TITLE: Confusion around the reflection equation algebra
QUESTION [6 upvotes]: 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:

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

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

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"?

REPLY [3 votes]: Let me first note that the reflection matrix, which you denote by $A$, is often called K-matrix, cf its graphical representation with | a 'wall' and < the 'worldline' of particle bouncing off the wall. The graphical form of the equation can already be found in Cherednik, Factorizing particles on a half-line and root systems (1984) https://link.springer.com/article/10.1007/BF01038545. The notation $K$ might be due to Sklyanin, Boundary conditions for integrable quantum systems (1988), https://iopscience.iop.org/article/10.1088/0305-4470/21/10/015.
The reflection (equation) algebra is the reflection-equation analogue of the Yang--Baxter algebra: to any choice of finite-dimensional vector space and R-matrix obeying the Yang--Baxter equation (and suitable other properties, such as braiding unitarity and an 'initial condition') one can associate a unital associative algebra generated by the operator-valued (noncommutative) entries of the K-matrix obeying the reflection equation.
If one would replace the reflection (`$RKRK$') equation by the $RLL$-equation one instead arrives at the Yang--Baxter algebra, which is the operator algebra closely related to the FRT (or R-matrix) presentation of quantum affine algebras.
Re 3: The FRT presentation says nothing about the quantum determinant, so to get $SL_n$ you need to impose $qdet = 1$ separately, which is your last equation in 3. The version of the reflection equation that you give there can sometimes be simplified: Suppose that the R-matrix is symmetric in the sense that $P R P = R$ with $P$ the permutation. Then $R_{21} = R_{12}$ in the usual tensor-leg notation. In such cases all R-matrices in the reflection equation can be written using just $R$. (Graphically the need for $R_{21}$ is clear, though.)
Re 2: These authors work with the braid-like version of the R-matrix, often denoted by $\check{R}$. Namely, suppose that $R$ obeys the YBE
$$ R_{12}(u,v) \ R_{13}(u,w) \ R_{23}(v,w) = R_{23}(v,w) \ R_{13}(u,w) \ R_{12}(u,v) \ , $$
where I have assumed that the R-matrix might depend on a spectral parameter associated to each copy of the auxiliary space in general. (This is for the affine case, but helps highlighting the structure of the equation.)
Then both of $P \ R$ and $R \ P$ obey the braid-like version of the YBE
$$ \check{R}_{12}(u,v) \ \check{R}_{23}(u,w) \ \check{R}_{12}(v,w) = \check{R}_{23}(v,w) \ \check{R}_{12}(u,w) \ \check{R}_{23}(u,v) \ . $$
You always have to check which version is used. In the paper you cite in 2 it's the latter, which is why both $A$s have the same subscript.
Re 1: I believe that the proper algebraic interpretation of Sklyanin's construction of representations of the K-matrix as the double-row monodromy matrix, constructed from a K-matrix with scalar entries and an L-operator, is as a coideal subalgebra, see Kolb and Stokman, Reflection equation algebras, coideal subalgebras, and their centres, https://arxiv.org/abs/0812.4459.
You might also be interested in the recent paper by Appel and Vlaar, Universal k-matrices for quantum Kac-Moody algebras, https://arxiv.org/abs/2007.09218