H=\Bbbk G\) with \(u\in Z(G)\), \(u^2=1\) | \(R_u=\frac12(1\otimes1+1\otimes u+u\otimes1-u\otimes u)\) | On \(\Bbbk L\), all \(K\)-matrices are \(a\otimes1\) with \(a\in C_G(L)\) [2508.19845] |
| Sweedler algebra \(H_4\) | Quasitriangular forms \(R_\lambda\) | For \(\lambda\neq0\), only nontrivial \(K\)-matrix on \(\Bbbk\{1,g\}\) is \(1\otimes1\); for \(\lambda=0\), also \(g\otimes1\) [2508.19845] |
| \(U_q(\mathfrak{sl}_2)\) toy example | \(A=\langle K^{\pm1},E\rangle\) | Reflection-equation solutions of form \(K=K\otimes1+\alpha F\otimes E+\beta FK^{-1}\otimes EK\) [2508.19845] |

In the group algebra case, every coideal subalgebra is \(\Bbbk L\) for a subgroup \(L\le G\), and two quasitriangular structures \((\Bbbk L,a\otimes1)\) and \((\Bbbk L',a'\otimes1)\) are braided Morita equivalent if and only if there exists \(g\in G\) such that
\[
L'=gLg^{-1},\qquad a'=gag^{-1}
\]
[2508.19845]. Thus conjugacy of pairs \((L,a)\) exactly matches braided Morita equivalence [2508.19845].

In the Sweedler case, the proper left coideal subalgebras are exactly \(\Bbbk\), \(\Bbbk\{1,g\}\), \(\Bbbk\{1,gx\}\), and \(H_4\); no nontrivial \(K\)-matrices exist on the other proper subalgebras, and the possible quasitriangular comodule algebras fall into distinct braided Morita-equivalence classes [2508.19845].

For the \(U_q(\mathfrak{sl}_2)\) toy example, the action of the generator \(t\in \mathsf{Br}_2^{\mathrm{BC}}\) on \(H\otimes A\cong H_{\mathrm{reg}}\otimes A_{\mathrm{reg}}\) is
\[
\rho_2(t)(h\otimes a)=\sum_j hK_j\otimes K^j a.
\]
The trace \(\mathrm{tr}\,\rho_2(t)\) and the eigenvalues of \(\rho_2(\sigma_1)\) are described as spectral invariants of the braided Morita class [2508.19845].

The broader outlook formulated in this work is that, for families such as quantum symmetric pairs, reflection equation algebras, and semisimple Hopf algebras, these boundary braid-group invariants may refine the usual Drinfeld–Reshetikhin–Turaev invariants and support a systematic classification of braided module categories [2508.19845].

## 7. Related constructions: doubles, cocycle twisting, and coquasitriangular analogues

Quasitriangular comodule algebras are also produced by crossed-product constructions over Drinfeld doubles. If \((A,B)\) are dually paired bialgebras and \(C\) is a right \(A\)-comodule algebra with coaction \(\rho(c)=c_{(0)}\otimes c_{(1)}\), then one forms
\[
C\# B=C\otimes B
\]
with multiplication
\[
(c\# b)(c'\# b')=c\,c'_{(0)}\,\langle c'_{(1)},b_{(1)}\rangle \# b_{(2)}b'
\]
[1708.02641]. When \(A\) and \(B\) are Hopf algebras and one forms their Drinfeld double \(D(A,B)\), the algebra \(C\# B\) carries a natural \(D(A,B)\)-coaction by a conjugation-by-\(R\) formula, making it a left \(D(A,B)\)-comodule algebra [1708.02641]. The same mechanism extends to braided Drinfeld doubles in \(\mathrm{Mod}(H)\), where braided bialgebras \(B\) and \(C\) lead to a braided crossed product \(C\rtimes B\) that again carries a natural double coaction [1708.02641].

These constructions are compatible with 2-cocycle twisting. If \(\sigma_A\in H^2(A,\Bbbk)\) and \(\sigma_B\in H^2(B,\Bbbk)\), one obtains a cocycle \(\sigma_D\) on \(D(A,B)\), and the double’s quasitriangular structure twists by
\[
R\longmapsto (\sigma_D^{-1})_{21}\,R\,\sigma_D
\]
[1708.02641]. A comodule algebra \(C\) can likewise be deformed compatibly under the induced cocycle [1708.02641]. This provides a method for generating new quasitriangular or braided comodule-algebra examples from existing ones.

There is also a dual, coquasitriangular theory. If \((A,r)\) is a coquasitriangular Hopf algebra, then the category of right \(A\)-comodules \(\mathcal M^A\) is braided, with braiding
\[
\Psi(b\otimes c)=c_{(0)}\otimes b_{(0)}\,r(b_{(1)},c_{(1)})
\]
for right \(A\)-comodules [1703.03456]. Finite-dimensional braided Hopf algebras \(B\in\mathcal M^A\) can be assembled with \(A\) and the braided dual \(B^*\) into the co-double bosonization
\[
H=B^{\underline{\mathrm{op}}}\!\triangleright\!\!\!<\,A\,\triangleright\!>\!<\,B^*
\]
which is itself an ordinary coquasitriangular Hopf algebra [1703.03456]. The resulting constructions yield explicit models of \(c_q[SL_2]\) and \(c_q[SL_3]\) at odd roots of unity, with monomials in the new generators forming, up to explicit \(q\)-factor normalizations, bases dual to standard PBW bases of the corresponding reduced quantum enveloping algebras [1703.03456].

Although this is a dual theory rather than the direct theory of quasitriangular comodule algebras, it shows that boundary-type and braided-comodule constructions have natural counterparts under Hopf duality. This suggests a broader landscape in which quasitriangular and coquasitriangular structures, doubles and reflective centers, and \(R\)- and \(K\)-matrices are related by systematic dualization and transmutation procedures.

Source: https://www.emergentmind.com/topics/quasitriangular-comodule-algebras