I have encountered several occurrences of the so called reflection equation algebra (REA) but depending on where I find them, I feel like I get slightly different objects. In all cases there is a quasi-triangular Hopf algebra lurking in the background. In what follows $V$ will always be a vector space of dimension $n$. Here is a list of the different occurrences I encountered:


1. Let $H$ be a quasi-triangular Hopf algebra with $R \in H \otimes H$ its universal $R$-matrix (here we possibly have completions but it does not really matters). The reflection algebra is then as vector space the restricted dual $H^\circ$. That is the subalgebra of the full dual spanned by the so called matrix coefficients. The algebra structure comes from the algebra structure of the full dual but twisted by the universal $R$-matrix. I think this is sometimes called the braided dual of $H$.
See for example definition 4.12 of <https://arxiv.org/pdf/math/0204295.pdf>
2. Let $R: V \otimes V \rightarrow V \otimes V$ be an endomorphism of $V \otimes V$ satisfying the Yang-Baxter equation. Then the reflection equation algebra if the algebra generated by elements $(a\_{ij})\_{1 \leq i,j \leq n}$ with relation
$$RA\_1RA\_1 = A\_1RA\_1R$$
where $A$ is the matrix $n \times n$ having the generating elements as coefficients and $A\_1 = A \otimes Id$.
I think here the generating elements are somewhat thought to be elements of $V^{\ast} \otimes V$. This was found at the very beginning of the introduction of <https://arxiv.org/pdf/1806.10219.pdf>
3. This one is a special example. Here the Hopf algebra lurking in the background is $U\_q(\frak{sl\_2})$ and the $R$-matrix is given by
$$
\begin{pmatrix}
q & 0 & 0 & 0\\
0 & 1 & q-q^{-1} & 0 \\
0 & 0 & 1 & 0\\
0 & 0 & 0& q
\end{pmatrix}.$$ In this case it is the algebra generated by the elements $(a\_{ij})\_{1 \leq i,j \leq 2}$ with relation :
$$R\_{21}A\_1RA\_2 = A\_2R\_{21}A\_1R$$ and also
$$ a\_{11}a\_{22}- q^2a\_{12}a\_{21} = 1.$$
This algebra is often denoted by $\mathcal{O}\_q(SL\_2)$ or sometimes $\mathcal{O}(Rep\_q(SL\_2))$. This appeared as Example 1.23 in <https://arxiv.org/pdf/1908.05233.pdf> and also as Definition 2.1. in <https://arxiv.org/pdf/1811.09293.pdf> (be aware of the foot note to get back what I wrote).


I can see how some of those are related, for example the third one is almost a specific case of the second one but there is one more relations.


In the first one matrix elements can be thought as being in $W^{\ast} \otimes W$ for any representation $W$ of $H$. In the case where any finite dimensional representation of $H$ can be seen as subrepresentation of a tensor product of the standard representation $V$, then it is actually generated only by the matrix coefficients coming from $V$.It then looks a lot like what we have in 2). However, there is still the a relation missing if one specializes to the case $H = U\_q(\frak{sl2})$ to get the same as in 3). And what if there is a representation of $H$ that is not a subrepresentation of a tensor product of the standard one?


**QUESTION:**
Are all of those actually the same thing or am I missing something? I am a bit confused on what people actually call the reflection equation algebra. Is there some kind of nice definition for any quasi-triangular Hopf algebra $H$ that englobes all the above "examples"?



Confusion around the reflection equation algebra
1. The only reasonable definition of the REA associated with a quasi-triangular Hopf algebra is 1). This is, of course, a somewhat abstract definition but provides a solution of the RE which is universal in a precise sense.
2. is reminiscent of the so-called Faddeev-Reshetikhin-Takhtajan (usually abbreviated as FRT) construction. Its main advantage is that it does not require an Hopf algebra to start with (rather, in the original FRT construction, the goal was to produce a Hopf algebra starting from an arbitrary solution of the QYBE). EVen if $R$ do come from a quasi-triangular Hopf algebra, it won't give the same answer as 1) except in the case of $U\_q(\mathfrak{gl}\_n)$ (even then this is not quite true, you get some deformation of $\mathcal O(\mathfrak{gl}\_n)$ rather than $\mathcal O(GL\_n)$). In general there will be a map from 2) to 1).
3. On the other hand as you say, you can run this construction in the case $V$ is some representation that generates all the other. Indeed this approach is useful to find a presentation of the REA, since it is indeed generated by matrix coefficients: roughly speaking this will give you a set of generators but **not** in general all the relations. This is what happens here: if you run the FRT-like reconstruction for the R-matrix of $\mathfrak{sl}\_n$ you get some algebra, but then you need to add this extra relation you mention which, as you probably know, is nothing but a $q$-analog of $\det(A)=1$. Again this already shows up in the original situation, see Definition 4 in [http://math.soimeme.org/~arunram/Resources/Reshetikhin/QuantizationOfLieGroupsAndLieAlgebras.html](http://math.soimeme.org/%7Earunram/Resources/Reshetikhin/QuantizationOfLieGroupsAndLieAlgebras.html).


**Edit** It's useful to think about universal properties: 1) is universal for algebras $A$ with a solution of the RE in $A\otimes H$, while 2) is universal for algebras $A$ with a solution of the RE in $A\otimes End(V)$. Of course, composing with the algebra map $H\rightarrow End(V)$ given by the action of $H$ on $V$ every solution of the first equation gives you a solution to the second one, so applying this to the case $A$ is the REA itself you get a map from the algebra constructed in 2) to the one constructed in 1).


