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We now make the following definitions111111These are the modified forms of the
R-matrices that appear in the corresponding left reflection equations, see
[Sk88, Eq. (13)].:

(8.1)
$\widetilde{{\mathcal{R}}}_{\varrho{\bar{\varrho}}}(z):={\mathcal{R}}_{\varrho{\bar{\varrho}}}(q^{2}z)^{-1},\qquad\widetilde{{\mathcal{R}}}_{\upsilon\phi}(z):={\mathcal{R}}_{\upsilon\phi}(q^{2}z)^{-1}.$

and represent these modified R-matrices by the following pictures:

$\widetilde{{\mathcal{R}}}_{\varrho{\bar{\varrho}}}(z){\mathcal{P}}=$$z_{2}$$z_{1}$
$\widetilde{{\mathcal{R}}}_{\upsilon\phi}(z){\mathcal{P}}=$$z_{2}$$z_{1}$

The various right-boundary K-matrices are represented as follows:

${\mathcal{K}}_{\rho}(z)=$$z$$z^{-1}$
${\mathcal{K}}_{{\bar{\varrho}}}(z)=$$z$$z^{-1}$
${\mathcal{K}}_{\upsilon}(z)=$$z$$z^{-1}$
${\mathcal{K}}_{\phi}(z)=$$z$$z^{-1}$

The left-boundary K-matrices defined in Section 6.2 are represented by the
natural analogues of these pictures. For example:

$\widetilde{\mathcal{K}}_{\rho}(z)=$$z$$z^{-1}$

Making use of these pictures, we see that Theorem 5.2 and Corollary 5.3 are
represented by

For the compatibility with the right boundary we claim that

$q^{\mu/2}z$$q^{-\mu/2}z$$z$$z^{-1}$$z$$z^{-1}$${\mathcal{O}}$=$z^{-1}$$z^{-1}$$q^{-\mu/2}z^{-1}$$q^{\mu/2}z$$q^{-\mu/2}z$$q^{\mu/2}z^{-1}$${\mathcal{O}}_{21}$

which corresponds to the following identity in ${\mathcal{A}}^{(2)}$:

(8.2)
${\mathcal{K}}_{\upsilon}(z)_{1}{\mathcal{R}}_{\upsilon\phi}(z^{2}){\mathcal{K}}_{\phi}(z)_{2}\,{\mathcal{O}}={\mathcal{O}}\,{\mathcal{K}}_{\varrho}(q^{-\mu/2}z)_{1}{\mathcal{R}}_{\varrho{\bar{\varrho}}}(z^{2}){\mathcal{K}}_{\bar{\varrho}}(q^{\mu/2}z)_{2},$

which we call the _right boundary factorization identity_. The diagrams above
serve as a motivation for the identity, which we now prove using results from
Section 3 (an alternative computational proof of Theorem 8.1 is given in
Appendix C).

######  Theorem 8.1.

For all $\mu\in\mathbb{C}$, all $q\in\mathbb{C}^{\times}$ not a root of unity
and all $\xi\in\mathbb{C}^{\times}$, relation (8.2) is satisfied.

###### Proof.

The proof is analogous to the proof of Theorem 5.2. We first note that

$\displaystyle\big{(}\varrho_{q^{-\mu/2}z}\otimes{\bar{\varrho}}_{q^{\mu/2}z}\big{)}\big{(}({\sf
id}\otimes\psi)({\mathcal{R}})\big{)}$
$\displaystyle=\big{(}\varrho_{q^{-\mu/2}z}\otimes{\bar{\varrho}}^{-}_{q^{-\mu/2}z^{-1}}\big{)}({\mathcal{R}})$
$\displaystyle\propto\;{\mathcal{R}}_{\varrho{\bar{\varrho}}}(z^{2}),$
$\displaystyle\big{(}\upsilon_{z}\otimes\phi_{z}\big{)}\big{(}({\sf
id}\otimes\psi)({\mathcal{R}})\big{)}$
$\displaystyle=\big{(}\upsilon_{z}\otimes\phi^{-}_{z^{-1}}\big{)}({\mathcal{R}})$
$\displaystyle\propto\;{\mathcal{R}}_{\upsilon\phi}(z^{2}).$

Noting the coproduct formula (3.19), we obtain

$\displaystyle{\mathcal{K}}_{\varrho}(q^{-\mu/2}z)_{1}{\mathcal{R}}_{\varrho{\bar{\varrho}}}(z^{2}){\mathcal{K}}_{\bar{\varrho}}(q^{\mu/2}z)_{2}$
$\displaystyle\propto\quad\big{(}\varrho_{q^{-\mu/2}z}\otimes{\bar{\varrho}}_{q^{\mu/2}z}\big{)}(\Delta({\mathcal{K}})),$
$\displaystyle{\mathcal{K}}_{\upsilon}(z)_{1}{\mathcal{R}}_{\upsilon\phi}(z^{2}){\mathcal{K}}_{\phi}(z)_{2}$
$\displaystyle\propto\quad\big{(}\upsilon_{z}\otimes\phi_{z}\big{)}(\Delta({\mathcal{K}})).$

Now Theorem 4.4 implies (8.2) up to a scalar. The fact that all factors fix
$w_{0}\otimes w_{0}$ shows that the scalar is 1. ∎

Compatibility with the left boundary requires that

$z^{-1}$$z^{-1}$${\mathcal{O}}_{21}^{-1}$$q^{\mu/2}z$$q^{-\mu/2}z$$q^{\mu/2}z^{-1}$$q^{-\mu/2}z^{-1}$$=$$z^{-1}$$z^{-1}$$z$$z$$q^{\mu/2}z$$q^{-\mu/2}z$${\mathcal{O}}^{-1}$

The identity in ${\mathcal{A}}^{(2)}$ corresponding to this is

(8.3)
${\widetilde{\mathcal{K}}}_{\bar{\varrho}}(q^{\mu/2}z,\widetilde{\xi})_{2}\widetilde{{\mathcal{R}}}_{\varrho{\bar{\varrho}}}(z^{2}){\widetilde{\mathcal{K}}}_{\varrho}(q^{-\mu/2}z,\widetilde{\xi})_{1}{\mathcal{O}}^{-1}={\mathcal{O}}^{-1}{\widetilde{\mathcal{K}}}_{\phi}(z,\widetilde{\xi})_{2}\widetilde{{\mathcal{R}}}_{\upsilon\phi}(z^{2}){\widetilde{\mathcal{K}}}_{\upsilon}(z,\widetilde{\xi})_{1}.$

######  Theorem 8.2.

Relation (8.3) is satisfied.

###### Proof.

Given the definitions (6.12) and (8.1), this follows straightforwardly by
inverting (8.2) and replacing $(z,\xi)\mapsto(qz,\widetilde{\xi}^{-1})$. ∎

##  9\. Discussion

The main result of this paper is Theorem 8.1 which can be viewed as a boundary
analogue of Theorem 5.2. To establish this result, we needed to first show
that all R and K-operators involved in Equation (8.2) are well-defined actions
of the universal elements ${\mathcal{R}}$ and ${\mathcal{K}}$ on the infinite-
dimensional $U_{q}(\widehat{\mathfrak{b}}^{+})$-modules involved. The key fact
that allows for this is that ${\mathcal{R}}$ and ${\mathcal{K}}$ live in
completions of $U_{q}(\widehat{\mathfrak{b}}^{+})\otimes
U_{q}(\widehat{\mathfrak{b}}^{-})$ and of $U_{q}(\widehat{\mathfrak{b}}^{+})$,
respectively. This is very familiar for ${\mathcal{R}}$ but for
${\mathcal{K}}$ relies on the recent works [AV22a, AV22b]. Introducing the
$U_{q}(\widehat{\mathfrak{b}}^{+})$-intertwiner ${\mathcal{O}}$ and the
formula for $\Delta({\mathcal{K}})$ given by (3.19), relation (8.2) follows
immediately from the intertwining property of ${\mathcal{O}}$.

The open Q-operator ${\mathcal{Q}}(z)$ of [VW20] is the trace of a product of
R and K-operators over the $U_{q}(\widehat{\mathfrak{b}}^{+})$-module
$(W,\varrho_{z})$ and there is a similar construction of an open Q-operator
$\overline{{\mathcal{Q}}}(z)$. In a future paper, the authors will present
this construction and the use of Theorem 8.2 in deriving a boundary analogue
of the factorization relation
$\mathcal{T}_{\mu}(z)\>\propto\>\mathcal{Q}(zq^{-\mu/2})\overline{\mathcal{Q}}(zq^{\mu/2})$.
They will also develop the analogous theory for different coideal subalgebras,
in particular those for which non-diagonal solutions of the reflection
equation are intertwiners.

##  A Deformed Pochhammer symbols and exponentials

This appendix is independent from the main text, but provides identities which
are used there. We review some basic theory of deformed Pochhammer symbols and
exponentials (as formal power series) with a deformation parameter
$p\in\mathbb{C}^{\times}$, which corresponds to $q^{2}$ in the main text.

###  A.1. Deformed Pochhammer symbols

Let $x$ be a formal variable. For $n\in\mathbb{Z}$, the (finite) deformed
Pochhammer symbol $(x;p)_{n}\in\mathbb{C}[[x]]$ is defined by

(A.1)
$(x;p)_{n}:=\begin{cases}\displaystyle\prod_{m=0}^{n-1}(1-xp^{m})&\text{if
}n\geq 0,\\\ \displaystyle\prod_{m=n}^{-1}(1-xp^{m})^{-1}&\text{if
}n<0\end{cases}$

(the definition for $n<0$ is understood as a product of geometric series);
since its constant coefficient is nonzero, it is invertible. For all
$p\in\mathbb{C}^{\times}$ and $n\in\mathbb{Z}_{\geq 0}$ we have the following
basic identity, see [GR90, (I.2), (I.3)]:

(A.2)
$(x;p)_{-n}=(p^{-n}x;p)_{n}^{-1}=(x/p;p^{-1})_{n}^{-1}=(-x)^{-n}p^{n(n+1)/2}(p/x;p)_{n}^{-1}.$

Assuming $|p|<1$, the infinite deformed Pochhammer symbol

(A.3) $(x;p)_{\infty}:=\prod_{m=0}^{\infty}(1-xp^{m})$

is an invertible formal power series with well-defined coefficients in
$\mathbb{C}$. The following identity holds in $\mathbb{C}[[x]]$, see [GR90,
(I.5)]:

(A.4) $(x;p)_{n}=\frac{(x;p)_{\infty}}{(p^{n}x;p)_{\infty}}.$

###  A.2. Deformed exponentials

From now on we assume that $p$ is not a root of unity. In particular,
$(p;p)_{k}\neq 0$ for all $k\in\mathbb{Z}_{\geq 0}$. The _deformed
exponential_ is the invertible formal power series

(A.5)
$e_{p}(x):={}_{1}\phi_{0}(0;-;p,x)=\sum_{k=0}^{\infty}\frac{x^{k}}{(p;p)_{k}}.$

The ordinary exponential formal power series arises as the limit $\lim_{p\to
1}e_{p}((1-p)x)={\mathrm{e}}^{x}$. This series satisfies the functional
relation

(A.6) $e_{p}(px)=(1-x)e_{p}(x),$

see [GR90, Sec. 1.3]. Using the fact that constants are the only formal power
series which are invariant under $x\mapsto px$, by an inspection of constant
coefficients we obtain from (A.6) the identity

(A.7) $e_{p}(x)=\frac{1}{(x;p)_{\infty}}\qquad\text{if }|p|<1.$

Similarly we consider the invertible formal power series

(A.8)
$E_{p}(x):={}_{0}\phi_{0}(-;-;p,-x)=\sum_{k=0}^{\infty}\frac{p^{k(k-1)/2}x^{k}}{(p;p)_{k}}.$

Then $E_{p}(-x)^{-1}$ also satisfies (A.6) and by comparing constant
coefficients again we deduce $e_{p}(x)=E_{p}(-x)^{-1}$. By evaluating (A.2) at
$x=1$, we obtain $E_{p}(-x)=e_{p^{-1}}(p^{-1}x)$ and hence

(A.9) $e_{p}(x)=e_{p^{-1}}(p^{-1}x)^{-1}\quad\in\mathbb{C}[[x]].$

Deformed exponentials in $x$ and $y$ satisfy various useful identities in
particular deformations of the commutative algebra $\mathbb{C}[[x,y]]$. For
instance, in any algebra generated by the symbols $x$ and $y$ such that
$yx=\gamma xy$ for $\gamma\in\mathbb{C}$, the definition implies the following
identity:

(A.10) $ye_{p}(x)=e_{p}(\gamma x)y$

which we will use repeatedly. For a survey of product formulas analogous to
$\exp(x)\exp(y)=\exp(x+y)$, see [Ko97]. We will need the following result.

######  Lemma A.1.

Let $x,y$ be elements of an algebra such that $yx=pxy$. The following
identities hold as formal power series in $x,y$:

(A.11) $\displaystyle e_{p}(x)e_{p}(y)$ $\displaystyle=e_{p}(x+y),$ (A.12)
$\displaystyle e_{p}(y)e_{p}(x)$
$\displaystyle=e_{p}\big{(}x(1-y)\big{)}e_{p}(y)=e_{p}(x)e_{p}(-xy)e_{p}(y)=e_{p}(x)e_{p}\big{(}(1-x)y\big{)}.$

###### Proof.

(A.11) is a direct consequence of the well-known q-binomial formula, see e.g.
[GR90, Ex. 1.35]. For (A.12) see [Ko97, Prop. 3.2]. ∎

###  A.3. Deformed exponentials as linear maps

Let $V$ be a $\mathbb{C}$-linear space. Call an operator $f$ on $V$ _locally
nilpotent_ if for all $v\in V$ there exists $o(v)\in\mathbb{Z}_{\geq 0}$ such
that $f^{o(v)}(v)=0$ (note that nilpotent operators are locally nilpotent and
if $V$ is finite-dimensional the converse is true). The deformed exponential
$e_{p}(f)$ defines an invertible map on $V$. More generally, if $y$ is an
indeterminate then $e_{p}(yf)$ is a well-defined invertible element of
$\operatorname{End}(V)[[y]]$.

Recall the infinite-dimensional vector space $W$ and its linear operators $a$,
$a^{\dagger}$, ${\bar{a}}^{\dagger}$, $f(D)$ ($f\in{\mathcal{F}}$) from
Section 4.2. In the case $V=W\otimes W$, we have the following commutation
relations for linear-operator valued formal series.

######  Lemma A.2.

Let $y$ be a formal variable. In $\operatorname{End}(W\otimes W)[[y]]$ the
following identities hold:

(A.13)
$\big{[}e_{p}(ya_{1}{\bar{a}}^{\dagger}_{2}),f(D_{1}+D_{2})\big{]}=\big{[}e_{p}(ya_{1}{\bar{a}}^{\dagger}_{2}),a_{1}\big{]}=\big{[}e_{p}(ya_{1}{\bar{a}}^{\dagger}_{2}),{\bar{a}}^{\dagger}_{2}\big{]}=0$

for all $f\in{\mathcal{F}}$ and

(A.14)
$\displaystyle\big{[}e_{p}(ya_{1}{\bar{a}}^{\dagger}_{2}),a^{\dagger}_{1}\big{]}$
$\displaystyle=yp^{D_{1}}{\bar{a}}^{\dagger}_{2}e_{p}(ya_{1}{\bar{a}}^{\dagger}_{2}),$
(A.15)
$\displaystyle\big{[}e_{p}(ya_{1}{\bar{a}}^{\dagger}_{2}),p^{-D_{1}}a_{2}\big{]}$
$\displaystyle=ye_{p}(ya_{1}{\bar{a}}^{\dagger}_{2})a_{1}p^{-D_{1}}.$

###### Proof.

Note that (A.13) follows directly from the definition of the deformed
exponential. A straightforward inductive argument using (4.4) yields

(A.16) $\displaystyle[a^{k+1},a^{\dagger}]$
$\displaystyle=(1-p^{k+1})p^{D}a^{k},$ (A.17)
$\displaystyle[({\bar{a}}^{{\dagger}})^{k+1},a]_{p^{k+1}}$
$\displaystyle=(1-p^{k+1})({\bar{a}}^{{\dagger}})^{k},$

for all $k\in\mathbb{Z}_{\geq 0}$, which imply (A.14) and (A.15),
respectively. ∎

##  B Explicit expressions for R-operators

In this appendix we derive explicit formulas for
${\mathcal{R}}_{\varrho{\bar{\varrho}}}(z)$ and
${\mathcal{R}}_{\upsilon\phi}(z)$, defined by (5.10) as images of the
universal R-matrix ${\mathcal{R}}$ fixing $w_{0}\otimes w_{0}$. We expect that
these will be useful in further studies of Baxter’s Q-operators for the open
XXZ spin chain; for now they will allow us to give a proof of the boundary
factorization identity which does not rely on the universal K-matrix
formalism. First we note that, by the second part of Theorem 2.4,
${\mathcal{R}}_{\varrho{\bar{\varrho}}}(z)$ and
${\mathcal{R}}_{\upsilon\phi}(z)$ lie in the centralizer

(B.1)
${\mathcal{A}}^{(2)}_{0}:=\Big{\\{}X\in{\mathcal{A}}^{(2)}\,\Big{|}\,\big{[}X,q^{D_{1}+D_{2}}\big{]}=0\Big{\\}}.$

One straightforwardly verifies that ${\mathcal{A}}^{(2)}_{0}$ is generated by
elements of the form

(B.2) $\sum_{k\geq
0}({\bar{a}}^{\dagger}_{2})^{k}f_{k}(D_{1},D_{2})a_{1}^{k},\qquad\sum_{k\geq
0}(a^{\dagger}_{1})^{k}f_{k}(D_{1},D_{2})a_{2}^{k},\qquad\qquad
f_{k}\in{\mathcal{F}}^{(2)}.$

Hence, elements of ${\mathcal{A}}^{(2)}_{0}$ in fact commute with all elements
of the form $f(D_{1}+D_{2})$ ($f\in{\mathcal{F}}$).

###  B.1. Explicit expression for $R_{\upsilon\phi}(z)$

We first state and prove an explicit formula for $R_{\upsilon\phi}(z)$. We
keep using the shorthand notation $p=q^{2}$.

######  Theorem B.1.

For all $z\in\mathbb{C}$ we have

(B.3)
${\mathcal{R}}_{\upsilon\phi}(z)=e_{p}(za^{\dagger}_{1}a_{2})q^{(\mu-1)(D_{2}-D_{1})-2D_{1}(D_{2}+1)}.$

###### Proof.

From Proposition 2.6 we deduce that ${\mathcal{R}}_{\upsilon\phi}(z)$ is a
solution of the linear relation

(B.4)
$X(\upsilon_{z}\otimes\phi^{-})(\Delta(u))=(\upsilon_{z}\otimes\phi^{-})(\Delta^{\rm
op}(u))X\quad\text{for all }u\in U_{q}(\widehat{\mathfrak{b}}^{-}).$

First of all, note that the element in the right-hand side of (B.3) satisfies
(B.4) with $u\in\\{k_{0},k_{1}\\}$ and so it suffices to prove that the vector
space

(B.5) $\mathcal{X}=\Big{\\{}X\in{\mathcal{A}}^{(2)}_{0}\,\Big{|}\,X\text{
satisfies }\eqref{X:equation:1}\text{ for }u\in\\{f_{0},f_{1}\\}\Big{\\}}$

is spanned by
$e_{p}(z^{2}a^{\dagger}_{1}a_{2})q^{(\mu-1)(D_{2}-D_{1})-2D_{1}(D_{2}+1)}$.
Using the explicit formulas (2.2), (4.9) and (4.17), we obtain that (B.4) is
equivalent to the system

$\displaystyle
X\Big{(}z^{-1}a_{1}(q^{-\mu}-q^{\mu-2D_{1}})q^{-\mu-2D_{2}-1}+q^{-1}a_{2}\Big{)}$
$\displaystyle=\Big{(}z^{-1}a_{1}(q^{-\mu}-q^{\mu-2D_{1}})+q^{\mu-2(D_{1}+1)}a_{2}\Big{)}X,$
$\displaystyle Xa^{\dagger}_{1}q^{\mu+1+2D_{2}}$
$\displaystyle=a^{\dagger}_{1}X.$

Without loss of generality we may write
$X=\widetilde{X}q^{(\mu-1)(D_{2}-D_{1})-2D_{1}(D_{2}+1)}$ with
$\widetilde{X}\in{\mathcal{A}}^{(2)}_{0}$. Hence (B.4) is equivalent to

(B.6) $z^{-1}[\widetilde{X},a_{1}(1-p^{\mu-D_{1}})]=p^{\mu-
D_{1}-1}a_{2}\widetilde{X}-\widetilde{X}p^{D_{1}}a_{2},\qquad\qquad[\widetilde{X},a^{\dagger}_{1}]=0.$

It is straightforward to check that the centralizer in
${\mathcal{A}}^{(2)}_{0}$ of $a^{\dagger}_{1}$ is the subalgebra generated by
elements of the form $\sum_{k\geq
0}(a^{\dagger}_{1})^{k}f_{k}(D_{2})a_{2}^{k}$ with $f_{k}\in{\mathcal{F}}$. It
follows that $\widetilde{X}$ is of this form. Therefore (B.4) is equivalent to
the single equation

$\sum_{k\geq 0}\big{[}(a^{\dagger}_{1})^{k},a_{1}(1-p^{\mu-
D_{1}})\big{]}f_{k}(D_{2})a_{2}^{k}=z\sum_{k\geq
0}(a^{\dagger}_{1})^{k}\big{(}p^{\mu-
D_{1}-k-1}f_{k}(D_{2}+1)-p^{D_{1}}f_{k}(D_{2})\big{)}a_{2}^{k+1}.$

The commutator vanishes if $k=0$ so in the left-hand side we replace $k$ by
$k+1$. For $k\geq 0$ we have

$\displaystyle\big{[}(a^{\dagger})^{k+1},a(1-p^{\mu-D})\big{]}$
$\displaystyle=(a^{\dagger})^{k}(1-p^{k+1})(p^{\mu-D-k-1}-p^{D}).$

Hence (B.4) is equivalent to the recurrence relation

$(1-p^{k+1})\big{(}p^{\mu-
D_{1}-k-1}-p^{D_{1}}\big{)}f_{k+1}(D_{2})=z\big{(}p^{\mu-
D_{1}-k-1}f_{k}(D_{2}+1)-p^{D_{1}}f_{k}(D_{2})\big{)}.$

Viewing ${\mathcal{F}}^{(2)}(D_{1},D_{2})$ as an
${\mathcal{F}}(D_{2})$-module, the elements $p^{\pm D_{1}}$ are linearly
independent. Hence the above recurrence relation is equivalent to the system

$(1-p^{k+1})f_{k+1}(D)=zf_{k}(D+1),\qquad f_{k}(D+1)=f_{k}(D).$

This is in turn equivalent to $f_{k}(D)\in(p;p)_{k}^{-1}z^{k}\mathbb{C}$ for
$k\in\mathbb{Z}_{>0}$, as required. ∎

###  B.2. The automorphism $\chi$ and the q-oscillator subalgebra
$\widetilde{{\mathcal{A}}}$

To obtain an expression for $R_{\upsilon\phi}(z)$ in terms of deformed
exponentials, it is very convenient to point out an additional automorphism
$\chi$. It cannot be defined on all of ${\mathcal{A}}$ so we will consider a
subalgebra $\widetilde{{\mathcal{A}}}$. First, consider the subalgebra
$\widetilde{{\mathcal{F}}}(D)\subset{\mathcal{F}}(D)$ generated by

$p^{\pm D(D+1)/2},\qquad\gamma^{D},\qquad(p\widetilde{\gamma};p)_{D}^{\pm
1},\qquad(p\gamma z^{2};p)_{D},\qquad(-\gamma
z^{2})^{-D}(p\gamma^{-1}z^{-2};p)_{D}^{-1}$

for all $\gamma\in\mathbb{C}^{\times}$ and
$\widetilde{\gamma}\in\mathbb{C}^{\times}\backslash p^{\mathbb{Z}}$.

For elements of $\mathcal{G}(D)$, unlike general elements of $\mathcal{F}(D)$,
the symbol $D$ can be formally evaluated at negative integers. Accordingly, we
define an involutive automorphism $\chi$ of $\widetilde{{\mathcal{F}}}(D)$
accomplishing the formal replacement $D\mapsto-D-1$. To be more precise, we
set

(B.7) $\displaystyle$\chi\big{(} p^{\pm D(D+1)/2} \big{)} = p^{\pm D(D+1)/2},
\qquad\qquad\chi\big{(}\gamma^D \big{)} = \gamma^{-D-1}$,$
$\displaystyle\chi\big{(}(p\widetilde{\gamma};p)_{D}^{\pm 1}\big{)}$
$\displaystyle=(1-\widetilde{\gamma})^{\mp 1}p^{\pm
D(D+1)/2}(-\widetilde{\gamma})^{\mp D}(p\widetilde{\gamma}^{-1};p)_{D}^{\mp
1},$ $\displaystyle\chi\big{(}(p\gamma z^{2};p)_{D}\big{)}$
$\displaystyle=(1-\gamma z^{2})^{-1}p^{D(D+1)/2}(-\gamma
z^{2})^{-D}(p\gamma^{-1}z^{-2};p)_{D}^{-1},$ $\displaystyle\chi\big{(}(-\gamma
z^{2})^{-D}(p\gamma^{-1}z^{-2};p)_{D}^{-1}\big{)}$ $\displaystyle=(1-\gamma
z^{2})p^{-D(D+1)/2}(p\gamma z^{2};p)_{D}.$

We denote the subalgebra of $\operatorname{End}(W)$ generated by
$a^{\dagger}$, $a$ and $\mathcal{G}(D)$ by $\widetilde{{\mathcal{A}}}$. It is
straightforward to check that $\chi$ extends to a (non-involutive) algebra
automorphism of $\widetilde{{\mathcal{A}}}$ by means of the assignments

(B.8) $\chi(a)={\bar{a}}^{\dagger},\qquad\chi(a^{\dagger})=a.$

We can formulate a completion of the tensor product
$\widetilde{{\mathcal{A}}}\otimes\widetilde{{\mathcal{A}}}$ in a similar way
as for ${\mathcal{A}}\otimes{\mathcal{A}}$. More precisely, we consider the
subalgebra $\widetilde{{\mathcal{F}}}^{(2)}$ of ${\mathcal{F}}^{(2)}$
generated by the subsets $\widetilde{{\mathcal{F}}}(D_{1})$,
$\widetilde{{\mathcal{F}}}(D_{2})$ and the special elements $p^{\pm
D_{1}(D_{2}+1)}$. The completed tensorial square of
$\widetilde{{\mathcal{A}}}$ is defined to be the subalgebra
$\widetilde{{\mathcal{A}}}^{(2)}$ of $\operatorname{End}(W\otimes W)$
generated by the elements (4.8) with
$g_{k,\ell},\,h_{k,\ell}\in\widetilde{{\mathcal{F}}}^{(2)}$. Note that the
boundary factorization identity (8.2) is an identity in the subalgebra
$\widetilde{{\mathcal{A}}}^{(2)}\subset\operatorname{End}(W\otimes W)[[z]]$.

The automorphism

(B.9) $\chi^{(2)}:=\sigma\circ(\chi\otimes\chi^{-1})$

of $\widetilde{{\mathcal{A}}}\otimes\widetilde{{\mathcal{A}}}$ naturally
extends to an automorphism of $\widetilde{{\mathcal{A}}}^{(2)}$, fixing
$p^{\pm D_{1}(D_{2}+1)}$ and acting termwise on power series in locally
nilpotent operators.

######  Remark B.2.

The map $\chi$ can be seen as an infinite-dimensional version of conjugation
by anti-diagonal matrices; certain
$U_{q}(\widehat{\mathfrak{b}}^{+})$-representations are naturally related this
way. For instance, for the 2-dimensional representation $\Pi$, note that
$\operatorname{Ad}(J)\circ\Pi=\Pi\circ\Phi$ where $\operatorname{Ad}$ denotes
‘conjugation by’ and $J=\big{(}\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}\big{)}$. In the same way, $\chi$ relates the
prefundamental representations $\varrho$ and ${\bar{\varrho}}$ up to a twist
by the diagram automorphism $\Phi$:
$\chi\circ\varrho={\bar{\varrho}}\circ\Phi$. Hence, the condition (2.19) and
the 1-dimensionality of the solution space of the relevant linear equation
implies
$(\operatorname{Ad}(J)\otimes\chi)({\mathcal{L}}_{\varrho}(z))={\mathcal{L}}_{\bar{\varrho}}(z)$.
At the same time, a suitable scalar multiple of ${\mathcal{R}}_{\Pi\,\Pi}(z)$,
i.e. the R-matrix for the XXZ chain, is fixed by $\operatorname{Ad}(J\otimes
J)$ and we will see in Section B.3 that the same statement is true for
${\mathcal{R}}_{\varrho{\bar{\varrho}}}(z)$ and $\chi^{(2)}$.

From (3.5) it follows that
$\Phi(U_{q}(\mathfrak{k}))=U_{q}(\mathfrak{k})|_{\xi\mapsto\xi^{-1}}$. Hence,
the boundary counterparts of the above relations also involve inversion of the
free parameter $\xi$:

$\operatorname{Ad}(J)\big{(}K_{\Pi}(z)\big{)}|_{\xi\mapsto\xi^{-1}}=-\xi\,K_{\Pi}(z),\qquad\chi({\mathcal{K}}_{\varrho}(z))|_{\xi\mapsto\xi^{-1}}=q^{-1}(z^{2}-\xi^{-1})^{-1}{\mathcal{K}}_{\bar{\varrho}}(z).$

In fact, applying $\chi\otimes\operatorname{Ad}(J)$ to the reflection equation
(6.4) with $\pi=\varrho$ and inverting $\xi$ we see that

${\mathcal{K}}_{\varrho}(z)\mapsto\chi({\mathcal{K}}_{\varrho}(z))|_{\xi\mapsto\xi^{-1}}$

defines a bijection: ${\sf RE}_{\varrho}\to{\sf RE}_{\bar{\varrho}}$ of the
solution spaces defined in (6.7). $\diameter$

We can use the map $\chi^{(2)}$ to generate further relations similar to those
in Lemma A.2.

######  Lemma B.3.

Let $y$ be a formal parameter. In $\operatorname{End}(W\otimes W)[[y]]$ the
following identities hold:

(B.10) $\displaystyle[{\bar{a}}^{\dagger}_{2},e_{p}(ya^{\dagger}_{1}a_{2})]$
$\displaystyle=ye_{p}(ya^{\dagger}_{1}a_{2})a^{\dagger}_{1}p^{-D_{2}-1},$
(B.11)
$\displaystyle[{\bar{a}}^{\dagger}_{1}a_{2},e_{p}(ya_{1}{\bar{a}}^{\dagger}_{2})]$
$\displaystyle=y\big{(}e_{p}(ya_{1}{\bar{a}}^{\dagger}_{2})p^{-D_{1}-1}-p^{-D_{2}-1}e_{p}(ya_{1}{\bar{a}}^{\dagger}_{2})\big{)}.$

###### Proof.

In this proof we consider the algebra ${\mathcal{A}}$ as a subalgebra of
$\operatorname{End}(W)[[y]]$ instead of $\operatorname{End}(W)[[z]]$, and
similarly for ${\mathcal{A}}^{(2)}$. To prove (B.10), first we apply
$\chi^{(2)}$ to (A.14), obtaining

(B.12)
$[e_{p}(ya_{1}{\bar{a}}^{\dagger}_{2}),a_{2}]=ya_{1}p^{-D_{2}-1}e_{p}(ya_{1}{\bar{a}}^{\dagger}_{2}).$

Now consider the unique involutive algebra anti-automorphism
$\eta:{\mathcal{A}}\to{\mathcal{A}}$ which exchanges $a$ and $a^{\dagger}$ and
fixes $f(D)$ for all $f\in{\mathcal{F}}$ and the unique involutive algebra
anti-automorphism $\overline{\eta}:{\mathcal{A}}\to{\mathcal{A}}$ which
exchanges $a$ and ${\bar{a}}^{\dagger}$ and fixes $f(D)$ for all
$f\in{\mathcal{F}}$. Then $\eta^{(2)}:=\eta\otimes\overline{\eta}$ is an
algebra antiautomorphism of ${\mathcal{A}}\otimes{\mathcal{A}}$. It extends in
a natural way to an algebra antiautomorphism of ${\mathcal{A}}^{(2)}$. By
applying $\eta^{(2)}$ to (B.12) we obtain (B.10).

Finally, to prove (B.11), upon right-multiplying (A.15) by $p^{D_{1}+D_{2}+1}$
we obtain

(B.13)
$[e_{p}(ya_{1}{\bar{a}}^{\dagger}_{2}),a_{1}p^{D_{2}}]=ye_{p}(ya_{1}{\bar{a}}^{\dagger}_{2})a_{1}p^{D_{2}}.$

From (A.14) and (B.13) it follows that

(B.14)
$\displaystyle=y\Big{(}{\bar{a}}^{\dagger}_{2}p^{D_{1}}e_{p}(ya_{1}{\bar{a}}^{\dagger}_{2})a_{2}+a^{\dagger}_{1}e_{p}(ya_{1}{\bar{a}}^{\dagger}_{2})a_{1}\Big{)}p^{D_{2}}$
$\displaystyle=y\Big{(}p^{D_{1}}e_{p}(ya_{1}{\bar{a}}^{\dagger}_{2})\big{(}p^{D_{2}}-1\big{)}+\big{(}1-p^{D_{1}}\big{)}e_{p}(ya_{1}{\bar{a}}^{\dagger}_{2})p^{D_{2}}\Big{)}$
$\displaystyle=y\big{(}e_{p}(ya_{1}{\bar{a}}^{\dagger}_{2})p^{D_{2}}-p^{D_{1}}e_{p}(ya_{1}{\bar{a}}^{\dagger}_{2})\big{)}.$

Now (B.11) follows as the $\chi^{(2)}$-image of (B.14). ∎

###  B.3. Explicit expression for ${\mathcal{R}}_{\varrho{\bar{\varrho}}}(z)$

To aid the computation of ${\mathcal{R}}_{\varrho{\bar{\varrho}}}(z)$,
consider the subalgebra
$\widetilde{{\mathcal{A}}}^{(2)}_{0}=\widetilde{{\mathcal{A}}}^{(2)}\cap{\mathcal{A}}^{(2)}_{0}$,
which is preserved by $\chi^{(2)}$.

######  Lemma B.4.

${\mathcal{R}}_{\varrho{\bar{\varrho}}}(z)$ is a
$\widetilde{{\mathcal{A}}}^{(2)}_{0}$-valued formal power series whose
coefficients are fixed by $\chi^{(2)}$.

###### Proof.

It is clear from (4.11) and (4.17) that $\varrho\otimes{\bar{\varrho}}^{-}$
takes values in
$\widetilde{{\mathcal{A}}}\otimes\widetilde{{\mathcal{A}}}\subset\widetilde{{\mathcal{A}}}^{(2)}$.
Now recall (2.20) and note that the factor $\kappa$ acts as
$p^{D_{1}(D_{2}+1)}$. Furthermore, noting the form of $(\Sigma_{z}\otimes{\sf
id})(\Theta)$ given by (2.26) with the components $\Theta_{\lambda}$ lying in
$U_{q}(\widehat{\mathfrak{n}}^{+})_{\lambda}\otimes
U_{q}(\widehat{\mathfrak{n}}^{+})_{-\lambda}$ ($\lambda\in\widehat{Q}^{+}$),
we obtain that the action of $\mathcal{R}(z)$ on $(W\otimes
W,\varrho\otimes{\bar{\varrho}}^{-})$ is by an element of
$\widetilde{{\mathcal{A}}}^{(2)}_{0}$. For the second part, note that

$\chi^{(2)}\circ(\varrho\otimes{\bar{\varrho}}^{-})=(\chi^{-1}\otimes\chi)\circ({\bar{\varrho}}^{-}\otimes\varrho)\circ\sigma=(\varrho\otimes{\bar{\varrho}}^{-})\circ(\omega\otimes\omega)\circ\sigma.$

Applying this to ${\mathcal{R}}(z)$, making use of (2.27), (2.24) and (2.18),
we obtain
$\chi^{(2)}({\mathcal{R}}_{\varrho{\bar{\varrho}}}(z))={\mathcal{R}}_{\varrho{\bar{\varrho}}}(z)$.
∎

In the derivation of the formula for
${\mathcal{R}}_{\varrho{\bar{\varrho}}}(z)$, we rely on the following result.

######  Lemma B.5.

The centralizer of the subset $\\{a^{\dagger}_{1},{\bar{a}}^{\dagger}_{2}\\}$
in ${\mathcal{A}}^{(2)}$ is equal to $\mathbb{C}[[z]]$.

###### Proof.

This centralizer is the intersection of the centralizer of $a^{\dagger}_{1}$
and the centralizer of ${\bar{a}}^{\dagger}_{2}$, which are easily found to be
equal to

$\bigg{\\{}\sum_{k,\ell\geq
0}(a^{\dagger}_{1})^{k}f_{k,\ell}(D_{2})a_{2}^{\ell}\,\bigg{|}\,f_{k,\ell}\in{\mathcal{F}}\bigg{\\}},\qquad\bigg{\\{}\sum_{k,\ell\geq
0}({\bar{a}}^{\dagger}_{2})^{k}g_{k,\ell}(D_{1})a_{1}^{\ell}\,\bigg{|}\,g_{k,\ell}\in{\mathcal{F}}\bigg{\\}},$

respectively. Clearly their intersection is trivial. ∎

Now we are ready to state and prove a formula for
${\mathcal{R}}_{\varrho{\bar{\varrho}}}(z)$ in terms of deformed exponentials.

######  Theorem B.6.

For all $z$ we have

(B.15)
${\mathcal{R}}_{\varrho{\bar{\varrho}}}(z)=e_{q^{2}}(q^{3}za_{1}{\bar{a}}^{\dagger}_{2})e_{q^{2}}(q^{-1}za^{\dagger}_{1}a_{2})q^{-2D_{1}(D_{2}+1)}.$

###### Proof.

Clearly, $w_{0}\otimes w_{0}$ is fixed by the expression on the right-hand
side of (B.15). In the following we initially work over the ring
$\mathbb{C}[[z,z_{2}]]$ for some new indeterminate $z_{2}$ and write
$z_{1}=zz_{2}$. By applying
$\varrho_{z_{1}}\otimes\Pi_{1}\otimes{\bar{\varrho}}^{-}_{z_{2}}$ to (2.17)
and left and right-multiplying by
${\mathcal{L}}^{-}_{{\bar{\varrho}},23}(z_{2}^{-1})^{-1}$ we obtain

(B.16)
${\mathcal{R}}_{\varrho{\bar{\varrho}}}(z)_{12}{\mathcal{L}}_{\varrho}(z_{1})_{13}{\mathcal{L}}^{-}_{{\bar{\varrho}}}(z_{2}^{-1})^{-1}_{32}={\mathcal{L}}^{-}_{{\bar{\varrho}}}(z_{2}^{-1})^{-1}_{32}{\mathcal{L}}_{\varrho}(z_{1})_{13}{\mathcal{R}}_{\varrho{\bar{\varrho}}}(z)_{12}$

an equation in
$(\widetilde{{\mathcal{A}}}^{(2)}\otimes\operatorname{End}({\mathbb{C}^{2}}))[[z_{2}]]$.
By a direct computation we obtain

(B.17)
${\mathcal{L}}^{-}_{\bar{\varrho}}(z_{2}^{-1})^{-1}=\frac{1}{z_{2}^{2}-1}\begin{pmatrix}q^{-D-1}z_{2}^{2}&{\bar{a}}^{\dagger}q^{-D-1}z_{2}\\\
aq^{D-1}z_{2}&q^{D+1}z_{2}^{2}-q^{-D-1}\end{pmatrix}\in\operatorname{End}({\mathbb{C}^{2}})\otimes\widetilde{{\mathcal{A}}}.$

Now we consider the equation

(B.18)
$(z_{2}^{2}-1)X_{12}{\mathcal{L}}_{\varrho}(z_{1})_{13}{\mathcal{L}}^{-}_{{\bar{\varrho}}}(z_{2}^{-1})^{-1}_{32}=(z_{2}^{2}-1){\mathcal{L}}^{-}_{{\bar{\varrho}}}(z_{2}^{-1})^{-1}_{32}{\mathcal{L}}_{\varrho}(z_{1})_{13}X_{12}$

in
$(\widetilde{{\mathcal{A}}}^{(2)}\otimes\operatorname{End}({\mathbb{C}^{2}}))[[z_{2}]]$,
for some $X\in\widetilde{{\mathcal{A}}}^{(2)}_{0}$ such that
$\chi^{(2)}(X)=X$. It suffices to prove that

(B.19)
$\mathcal{X}=\Big{\\{}X\in\widetilde{{\mathcal{A}}}^{(2)}_{0}\,\Big{|}\,X\text{
satisfies }\eqref{X:equation:2}\text{ and is fixed by }\chi^{(2)}\Big{\\}},$

which by Lemma B.4 contains
$(\varrho_{z}\otimes{\bar{\varrho}}^{-})({\mathcal{R}})$, is spanned by the
element given in the right-hand side of (B.15).

By considering explicit expressions for
$(z_{2}^{2}-1){\mathcal{L}}_{\varrho}(z_{1})_{13}{\mathcal{L}}^{-}_{{\bar{\varrho}}}(z_{2}^{-1})_{32}^{-1}$
and
$(z_{2}^{2}-1){\mathcal{L}}^{-}_{{\bar{\varrho}}}(z_{2}^{-1})_{32}^{-1}{\mathcal{L}}_{\varrho}(z_{1})_{13}$,
we obtain that (B.18) amounts to the system

$\displaystyle
X\big{(}q^{D_{1}-D_{2}-1}-a^{\dagger}_{1}a_{2}q^{-D_{1}+D_{2}-2}z\big{)}=\big{(}q^{D_{1}-D_{2}-1}-a_{1}{\bar{a}}^{\dagger}_{2}q^{D_{1}-D_{2}}z\big{)}X,$
$\displaystyle
X\Big{(}\big{(}{\bar{a}}^{\dagger}_{2}q^{D_{1}-D_{2}-1}+a^{\dagger}_{1}q^{-D_{1}-D_{2}-2}z\big{)}-a^{\dagger}_{1}q^{-D_{1}+D_{2}}zz_{2}^{2}\Big{)}=$
$\displaystyle\qquad\qquad=\Big{(}{\bar{a}}^{\dagger}_{2}q^{-D_{1}-D_{2}-1}-\big{(}a^{\dagger}_{1}q^{-D_{1}-D_{2}-2}+{\bar{a}}^{\dagger}_{2}q^{D_{1}-D_{2}+1}z\big{)}zz_{2}^{2}\Big{)}X,$
$\displaystyle
X\Big{(}a_{2}q^{-D_{1}+D_{2}-1}-\big{(}a_{1}q^{D_{1}-D_{2}}+a_{2}q^{D_{1}+D_{2}+1}z\big{)}zz_{2}^{2}\Big{)}=$
$\displaystyle\qquad\qquad=\Big{(}\big{(}a_{2}q^{D_{1}+D_{2}-1}+a_{1}q^{D_{1}-D_{2}}z\big{)}-a_{1}q^{D_{1}+D_{2}+2}zz_{2}^{2}\Big{)}X,$
$\displaystyle
X\Big{(}q^{-D_{1}+D_{2}+1}+q^{D_{1}-D_{2}+1}z^{2}-a_{1}{\bar{a}}^{\dagger}_{2}q^{D_{1}-D_{2}}z\Big{)}=\Big{(}q^{-D_{1}+D_{2}+1}+q^{D_{1}-D_{2}+1}z^{2}-a^{\dagger}_{1}a_{2}q^{-D_{1}+D_{2}-2}z\Big{)}X$

for $X\in\widetilde{{\mathcal{A}}}_{0}^{(2)}$ fixed by $\chi^{(2)}$. Since
$\mathbb{C}[[z,z_{2}]]=(\mathbb{C}[[z]])[[z_{2}]]$, considering expansion
coefficients with respect to $z_{2}$, we can use $[X,q^{D_{1}+D_{2}}]=0$ to
deduce that the above system is equivalent to

(B.20) $\displaystyle\begin{aligned}
Xa_{2}q^{-2D_{1}}&=\big{(}a_{2}+a_{1}q^{-2D_{2}+1}z\big{)}X,\qquad&a_{1}X&=X\big{(}a_{1}q^{-2(D_{2}+1)}+q^{-1}a_{2}z\big{)},\\\
Xa^{\dagger}_{1}q^{2(D_{2}+1)}&=\big{(}a^{\dagger}_{1}+{\bar{a}}^{\dagger}_{2}q^{2D_{1}+3}z\big{)}X,&{\bar{a}}^{\dagger}_{2}X&=X\big{(}{\bar{a}}^{\dagger}_{2}q^{2D_{1}}+q^{-1}a^{\dagger}_{1}z\big{)},\end{aligned}$
(B.21) $\displaystyle\begin{aligned}
\big{[}X,q^{2D_{1}}\big{]}&=\big{(}Xa^{\dagger}_{1}a_{2}q^{2D_{2}-1}-a_{1}{\bar{a}}^{\dagger}_{2}q^{2D_{1}+1}X\big{)}z,\\\
\big{[}X,q^{2D_{2}}+q^{2D_{1}}z^{2}\big{]}&=\big{(}Xa_{1}{\bar{a}}^{\dagger}_{2}q^{2D_{1}-1}-a^{\dagger}_{1}a_{2}q^{2D_{2}-3}X\big{)}z.\end{aligned}$

Note that $q^{-2D_{1}(D_{2}+1)}\in\widetilde{{\mathcal{A}}}^{(2)}_{0}$ is
fixed by $\chi^{(2)}$. Hence without loss of generality we may write

(B.22) $X=\widetilde{X}q^{-2D_{1}(D_{2}+1)},$

for some $\widetilde{X}\in\widetilde{{\mathcal{A}}}^{(2)}_{0}$ fixed by
$\chi^{(2)}$. The system (B.20-B.21) is equivalent to

(B.23) $\displaystyle[\widetilde{X},a_{2}]$
$\displaystyle=q^{-2D_{2}+1}a_{1}\widetilde{X}z,$
$\displaystyle[a_{1},\widetilde{X}]$
$\displaystyle=\widetilde{X}q^{2D_{1}-1}a_{2}z,$ (B.24)
$\displaystyle[\widetilde{X},a^{\dagger}_{1}]$
$\displaystyle={\bar{a}}^{\dagger}_{2}q^{2D_{1}+3}\widetilde{X}z,$
$\displaystyle[{\bar{a}}^{\dagger}_{2},\widetilde{X}]$
$\displaystyle=\widetilde{X}a^{\dagger}_{1}q^{-2D_{2}-3}z,$ (B.25)
$\displaystyle\big{[}\widetilde{X},q^{2D_{1}}\big{]}$
$\displaystyle=\big{(}\widetilde{X}a^{\dagger}_{1}a_{2}q^{2D_{1}-1}-a_{1}{\bar{a}}^{\dagger}_{2}q^{2D_{1}+1}\widetilde{X}\big{)}z,$
(B.26) $\displaystyle\big{[}\widetilde{X},q^{2D_{2}}+q^{2D_{1}}z^{2}\big{]}$
$\displaystyle=\big{(}\widetilde{X}a_{1}{\bar{a}}^{\dagger}_{2}q^{2D_{2}+3}-a^{\dagger}_{1}a_{2}q^{2D_{2}-3}\widetilde{X}\big{)}z.$

Since $\chi^{(2)}$ fixes $\widetilde{X}$, the equations in (B.23) and the
equations in (B.24) are pairwise equivalent. At the same time, the system
(B.23-B.24) implies (B.25) and (B.26). To show this, since
$[\widetilde{X},q^{2D_{1}}]=[a^{\dagger}_{1}a_{1},\widetilde{X}]$ from
(B.23-B.24) we obtain

$\displaystyle[\widetilde{X},q^{2D_{1}}]+a_{1}{\bar{a}}^{\dagger}_{2}q^{2D_{1}+1}\widetilde{X}z-\widetilde{X}a^{\dagger}_{1}a_{2}q^{2D_{1}-1}z=$
$\displaystyle\qquad=a_{1}{\bar{a}}^{\dagger}_{2}q^{2D_{1}+1}\widetilde{X}z-[\widetilde{X},a^{\dagger}_{2}]a_{1}+a^{\dagger}_{1}[a_{1},\widetilde{X}]-\widetilde{X}a^{\dagger}_{1}a_{2}q^{2D_{1}-1}z$
$\displaystyle\qquad=\big{(}{\bar{a}}^{\dagger}_{2}q^{2D_{1}+3}[a_{1},\widetilde{X}]-[\widetilde{X},a^{\dagger}_{1}]a_{2}q^{2D_{1}-1}\big{)}z,$

which vanishes, thereby recovering (B.25). Applying $\chi^{(2)}$ to (B.25) we
obtain
$[\widetilde{X},q^{-2D_{2}}]=\big{(}\widetilde{X}a^{\dagger}_{1}a_{2}q^{-2D_{2}-1}-a_{1}{\bar{a}}^{\dagger}_{2}q^{-2D_{2}+1}\widetilde{X}\big{)}z$.
Left-and-right multiplying this by $q^{2D_{2}}$ and using (B.23-B.24) to
rewrite the result we obtain

(B.27)
$[\widetilde{X},q^{2D_{2}}]=\big{(}{\bar{a}}^{\dagger}_{2}\widetilde{X}a_{1}q^{2D_{2}+3}-q^{2D_{2}-1}a^{\dagger}_{1}\widetilde{X}a_{2}\big{)}z.$

Finally, using (B.27) and again (B.23-B.24), we derive that

$\displaystyle[\widetilde{X},q^{2D_{2}}+q^{2D_{1}}z^{2}]-\widetilde{X}a_{1}{\bar{a}}^{\dagger}_{2}q^{2D_{2}+3}z+a^{\dagger}_{1}a_{2}q^{2D_{2}-3}\widetilde{X}z=$
$\displaystyle\qquad={\bar{a}}^{\dagger}_{2}\widetilde{X}a_{1}q^{2D_{2}+3}z-q^{2D_{2}-1}a^{\dagger}_{1}\widetilde{X}a_{2}z+[\widetilde{X},q^{2D_{1}}]z^{2}+$
$\displaystyle\qquad\qquad-({\bar{a}}^{\dagger}_{2}\widetilde{X}-\widetilde{X}a^{\dagger}_{1}q^{-2D_{2}-3}z)a_{1}q^{2D_{2}+3}z+a^{\dagger}_{1}q^{2D_{2}-1}(\widetilde{X}a_{2}-a_{1}q^{-2D_{2}+1}\widetilde{X}z)z$
$\displaystyle\qquad=\big{(}\widetilde{X}a^{\dagger}_{1}a_{1}+a^{\dagger}_{1}a_{1}\widetilde{X}+[\widetilde{X},1-a^{\dagger}_{1}a_{1}]\big{)}z^{2}$

which vanishes, thereby proving (B.26) as well.

We have obtained that the system (B.23-B.26) is equivalent to the system
(B.24). Writing $p=q^{2}$, without loss of generality we set

$\widetilde{X}=Ye_{p}(q^{3}za_{1}{\bar{a}}^{\dagger}_{2})e_{p}(q^{-1}za^{\dagger}_{1}a_{2})$

for some $Y\in\widetilde{{\mathcal{A}}}^{(2)}_{0}$ fixed by $\chi^{(2)}$,
noting that $e_{p}(q^{3}za_{1}{\bar{a}}^{\dagger}_{2})$ and
$e_{p}(q^{-1}za^{\dagger}_{1}a_{2})$ lie in
$\widetilde{{\mathcal{A}}}^{(2)}_{0}$ and are fixed by $\chi^{(2)}$. The
theorem now follows from the following claim.

_Claim:_ (B.24) is satisfied if and only if $Y\in\mathbb{C}[[z]]$.

In the special case $Y=1$, indeed (B.24) is indeed satisfied:

$\displaystyle[\widetilde{X},a^{\dagger}_{1}]-{\bar{a}}^{\dagger}_{2}q^{2D_{1}+3}z\widetilde{X}$
$\displaystyle=\Big{(}[e_{p}(q^{3}za_{1}{\bar{a}}^{\dagger}_{2}),a^{\dagger}_{1}]-{\bar{a}}^{\dagger}_{2}q^{2D_{1}+3}ze_{p}(q^{3}za_{1}{\bar{a}}^{\dagger}_{2})\Big{)}e_{p}(q^{-1}za^{\dagger}_{1}a_{2}),$
$\displaystyle[{\bar{a}}^{\dagger}_{2},\widetilde{X}]-\widetilde{X}a^{\dagger}_{1}q^{-2D_{2}-3}z$
$\displaystyle=e_{p}(q^{3}za_{1}{\bar{a}}^{\dagger}_{2})\Big{(}[{\bar{a}}^{\dagger}_{2},e_{p}(q^{-1}za^{\dagger}_{1}a_{2})]-e_{p}(q^{-1}za^{\dagger}_{1}a_{2})a^{\dagger}_{1}q^{-2D_{2}-3}z\Big{)},$

with the expressions in parentheses vanishing by virtue of (A.14) and (B.10)
(with $y=q^{-1}z$).

For general $Y$ we therefore have

$\displaystyle[\widetilde{X},a^{\dagger}_{1}]-{\bar{a}}^{\dagger}_{2}q^{2D_{1}+3}z\widetilde{X}$
$\displaystyle=[Y,a^{\dagger}_{1}]e_{p}(q^{3}za_{1}{\bar{a}}^{\dagger}_{2})e_{p}(q^{-1}za^{\dagger}_{1}a_{2}),$
$\displaystyle[{\bar{a}}^{\dagger}_{2},\widetilde{X}]-\widetilde{X}a^{\dagger}_{1}q^{-2D_{2}-3}z$
$\displaystyle=[{\bar{a}}^{\dagger}_{2},Y]e_{p}(q^{3}za_{1}{\bar{a}}^{\dagger}_{2})e_{p}(q^{-1}za^{\dagger}_{1}a_{2}).$

Both right-hand sides vanish, i.e. (B.24) is indeed satisfied, if and only if
$Y$ lies in the centralizer in $\widetilde{{\mathcal{A}}}^{(2)}$ of
$\\{a^{\dagger}_{1},{\bar{a}}^{\dagger}_{2}\\}$, which is trivial by Lemma
B.5. This proves the claim. ∎

##  C An alternative proof of the main theorem

In this part of the appendix we give a proof of the boundary factorization
identity (8.2) which is independent of the universal K-matrix formalism.
Before we state and prove a key lemma, note that expressions of the form
$e_{p}(\gamma^{D}y)$ where $\gamma\in\mathbb{C}^{\times}$ and $y$ is an
indeterminate give rise to well-defined $\operatorname{End}(W)$-valued formal
power series, sending $w_{j}$ to $e_{p}(\gamma^{j}y)w_{j}$.

######  Lemma C.1.

Let $y$ be a formal parameter and let $p$ be a nonzero complex number unequal
to a root of unity. In $\operatorname{End}(W\otimes W)[[y]]$ we have the
identities

(C.1) $\displaystyle e_{p}(pa_{1}{\bar{a}}^{\dagger}_{2})(y;p)_{D_{1}}$
$\displaystyle=(y;p)_{D_{1}}e_{p}(-a_{1}{\bar{a}}^{\dagger}_{2}p^{D_{1}}y)e_{p}(pa_{1}{\bar{a}}^{\dagger}_{2})$
(C.2) $\displaystyle
e_{p}(pa_{1}{\bar{a}}^{\dagger}_{2})(p^{1-D_{1}}y;p)_{D_{1}}^{-1}e_{p}(py{\bar{a}}^{\dagger}_{1}a_{2})$
$\displaystyle=e_{p}(py{\bar{a}}^{\dagger}_{1}a_{2})(p^{1-D_{2}}y;p)_{D_{2}}^{-1}e_{p}(pa_{1}{\bar{a}}^{\dagger}_{2}).$

###### Proof.

Note that

$W\otimes W=\bigoplus_{m\in\mathbb{Z}_{\geq 0}}(W\otimes
W)_{m},\qquad(W\otimes W)_{m}:=\bigoplus_{j,k\geq 0\atop
j+k=m}\mathbb{C}w_{j}\otimes w_{k}.$

Because each factor in (C.1-C.2) preserves each finite-dimensional subspace
$(W\otimes W)_{m}$, it suffices to prove the restrictions of (C.1-C.2) to
$(W\otimes W)_{m}$, where $m\in\mathbb{Z}_{\geq 0}$ is fixed but arbitrary.
Note that on $(W\otimes W)_{m}$ the operators appearing as arguments of the
deformed exponentials are nilpotent. Therefore the operators on the left- and
right-hand side of the restricted equations depend rationally on $p$ and hence
it suffices to prove them with $p$ restricted to an open subset of
$\mathbb{C}$.

We will prove the restriction of (C.1) to $(W\otimes W)_{m}$ for all
$p\in\mathbb{C}$ such that $|p|<1$. Combining (A.4) and (A.7) we obtain
$(y;p)_{D}=\frac{e_{p}(p^{D}y)}{e_{p}(y)}$; as a consequence, (C.1) is
equivalent to

(C.3)
$e_{p}(pa_{1}{\bar{a}}^{\dagger}_{2})e_{p}(p^{D_{1}}y)=e_{p}(p^{D_{1}}y)e_{p}(-a_{1}{\bar{a}}^{\dagger}_{2}p^{D_{1}}y)e_{p}(pa_{1}{\bar{a}}^{\dagger}_{2}).$

But this equation follows directly from (A.12) and the observation
$(a_{1}{\bar{a}}^{\dagger}_{2})(p^{D_{1}}y)=p(p^{D_{1}}y)(a_{1}{\bar{a}}^{\dagger}_{2})$.

On the other hand121212 We will need (C.2) with $|p|<1$, but we are not aware
of a direct proof of this. , we will prove the restricted version of (C.2) for
all $p\in\mathbb{C}^{\times}$ such that $|p|>1$. In this case, for all
$j\in\mathbb{Z}_{\geq 0}$ we have

$(p^{1-j}y;p)_{j}^{-1}=(y;p^{-1})_{j}^{-1}=\frac{(p^{-j}y;p^{-1})_{\infty}}{(y;p^{-1})_{\infty}}\in\mathbb{C}[[y]].$

From (A.7) and (A.9) we deduce the identity
$(p^{-j}y;p^{-1})_{\infty}=e_{p^{-1}}(p^{-j}y)^{-1}=e_{p}(p^{1-j}y)$ of formal
power series in $y$. Hence, we have
$(p^{1-D}y;p)_{D}^{-1}=(y;p^{-1})_{\infty}^{-1}e_{p}(p^{1-D}y)$ in
$\operatorname{End}(W)[[y]]$. As a consequence, (C.2) is equivalent to

(C.4)
$e_{p}(pa_{1}{\bar{a}}^{\dagger}_{2})e_{p}(p^{1-D_{1}}y)e_{p}(py{\bar{a}}^{\dagger}_{1}a_{2})=e_{p}(py{\bar{a}}^{\dagger}_{1}a_{2})e_{p}(p^{1-D_{2}}y)e_{p}(pa_{1}{\bar{a}}^{\dagger}_{2}).$

To prove (C.4), note that (B.11) can be evaluated at $y=p$, and the result can
be rewritten as

$e_{p}(pa_{1}{\bar{a}}^{\dagger}_{2})\big{(}p^{-D_{1}}+{\bar{a}}_{1}^{\dagger}a_{2}\big{)}=\big{(}p^{-D_{2}}+{\bar{a}}_{1}^{\dagger}a_{2}\big{)}e_{p}(pa_{1}{\bar{a}}^{\dagger}_{2}).$

By iteration we obtain

(C.5)
$e_{p}(pa_{1}{\bar{a}}^{\dagger}_{2})e_{p}\big{(}p^{1-D_{1}}y+py{\bar{a}}^{\dagger}_{1}a_{2}\big{)}=e_{p}\big{(}p^{1-D_{2}}y+py{\bar{a}}^{\dagger}_{1}a_{2}\big{)}e_{p}(pa_{1}{\bar{a}}^{\dagger}_{2}).$

Note that
$({\bar{a}}^{\dagger}_{1}a_{2})p^{1-D_{1}}=p\,p^{1-D_{1}}({\bar{a}}^{\dagger}_{1}a_{2})$
and
$p^{1-D_{2}}({\bar{a}}^{\dagger}_{1}a_{2})=p\,({\bar{a}}^{\dagger}_{1}a_{2})p^{1-D_{2}}$.
Applying (A.11), we obtain (C.4), as required. ∎

###### Alternative proof of Theorem 8.1..

By virtue of (4.13), the desired identity, viz.

(C.6)
${\mathcal{K}}_{\upsilon}(z)_{1}{\mathcal{R}}_{\upsilon\phi}(z^{2}){\mathcal{K}}_{\phi}(z)_{2}\,{\mathcal{O}}={\mathcal{O}}{\mathcal{K}}_{\varrho}(q^{-\mu/2}z)_{1}{\mathcal{R}}_{\varrho{\bar{\varrho}}}(z^{2}){\mathcal{K}}_{\bar{\varrho}}(q^{\mu/2}z)_{2}$

for arbitrary $\mu\in\mathbb{C}$ and $q,\xi\in\mathbb{C}^{\times}$ such that
$q$ is not a root of unity, is equivalent to

(C.7) $\displaystyle
e_{p}(pa_{1}{\bar{a}}^{\dagger}_{2}){\mathcal{K}}_{\upsilon}(z)_{1}{\mathcal{R}}_{\upsilon\phi}(z^{2}){\mathcal{K}}_{\phi}(z)_{2}e_{p}(pa_{1}{\bar{a}}^{\dagger}_{2})^{-1}=$
$\displaystyle\qquad=q^{\mu(D_{1}-D_{2})/2}{\mathcal{K}}_{\varrho}(q^{-\mu/2}z)_{1}{\mathcal{R}}_{\varrho{\bar{\varrho}}}(z^{2}){\mathcal{K}}_{\bar{\varrho}}(q^{\mu/2}z)_{2}q^{\mu(D_{2}-D_{1})/2}$

where $p=q^{2}$. The strategy of the proof is as follows. We move various
simple factors in ${\mathcal{F}}^{(2)}(D_{1},D_{2})$ to the right in both
sides of (C.7), thus bringing them to a similar form. Then more advanced
product formulas involving q-exponentials and finite q-Pochhammer symbols
yield the desired equality.

More precisely, we set $\gamma=pq^{-\mu}\xi^{-1}\in\mathbb{C}^{\times}$ and
from (A.2) deduce

$(\gamma
z^{-2};p)_{j}^{-1}=p^{j(1-j)/2}(-\gamma^{-1}z^{2})^{j}(p^{1-j}\gamma^{-1}z^{2};p)_{j}^{-1}$

for all $j\in\mathbb{Z}_{\geq 0}$. Using the identities
$q^{-D^{2}}a^{\dagger}=-q{\bar{a}}^{\dagger}q^{-D^{2}}$ and
$q^{-D^{2}}a=aq^{2D-1}q^{-D^{2}}$, we obtain, for the left-hand side of (C.7),

(C.8) $\displaystyle
e_{p}(pa_{1}{\bar{a}}^{\dagger}_{2}){\mathcal{K}}_{\upsilon}(z)_{1}{\mathcal{R}}_{\upsilon\phi}(z^{2}){\mathcal{K}}_{\phi}(z)_{2}e_{p}(pa_{1}{\bar{a}}^{\dagger}_{2})^{-1}=$
$\displaystyle\;=e_{p}(pa_{1}{\bar{a}}^{\dagger}_{2})\big{(}-\gamma^{-1}q^{1-D_{1}}\big{)}^{D_{1}}(\gamma
z^{2};p)_{D_{1}}(p^{1-D_{1}}\gamma^{-1}z^{2};p)_{D_{1}}^{-1}e_{p}(z^{2}a^{\dagger}_{1}a_{2})\cdot$
$\displaystyle\;\qquad\cdot
q^{(2\mu-1)D_{1}-2D_{2}-2D_{1}D_{2}-D_{2}^{2}}(-\xi)^{D_{2}}e_{p}(pa_{1}{\bar{a}}^{\dagger}_{2})^{-1}$
$\displaystyle\;=e_{p}(pa_{1}{\bar{a}}^{\dagger}_{2})(\gamma
z^{2};p)_{D_{1}}(p^{1-D_{1}}\gamma^{-1}z^{2};p)_{D_{1}}^{-1}e_{p}(p\gamma^{-1}z^{2}{\bar{a}}^{\dagger}_{1}a_{2})(-q^{-D_{1}-D_{2}-2}\xi)^{D_{1}+D_{2}}e_{p}(pa_{1}{\bar{a}}^{\dagger}_{2})^{-1}$
$\displaystyle\;=e_{p}(pa_{1}{\bar{a}}^{\dagger}_{2})(\gamma
z^{2};p)_{D_{1}}(p^{1-D_{1}}\gamma^{-1}z^{2};p)_{D_{1}}^{-1}e_{p}(p\gamma^{-1}z^{2}{\bar{a}}^{\dagger}_{1}a_{2})e_{p}(pa_{1}{\bar{a}}^{\dagger}_{2})^{-1}(-q^{-D_{1}-D_{2}-2}\xi)^{D_{1}+D_{2}}.$

Similarly, for the right-hand side of (C.7) we obtain

(C.9) $\displaystyle
q^{\mu(D_{1}-D_{2})/2}{\mathcal{K}}_{\varrho}(q^{-\mu/2}z)_{1}{\mathcal{R}}_{\varrho{\bar{\varrho}}}(z^{2}){\mathcal{K}}_{\bar{\varrho}}(q^{\mu/2}z)_{2}q^{\mu(D_{2}-D_{1})/2}=$
$\displaystyle\;=(\gamma
z^{2};p)_{D_{1}}q^{\mu(D_{1}-D_{2})/2-D_{1}^{2}}(-\xi)^{D_{1}}e_{p}(q^{3}z^{2}a_{1}{\bar{a}}^{\dagger}_{2})e_{p}(q^{-1}z^{2}a_{1}^{\dagger}a_{2})\cdot$
$\displaystyle\;\qquad\cdot(p^{1-D_{2}}\gamma^{-1}z^{2};p)_{D_{2}}^{-1}q^{\mu(D_{2}-D_{1})/2-2(D_{1}+D_{2})-2D_{1}D_{2}-D_{2}^{2}}(-\xi)^{D_{2}}=$
$\displaystyle\;=(\gamma
z^{2};p)_{D_{1}}e_{p}(-a_{1}{\bar{a}}^{\dagger}_{2}q^{2D_{1}}\gamma
z^{2})e_{p}(p\gamma^{-1}z^{2}{\bar{a}}^{\dagger}_{1}a_{2})(p^{1-D_{2}}\gamma^{-1}z^{2};p)_{D_{2}}^{-1}(-q^{-D_{1}-D_{2}-2}\xi)^{D_{1}+D_{2}}.$

Therefore (C.7) is equivalent to

(C.10) $\displaystyle e_{p}(pa_{1}{\bar{a}}^{\dagger}_{2})(\gamma
z^{2};p)_{D_{1}}(p^{1-D_{1}}\gamma^{-1}z^{2};p)_{D_{1}}^{-1}e_{p}(p\gamma^{-1}z^{2}{\bar{a}}^{\dagger}_{1}a_{2})e_{p}(pa_{1}{\bar{a}}^{\dagger}_{2})^{-1}=$
$\displaystyle\qquad\qquad=(\gamma
z^{2};p)_{D_{1}}e_{p}(-a_{1}{\bar{a}}^{\dagger}_{2}p^{D_{1}}\gamma
z^{2})e_{p}(p\gamma^{-1}z^{2}{\bar{a}}^{\dagger}_{1}a_{2})(p^{1-D_{2}}\gamma^{-1}z^{2};p)_{D_{2}}^{-1}.$

Applying (C.1) with $y=\gamma z^{2}$ and (C.2) with $y=\gamma^{-1}z^{2}$, we
deduce (C.10), as required. ∎

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