# Crystal limits of compact semisimple quantum groups as higher-rank graph
algebras

Marco Matassa  OsloMet – Oslo Metropolitan University, Oslo, Norway
<EMAIL_ADDRESS>and  Robert Yuncken  Institut Elie Cartan de Lorraine
– Université de Lorraine – 3 rue Augustin Fresnel, 57000 Metz, France
<EMAIL_ADDRESS>
###### Abstract.

Let $\mathcal{O}_{q}[K]$ denote the quantized coordinate ring over the field
$\mathbb{C}(q)$ of rational functions corresponding to a compact semisimple
Lie group $K$, equipped with its $*$-structure. Let
${\mathbf{A}_{0}}\subset\mathbb{C}(q)$ denote the subring of regular functions
at $q=0$. We introduce an ${\mathbf{A}_{0}}$-subalgebra
$\mathcal{O}_{q}^{\mathbf{A}_{0}}[K]\subset\mathcal{O}_{q}[K]$ which is stable
with respect to the $*$-structure, and which has the following properties with
respect to the crystal limit $q\to 0$.

The specialization of $\mathcal{O}_{q}[K]$ at each
$q\in(0,\infty)\setminus\\{1\\}$ admits a faithful $*$-representation
$\pi_{q}$ on a fixed Hilbert space, a result due to Soibelman. We show that
for every element $a\in\mathcal{O}_{q}^{\mathbf{A}_{0}}[K]$, the family of
operators $\pi_{q}(a)$ admits a norm-limit as $q\to 0$. These limits define a
$*$-representation $\pi_{0}$ of $\mathcal{O}_{q}^{\mathbf{A}_{0}}[K]$. We show
that the resulting $*$-algebra
$\mathcal{O}[K_{0}]=\pi_{0}(\mathcal{O}_{q}^{\mathbf{A}_{0}}[K])$ is a
Kumjian-Pask algebra, in the sense of Aranda Pino, Clark, an Huef and Raeburn.
We give an explicit description of the underlying higher-rank graph in terms
of crystal basis theory. As a consequence, we obtain a continuous field of
$C^{*}$-algebras $(C(K_{q}))_{q\in[0,\infty]}$, where the fibres at $q=0$ and
$\infty$ are explicitly defined higher-rank graph algebras.

###### Key words and phrases:

quantum groups, crystal basis, quantized coordiante rings, graph algebras,
higher-rank graph algebras

###### 1991 Mathematics Subject Classification:

Primary: 20G42, Secondary: 46L67, 17B3

###### Contents

  1. 1 Introduction
    1. 1.1 Algebraic and analytic crystal limits
    2. 1.2 Statement of main results
    3. 1.3 Structure of the paper
  2. 2 Background material
    1. 2.1 Ground fields and specialization
    2. 2.2 Quantized enveloping algebras
    3. 2.3 Crystal bases
    4. 2.4 Tensor products
  3. 3 The quantized coordinate ring
    1. 3.1 Definitions
    2. 3.2 Star structure and compact integral form
    3. 3.3 The gauge action
    4. 3.4 Flag manifolds and non-simply connected groups
  4. 4 The analytic limit of the quantized coordinate ring
    1. 4.1 The analytic limit for $SU_{q}(2)$
    2. 4.2 The analytic limit in higher rank
  5. 5 The Cartan braiding and the crystal limit
    1. 5.1 Braiding and commutation relations
    2. 5.2 The Cartan braiding
    3. 5.3 Crystal limit of the commutation relations
  6. 6 Properties of the Cartan braiding
    1. 6.1 Hexagon and braid relations
    2. 6.2 Partial action of the symmetric group
    3. 6.3 Left and right ends
  7. 7 The higher-rank graph algebra
    1. 7.1 Higher-rank graphs
    2. 7.2 Higher-rank graph algebras
    3. 7.3 Higher-rank graphs associated to quantum groups
    4. 7.4 Properties of the higher-rank graph
  8. 8 The crystal algebra
    1. 8.1 Definition of the algebra
    2. 8.2 Projections associated to crystal elements
    3. 8.3 Projections associated to sets of crystal elements
    4. 8.4 Graph algebra relations
  9. 9 Crystal limits are higher-rank graph algebras
  10. 10 Further properties and examples

##  1\. Introduction

###  1.1. Algebraic and analytic crystal limits

Let $K$ be a compact semisimple Lie group, $G$ its complex form. The quantized
coordinate ring on $G$ can be constructed in two different flavours. In the
algebraic approach we define a $\mathbb{C}(q)$-algebra $\mathcal{O}_{q}[G]$
with $q$ an indeterminate, while in the analytic approach we define a
$\mathbb{C}$-algebra $\mathcal{O}[G_{q}]$, or a $*$-algebra
$\mathcal{O}[K_{q}]$ if we are interested in the compact form, with
$q\in(0,\infty)\setminus\\{1\\}$ being a numerical value. In this paper, we
study these algebras in the crystal limit $q\to 0$ (or $q\to\infty$). This is
defined quite differently in the two communities.

In the algebraic world, the $q=0$ limit is realized by the theory of crystal
bases, due to Kashiwara [Kas90, Kas91] and Lusztig [Lus90]. Beginning with the
quantized enveloping algebra $\mathcal{U}_{q}(\mathfrak{g})$ of
$\mathfrak{g}=\mathfrak{k}_{\mathbb{C}}$ over $\mathbb{C}(q)$ or
$\mathbb{Q}(q)$, one replaces the quantized Lie algebra generators $E_{i}$,
$F_{i}$ by algebraic renormalizations $\tilde{E}_{i}$, $\tilde{F}_{i}$, called
the Kashiwara operators, and then localizes the finite-dimensional modules at
$q=0$. The resulting localized modules admit nice “crystal” bases with an
elegant combinatorial structure. This approach was used by [Kas93, LT96,
Igl06] to define a crystal basis for the quantized coordinate rings
$\mathcal{O}_{q}[G]$.

In the analytical world, one is interested in the continuous fields of
quantized coordinate rings, in the $C^{*}$-algebraic sense of Woronowicz
[Wor87b, Wor87a] and Vaksman-Soibelman [VS88, Soĭ91], as well as the
subalgebras of continuous functions on quantized flag varieties. Here, we have
a handful of results, in particular due to Hong and Szymanski [HS02, HS03],
which show that in certain examples ($\mathrm{SU}_{q}(2)$, quantum projective
spaces, quantum spheres and quantum lens spaces), the continuous field of
$C^{*}$-algebras over $(0,\infty)$ extends to $[0,\infty]$, with fibres at the
boundaries given by graph $C^{*}$-algebras.111In fact, their results are
stronger, showing that their quantized function algebras are isomorphic for
all values of $q\neq 1$, although only in the $q=0$ limit do the graph algebra
generators coincide with matrix coefficients for the quantum group. A recent
study of the $q=0$ limit of $C(\mathrm{SU}_{q}(n))$ has also been undertaken
in [GP22], although without comparing the results to graph algebras. In fact,
it is known that groups with rank larger than one do not admit a description
as a graph algebra, see the remarks at the end of the introduction of [HS02].

To overcome this difficulty, one can look at the more general concept of
higher-rank graph algebras, due to Kumjian and Pask [KP00]. Roughly speaking,
a higher-rank graph is a graph whose edges are classed into $N$ colours, and
which is equipped with an equivalence relation on paths such that, for
instance, a path of two edges of colours red-blue can always be replaced by an
equivalent path of colours blue-red with the same start and end points. In
this way, paths $e$ in the graph are equipped with a _degree_ , or _coloured
length_ , $\mathsf{d}(e)\in\mathbb{N}^{N}$. For the precise definition, see
Section 7.

In a recent advance, Olof Giselsson [Gis] has shown that $C(SU_{q}(3))$ is
isomorphic to the $C^{*}$-algebra of a $2$-graph, which is reproduced in Fig.
1.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$ Figure 1. Giselsson’s
$2$-graph for $SU(3)$. Note that this is not just the Bruhat graph for
$SU(3)$—there are additional edges.

The main result of this article is that the coordinate ring of any quantized
compact semisimple Lie group admits a $q=0$ limit which is a higher-rank graph
algebra. The construction of the higher-rank graph, which is summarized below
and makes use of the theory of crystal bases, will make clear the link between
the algebraic and analytic approaches to the crystal limit.

The generators of the $C^{*}$-algebra of our higher-rank graphs can be
represented by tensor products of shift operators. They have a much simpler
algebra structure than the algebras $C(K_{q})$, even at $q=1$, since giving an
explicit formula for a product of matrix coefficients requires the use of
Clebsch-Gordan formulas, whereas the product rules for graph algebras are
purely combinatorial. In this way, our results can be seen as a non-
commutative geometric counterpart of the simplifications afforded by crystal
basis theory in the algebraic context.

###  1.2. Statement of main results

The quantum groups $\mathcal{O}[G_{q}]$ with $q$ specialized in
$(0,\infty)\setminus\\{1\\}$ can be represented faithfully on a Hilbert space
using the work of Soibelman [Soĭ91]. By composing this with the specialization
map222The specialization map is only partially defined, since not all elements
of $\mathcal{O}_{q}[G]$ can be specialized at any
$q\in(0,\infty)\setminus\\{1\\}$. We will ignore this detail in the
introduction. $\mathcal{O}_{q}[G]\to\mathcal{O}[G_{q}]$, we obtain for every
value $q\in(0,\infty)\setminus\\{1\\}$ a representation $\pi_{q}$ of
$\mathcal{O}_{q}[G]$ on the Hilbert space
$\mathsf{H}=\ell^{2}(\mathbb{N})^{\otimes l}\otimes L^{2}(T)$, where $l$ is
the length of the longest word in the Weyl group of $K$ and $T$ is the maximal
torus of $K$.

Using the theory of crystal bases, we can easily define a subring
$\mathcal{O}_{q}^{\mathbf{A}_{0}}[G]$ of the $\mathbb{C}(q)$-algebra
$\mathcal{O}_{q}[G]$, consisting of elements that can be "specialized at
$q=0$" (see for instance [Igl06, §6] for this construction, where it is
denoted $\widetilde{\mathcal{F}}$). We begin by proving the following analytic
result, which is probably known to experts.

######  Theorem 1.1.

For any $u\in\mathcal{O}_{q}^{\mathbf{A}_{0}}[G]$, the one-parameter family of
operators $\pi_{q}(u)$ admits a well-defined norm limit $\pi_{0}(u)$ as $q\to
0$.

However, an important subtlety arises when we introduce the $*$-structure. One
can define a $*$-structure on the $\mathbb{C}(q)$ algebra $\mathcal{O}_{q}[G]$
which specializes to the usual $*$-structure on $\mathcal{O}[G_{q}]$ when
$q\in(0,\infty)\setminus\\{1\\}$, and which defines the real form
$\mathcal{O}[K_{q}]$. However, the ${\mathbf{A}_{0}}$-subalgebra
$\mathcal{O}_{q}^{\mathbf{A}_{0}}[G]$ is not stable under the $*$, see Example
3.1, so the algebra
$\mathcal{O}_{q}^{\mathbf{A}_{0}}[G]/q\mathcal{O}_{q}^{\mathbf{A}_{0}}[G]$ at
$q=0$ doesn’t inherit a $*$-structure.

To resolve this issue, we define a new ${\mathbf{A}_{0}}$-algebra
$\mathcal{O}_{q}^{\mathbf{A}_{0}}[K]\subset\mathcal{O}_{q}[G]$ which is stable
under the $*$. Essentially, $\mathcal{O}_{q}^{\mathbf{A}_{0}}[K]$ is just the
${\mathbf{A}_{0}}$-algebra generated by $\mathcal{O}_{q}^{\mathbf{A}_{0}}[G]$
and $\mathcal{O}_{q}^{\mathbf{A}_{0}}[G]^{*}$, although for technical reasons
we define it slightly differently, see Definition 3.2. The limit $\pi_{0}$ of
the Soibelman representations extends to
$\mathcal{O}_{q}^{\mathbf{A}_{0}}[K]$, and this allows us to define a
$*$-subalgebra
$\mathcal{O}[K_{0}]=\pi_{0}(\mathcal{O}_{q}^{\mathbf{A}_{0}}[K])$ of
$\mathcal{B}(\mathsf{H})$. Our main task is to investigate the structure of
this $*$-algebra.

For this, a major role is played by crystal basis theory, as follows. Given a
dominant integral weight $\lambda\in\mathbf{P}^{+}$, we write $V(\lambda)$ for
the simple $\mathcal{U}_{q}(\mathfrak{g})$-module of highest weight $\lambda$,
and $\mathcal{B}(\lambda)$ for the associated crystal. Recall that if
$\mu,\mu^{\prime}\in\mathbf{P}^{+}$ are dominant integral weights with
$\mu+\mu^{\prime}=\lambda$, then there is a unique non-trivial morphism of
crystals
$\iota_{\mu,\mu^{\prime}}:\mathcal{B}(\lambda)\to\mathcal{B}(\mu)\otimes\mathcal{B}(\mu^{\prime})$.
The image of this inclusion is called the _Cartan component_ of the tensor
product $\mathcal{B}(\mu)\otimes\mathcal{B}(\mu^{\prime})$, namely the unique
irreducible component of highest weight $\mu+\mu^{\prime}$.

######  Definition 1.2.

Let $\lambda,\mu\in\mathbf{P}^{+}$ with $\mu\leq\lambda$. The _$\mu$ -right
end_ of a crystal element $b\in\mathcal{B}(\lambda)$ is the element
$\mathsf{R}(b)=b^{\prime\prime}\in\mathcal{B}(\mu)$ determined by
$\iota_{\lambda-\mu,\mu}:b\mapsto b^{\prime}\otimes b^{\prime\prime}$.

If ${\bm{\Pi}}=(\varpi_{1},\cdots,\varpi_{r})$ is the family of fundamental
weights of $K$ and $\lambda\geq\rho=\sum_{i}\varpi_{i}$, then we write

$\mathsf{R}_{\bm{\Pi}}(b)=\big{(}\mathsf{R}_{\varpi_{1}}(b),\cdots,\mathsf{R}_{\varpi_{r}}(b)\big{)}$

for the family of _fundamental right ends_ of $b\in\mathcal{B}(\lambda)$.

We can now define the higher-rank graph associated to the crystal limit of
$\mathcal{O}[K_{q}]$. We identify the set of dominant integral weights
$\mathbf{P}^{+}$ with the monoid $\mathbb{N}^{r}$ by identifying
$(n_{i})\in\mathbb{N}^{r}$ with $\sum_{i}n_{i}\varpi_{i}$.

######  Theorem 1.3.

We can define an $r$-graph $\Lambda_{\mathfrak{g}}$ as follows. The vertex set
is

$\Lambda_{\mathfrak{g}}^{0}=\\{\mathsf{R}_{\bm{\Pi}}(b)\mid
b\in\mathcal{B}(\rho)\\}$

namely the fundamental right ends of elements of $\mathcal{B}(\rho)$. The
paths are given by pairs
$(v,b)\in\Lambda_{\mathfrak{g}}^{0}\times\mathcal{B}(\lambda)$ where
$v=\mathsf{R}_{\bm{\Pi}}(c)$ is such that $c\otimes b$ is in the Cartan
component of $\mathcal{B}(\rho)\otimes\mathcal{B}(\lambda)$. The range and
source maps are given by

$\displaystyle\mathsf{s}(v,b)=v,$
$\displaystyle\mathsf{r}(v,b)=\mathsf{R}_{\bm{\Pi}}(c\otimes b).$

######  Remark 1.4.

It is part of the proposition that the above definitions depend only on the
vertex $v\in\Lambda_{\mathfrak{g}}^{0}$ and not on the choice of crystal
element $c\in\mathcal{B}(\rho)$ which represents it.

Our main theorem is the following. We write
$\mathrm{KP}_{\mathbb{C}}(\Lambda_{\mathfrak{g}})$ for the Kumjian-Pask
algebra of $\Lambda_{\mathfrak{g}}$ in the sense of [APCaHR13], and
$C^{*}(\Lambda_{\mathfrak{g}})$ for the higher rank graph $C^{*}$-algebra in
the sense of [KP00].

######  Theorem 1.5.

Let $K$ be a compact, connected, simply connected semisimple Lie group, with
complexified Lie algebra $\mathfrak{g}=\mathfrak{k}_{\mathbb{C}}$, and let
$\Lambda_{\mathfrak{g}}$ be the higher rank graph from Theorem 1.3. There is
an isomorphism of $*$-algebras
$\mathcal{O}[K_{0}]\cong\mathrm{KP}_{\mathbb{C}}(\Lambda_{\mathfrak{g}})$. Its
$C^{*}$-closure is $C(K_{0})\cong C^{*}(\Lambda_{\mathfrak{g}})$.

In fact, we obtain a much more general result, see Theorem 9.3. Firstly, we
may allow $K$ to be non-simply connected. Secondly, we consider the quantized
coordinate ring of the canonical torus bundle $Y_{S}\to X_{S}$ over any
generalized flag variety for $K$ associated to a set $S\subseteq{\bm{\Delta}}$
of simple roots, see Section 3.4 for the notation. The coordinate ring
$\mathcal{O}[Y_{S}]$ is generated by matrix coefficients for simple
$\mathfrak{g}$-modules with highest weights in a submonoid
$\mathbf{P}^{+}_{K,S}\subseteq\mathbf{P}^{+}$. We write
$\mathsf{C}=(\vartheta_{1},\cdots,\vartheta_{N})$ for the set of dominant
weights which generate $\mathbf{P}_{K,S}$. The quantized coordinate rings of
$Y_{S}$ and $X_{S}$ admit analytic limits at $q=0$.

We prove that by replacing ${\bm{\Pi}}$ by $\mathsf{C}$ in the definition of
$\Lambda_{\mathfrak{g}}$, we obtain another higher-rank graph
$\Lambda_{\mathfrak{g},\mathsf{C}}$ and show that we have an isomorphism
$\mathcal{O}[Y_{S,0}]\cong\mathrm{KP}_{\mathbb{C}}(\Lambda_{\mathfrak{g},\mathsf{C}})$.
The gauge-invariant subalgebra is then
$\mathcal{O}[X_{S,0}]\cong\mathrm{KP}_{\mathbb{C}}(\Lambda_{\mathfrak{g},\mathsf{C}})_{0}$.
These isomorphisms extend naturally to the $C^{*}$-algebras.

The results can be summarized in the following way.

######  Theorem 1.6.

Let $K$ be a compact connected semisimple Lie group, and let $Y_{S}$ be the
canonical torus bundle over the generalized flag variety $X_{S}$ associated to
a set $S\subset{\bm{\Delta}}$ of simple roots. There is a continuous field of
$C^{*}$-algebras $C(Y_{S,\bullet})$ over $[0,\infty]$ whose fibres are

$C(Y_{S,q})\cong\begin{cases}C(Y_{S}),&q=1,\\\
C(Y_{S,q}),&q\in(0,\infty)\setminus\\{1\\}\\\
C^{*}(\Lambda_{\mathfrak{g},\mathsf{C}}),&q\in\\{0,\infty\\}.\end{cases}$

where $\Lambda_{\mathfrak{g},\mathsf{C}}$ is the higher rank graph described
above (and spelled out in Theorem 7.8).

######  Remark 1.7.

The $C^{*}$-algebras $C(K_{q})$ are known to be abstractly isomorphic for all
$q\in(0,\infty)\setminus\\{1\\}$, see [Gis20]. It is not yet known if they are
also isomorphic to the graph algebra $C(K_{0})$, other than for the cases
$K=\mathrm{SU}_{q}(2)$ and $\mathrm{SU}_{q}(3)$, the latter due again to
Giselsson [Gis20], and a few examples in the case of homogeneous spaces, due
to Hong and Szymanski [HS02].

###  1.3. Structure of the paper

In Section 2 we review some background material on quantized enveloping
algebras and crystal bases. In Section 3 we review some standard material on
quantized coordinate rings, as well as defining the ${\mathbf{A}_{0}}$-form
$\mathcal{O}_{q}^{\mathbf{A}_{0}}[K]$ of $\mathcal{O}_{q}[K]$. In Section 4 we
show that the elements of $\mathcal{O}_{q}^{\mathbf{A}_{0}}[K]$ admit analytic
limits in terms of the Soibelman representation. This allows us to define the
$*$-algebra $\mathcal{O}[K_{0}]$, which is our main object of study. In
Section 5 we introduce the notion of Cartan braiding, which is the crystal
limit of a certain rescaled version of the braiding in the category of finite-
dimensional $\mathcal{U}_{q}(\mathfrak{g})$-modules. This allows us to obtain
various relations holding in $\mathcal{O}[K_{0}]$. In Section 6 we study the
Cartan braiding in more detail: we show that it satisfies the hexagon and
braid relations, and gives a partial action of the symmetric group on tensor
products. We also introduce the notion of right end of a crystal. In Section 7
we discuss higher-rank graph and their corresponding algebras. We prove, using
crystal bases, that one can associate a higher-rank graph
$\Lambda_{\mathfrak{g}}$ of rank $r$ to any complex semisimple Lie algebra
$\mathfrak{g}$ of rank $r$ (in fact, we prove this in a more general setting).
In Section 8 we introduce the crystal algebra $\mathcal{A}_{\mathfrak{k}}$ as
a useful tool to study the relation between $\mathcal{O}[K_{0}]$ and
$\mathrm{KP}_{\mathbb{C}}(\Lambda_{\mathfrak{g}})$, the $*$-algebra associated
to the higher-rank graph $\Lambda_{\mathfrak{g}}$. The main result here is
that we have surjective $*$-homomorphisms
$\mathrm{KP}_{\mathbb{C}}(\Lambda_{\mathfrak{g}})\to\mathcal{A}_{\mathfrak{k}}$
and $\mathcal{A}_{\mathfrak{k}}\to\mathcal{O}[K_{0}]$. In Section 9 we show
that the three $*$-algebras
$\mathrm{KP}_{\mathbb{C}}(\Lambda_{\mathfrak{g}})$,
$\mathcal{A}_{\mathfrak{k}}$ and $\mathcal{O}[K_{0}]$ are all $*$-isomorphic,
which gives our main result about the structure of the crystal limit
$\mathcal{O}[K_{0}]$. Finally in Section 10 we discuss some further properties
of the higher-rank graphs $\Lambda_{\mathfrak{g}}$, mainly related to the role
of the Weyl groups, as well as discussing some explicit examples.

### Acknowledgements

We would like to thank Olof Giselsson for sharing with us the example of
$\mathrm{SU}_{q}(3)$ as a $2$-graph algebra, as well as comparing his own
approach with ours. We would also like to thank Sergey Neshveyev for
discussions concerning continuous fields of $C^{*}$-algebras in the context of
compact quantum groups.

##  2\. Background material

###  2.1. Ground fields and specialization

As mentioned before, we will consider the quantized enveloping algebra
$\mathcal{U}_{q}(\mathfrak{g})$ and quantized coordinate ring
$\mathcal{O}_{q}[G]$ in two settings: 1) over the field of rational functions
$\mathbb{C}(q)$ with $q$ an indeterminate; 2) over $\mathbb{C}$ with
$q\in(0,\infty)\setminus\\{1\\}$.

We write $\mathbb{C}[q]$ for the polynomial ring in $q$, so that
$\mathbb{C}(q)$ is its field of fractions. We consider the three subrings

$\displaystyle{\mathbf{A}_{0}}$ $\displaystyle:=\\{g/h:g,h\in\mathbb{C}[q],\
h(0)\neq 0\\},$ $\displaystyle\mathbf{A}_{\infty}$
$\displaystyle:=\\{g/h:g,h\in\mathbb{C}[q],\ h(\infty)\neq 0\\},$ (2.1)
$\displaystyle\mathbf{A}$ $\displaystyle:=\mathbb{C}[q,q^{-1}].$

For every fixed $q\in[0,\infty]$, there is a partially defined _specialization
map_ $\operatorname{ev}_{q}:\mathbb{C}(q)\to\mathbb{C}$ given by evaluation at
$q$. We apologize for the reuse of $q$ as both formal parameter and
specialized value $q\in[0,\infty]$. The rings ${\mathbf{A}_{0}}$,
$\mathbf{A}_{\infty}$, $\mathbf{A}$ contain the elements which can be
evaluated at $0$, at $\infty$ and at all $q\in\mathbb{C}^{\times}$,
respectively.

###  2.2. Quantized enveloping algebras

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and fix a Cartan
subalgebra $\mathfrak{h}$. Write $\mathfrak{k}$ for the compact real form of
$\mathfrak{g}$ and put $\mathfrak{t}=\mathfrak{k}\cap\mathfrak{h}$. The
connected, simply connected Lie groups associated to $\mathfrak{g}$ and
$\mathfrak{k}$ are denoted by $G$ and $K$. The maximal torus of $K$ is
$T\cong\mathbb{T}^{r}$, where $r$ denotes the rank.

We write ${\bm{\Phi}}$ for the set of roots of $\mathfrak{g}$ and
${\bm{\Delta}}=\\{\alpha_{1},\cdots,\alpha_{r}\\}$ for a choice of simple
roots. The root lattice will be denoted by $\mathbf{Q}$ and the lattice of
integral weights by $\mathbf{P}$, generated by the fundamental weights
${\bm{\Pi}}=\\{\varpi_{1},\cdots,\varpi_{r}\\}$. The abelian semigroup
$\mathbf{P}^{+}=\mathbb{N}\cdot{\bm{\Pi}}\cong\mathbb{N}^{r}$ of dominant
weights will play an important role in the higher-rank graph to be defined
later. We put $\rho=\sum_{i=1}^{r}\varpi_{i}$ as usual. We write
$(\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}},\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}})$
for the invariant bilinear form on $\mathfrak{h}^{*}$ such that the short
roots $\alpha$ satisfy $(\alpha,\alpha)=2$.

The quantized enveloping algebra $\mathcal{U}_{q}(\mathfrak{g})$ is the Hopf
algebra with generators $E_{i}$, $F_{i}$ and $K_{i}^{\pm 1}$ for
$i=1,\cdots,r$, and with the standard algebra relations that can be found for
instance in [KS97, §6.1.2], or [Jan96, §4.3] with $K_{i}=K_{\alpha_{i}}$.
Again, we may consider this either over the field $\mathbb{C}(q)$ or over
$\mathbb{C}$ with $q\in(0,\infty)\setminus\\{1\\}$ a fixed parameter. There
are different possible choices for the coproduct. The one we consider is more
or less standard in the context of crystal bases333For the operator
algebraist, it is equivalent to coproduct from [NT13], but the opposite of the
coproduct from [KS97, VY20]. and is given by

$\Delta(K_{i})=K_{i}\otimes K_{i},\quad\Delta(E_{i})=E_{i}\otimes
K_{i}^{-1}+1\otimes E_{i},\quad\Delta(F_{i})=F_{i}\otimes 1+K_{i}\otimes
F_{i}.$ (2.2)

We write $\mathcal{U}_{q}(\mathfrak{h})$ for the Hopf subalgebra generated by
the $K_{i}^{\pm 1}$.

We also consider a $*$-structure on $\mathcal{U}_{q}(\mathfrak{g})$ which
corresponds to the compact real form $\mathfrak{k}$ of $\mathfrak{g}$. It acts
on the generators by

$\displaystyle K_{i}^{*}=K_{i},$ $\displaystyle
E_{i}^{*}=q_{i}F_{i}K_{i}^{-1},$ $\displaystyle
F_{i}^{*}=q_{i}^{-1}K_{i}E_{i},$ (2.3)

and $q^{*}=q$ when we work over $\mathbb{C}(q)$. Here
$q_{i}=q^{(\alpha_{i},\alpha_{i})/2}$. We write
$\mathcal{U}_{q}(\mathfrak{k})$ for $\mathcal{U}_{q}(\mathfrak{g})$ equipped
with this $*$-structure. We note that, since we are putting $q^{*}=q$, the
specialization map at $q\in\mathbb{C}^{\times}$ is a $*$-morphism only when
$q$ is real.

######  Remark 2.1.

We warn the reader that this is not the most common choice for the
$*$-structure in the operator algebraic picture. It is a convenient choice
when working with crystal bases because the formulas (2.3) coincide with those
of the standard anti-automorphism of $\mathcal{U}_{q}(\mathfrak{g})$ called
$\tau_{1}$ in [Jan96, §9.20 (3)], except that our involution $*$ is extended
$\mathbb{C}$-antilinearly.

We write $V(\lambda)$ for the irreducible finite-dimensional integrable
(_i.e._ , type 1) $\mathcal{U}_{q}(\mathfrak{g})$-module of highest weight
$\lambda\in\mathbf{P}^{+}$. Again, this can be constructed either in the
algebraic setting as a module over $\mathbb{C}(q)$, or in the analytic setting
as a $\mathbb{C}$-vector space. If necessary, we will denote the latter by
$V(\lambda)_{q}$ with $q\in(0,\infty)\setminus\\{1\\}$ a fixed parameter, but
generally we will suppress the subscript and interpret the
$\mathcal{U}_{q}(\mathfrak{g})$-module $V(\lambda)$ according to context. Once
again, there is a partially defined specialization map
$\operatorname{ev}_{q}:V(\lambda)\to V(\lambda)_{q}$ for each
$q\in(0,\infty)\setminus\\{1\\}$.

There is a unique $\mathbb{C}(q)$-valued inner product on $V(\lambda)$ such
that the $\mathcal{U}_{q}(\mathfrak{g})$-action is a $*$-representation and
the highest weight vector $v_{\lambda}$ has norm $1$, see [Jan96, Lemma 9.20
c)]. For convenience, we will refer to this as the _standard inner product_ on
$V(\lambda)$.

###  2.3. Crystal bases

In this section we briefly recall various facts associated with Kashiwara’s
theory of crystal bases [Kas90, Kas91]. Our main references are the textbooks
[HK02] and [Jan96].

We need the Kashiwara operators $\tilde{E}_{i}$ and $\tilde{F}_{i}$, which are
algebraic renormalizations of $E_{i}$ and $F_{i}$ with well-defined limits at
$q=0$. They play a similar role to the operator phases of $E_{i}$ and $F_{i}$
in the $C^{*}$-algebraic world. To define them, we first introduce the divided
powers

$\displaystyle E_{i}^{(k)}$ $\displaystyle:=E_{i}^{k}/[k]_{q_{i}}!,$
$\displaystyle F_{i}^{(k)}$ $\displaystyle:=F_{i}^{k}/[k]_{q_{i}}!,$

where $[k]_{q}:=\frac{q^{k}-q^{-k}}{q-q^{-1}}$ and
$[k]_{q}!:=[k]_{q}[k-1]_{q}\cdots[1]_{q}$. The Kashiwara operators are defined
by imposing that, for any weight vector $u_{0}$ in an integrable
$\mathcal{U}_{q}(\mathfrak{g})$-module $V$ with $E_{i}u_{0}=0$, we have

$\displaystyle\tilde{E}_{i}:F_{i}^{(k)}u_{0}$ $\displaystyle\mapsto
F_{i}^{(k-1)}u_{0},$ $\displaystyle\tilde{F}_{i}:F_{i}^{(k)}u_{0}$
$\displaystyle\mapsto F_{i}^{(k+1)}u_{0},$

with the convention $F_{i}^{(-1)}u_{0}=0$.

We can now define crystal bases.

######  Definition 2.2.

A _crystal lattice_ in $V$ is an ${\mathbf{A}_{0}}$-submodule $\mathcal{L}$ of
$V$ such that:

  1. (1)

$\mathcal{L}$ is finitely generated over ${\mathbf{A}_{0}}$ and generates $V$
as a vector space over $\mathbb{C}(q)$,

  2. (2)

$\mathcal{L}=\bigoplus_{\mu\in\mathbf{P}}\mathcal{L}_{\mu}$, where
$\mathcal{L}_{\mu}=\mathcal{L}\cap V_{\mu}$,

  3. (3)

$\tilde{E}_{i}\mathcal{L}\subset\mathcal{L}$ and
$\tilde{F}_{i}\mathcal{L}\subset\mathcal{L}$ for all $i$.

######  Definition 2.3.

A _crystal basis_ of $V$ is a pair $(\mathcal{L},\mathcal{B})$, where
$\mathcal{L}$ is a crystal lattice and $\mathcal{B}$ is a $\mathbb{C}$-basis
of the quotient $\mathcal{L}/q\mathcal{L}$ such that:

  1. (1)

$\mathcal{B}=\bigcup_{\mu}\mathcal{B}_{\mu}$, where
$\mathcal{B}_{\mu}=\mathcal{B}\cap(\mathcal{L}_{\mu}/q\mathcal{L}_{\mu})$,

  2. (2)

$\tilde{E}_{i}\mathcal{B}\subset\mathcal{B}\sqcup\\{0\\}$ and
$\tilde{F}_{i}\mathcal{B}\subset\mathcal{B}\sqcup\\{0\\}$ for all $i$,

  3. (3)

for any $b,b^{\prime}\in\mathcal{B}$ and $i$ we have
$b=\tilde{E}_{i}b^{\prime}\Longleftrightarrow b^{\prime}=\tilde{F}_{i}b$.

In this way, we replace the action of $\mathcal{U}_{q}(\mathfrak{g})$ on a
finite-dimensional module $V$ by a purely combinatorial action of the
Kashiwara operators $\tilde{E}_{i},\tilde{F}_{i}$ on the finite set
$\mathcal{B}\sqcup\\{0\\}$. Kashiwara proved that every finite-dimensional
integrable $\mathcal{U}_{q}(\mathfrak{g})$-module admits a crystal basis,
which is unique up to isomorphism. We denote the crystal basis of the simple
module $V(\lambda)$ by $(\mathcal{L}(\lambda),\mathcal{B}(\lambda))$.

The crystal basis is orthonormal in the following sense, compare [HK02, Lemma
5.1.6], [Jan96, Theorem 9.25].

######  Lemma 2.4.

Fix $\lambda\in\mathbf{P}^{+}$. Let
$\\{v_{1},\cdots,v_{d}\\}\subset\mathcal{L}(\lambda)$ be a family of lifts for
the crystal basis $\mathcal{B}(\lambda)=\\{b_{1},\cdots,b_{d}\\}$ of
$V(\lambda)$. Let
$\langle\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}},\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}\rangle$
be the standard inner product on $V(\lambda)$. Then $\langle
v_{a},v_{b}\rangle\in\delta_{ab}+q{\mathbf{A}_{0}}$.

As a consequence, for any $u,v\in\mathcal{L}(\lambda)$ we have $\langle
u,v\rangle\in{\mathbf{A}_{0}}$.

From an analytic point of view, Lemma 2.4 means that for any
$u,v\in\mathcal{L}(\lambda)$, the limit of the specialized inner products

$\lim_{q\to 0}\operatorname{ev}_{q}\langle u,v\rangle$

exists. Note that $\operatorname{ev}_{q}\langle u,v\rangle$ does not
necessarily exist for all $q>0$, but since $\langle
u,v\rangle\in{\mathbf{A}_{0}}$, it at least exists in some neighbourhood of
$0$. We will use similar observations frequently in this work, and as above
will use the notation $\lim_{q\to 0}$ when strictly speaking we mean
$\lim_{q\to 0^{+}}$.

###  2.4. Tensor products

A major feature of crystal bases is the simplicity of their tensor products,
and this will play a major role in what follows.

Let $V_{1}$ and $V_{2}$ be finite-dimensional integrable
$\mathcal{U}_{q}(\mathfrak{g})$-modules with crystal bases
$(\mathcal{L}_{1},\mathcal{B}_{1})$ and $(\mathcal{L}_{2},\mathcal{B}_{2})$
respectively. Then
$(\mathcal{L}_{1}\otimes_{{\mathbf{A}_{0}}}\mathcal{L}_{2},\mathcal{B}_{1}\times\mathcal{B}_{2})$
is a crystal basis of $V_{1}\otimes V_{2}$, see [HK02, Theorem 4.4.1]. We will
be lazy and write $\mathcal{L}_{1}\otimes\mathcal{L}_{2}$ to mean the crystal
lattice $\mathcal{L}_{1}\otimes_{\mathbf{A}_{0}}\mathcal{L}_{2}$.

To describe the action of the Kashiwara operators we first define

$\varepsilon_{i}(b):=\max\left\\{k\in\mathbb{N}_{0}:\tilde{E}_{i}^{k}b\neq
0\right\\},\quad\varphi_{i}(b):=\max\left\\{k\in\mathbb{N}_{0}:\tilde{F}_{i}^{k}b\neq
0\right\\}.$

Then the action on the tensor product is given by

$\displaystyle\tilde{E}_{i}(b\otimes b^{\prime})$
$\displaystyle=\begin{cases}\tilde{E}_{i}b\otimes
b^{\prime}&\varphi_{i}(b)\geq\varepsilon_{i}(b^{\prime})\\\
b\otimes\tilde{E}_{i}b^{\prime}&\varphi_{i}(b)<\varepsilon_{i}(b^{\prime})\end{cases},$
(2.4) $\displaystyle\tilde{F}_{i}(b\otimes b^{\prime})$
$\displaystyle=\begin{cases}\tilde{F}_{i}b\otimes
b^{\prime}&\varphi_{i}(b)>\varepsilon_{i}(b^{\prime})\\\
b\otimes\tilde{F}_{i}b^{\prime}&\varphi_{i}(b)\leq\varepsilon_{i}(b^{\prime})\end{cases}.$
(2.5)

Here, as is customary, we denote the element
$(b_{1},b_{2})\in\mathcal{B}_{1}\times\mathcal{B}_{2}$ by $b_{1}\otimes
b_{2}$, and we use the conventions $b_{1}\otimes 0=0\otimes b_{2}=0$.

##  3\. The quantized coordinate ring

###  3.1. Definitions

Let $V$ be a finite-dimensional integrable
$\mathcal{U}_{q}(\mathfrak{g})$-module over $\mathbb{C}(q)$ and let
$V^{*}=\operatorname{Hom}_{\mathbb{C}(q)}(V,\mathbb{C}(q))$ be the dual
module. A _matrix coefficient_ for the module $V$ is a $\mathbb{C}(q)$-linear
map $c^{V}_{f,v}$ on $\mathcal{U}_{q}(\mathfrak{g})$ of the form

$c^{V}_{f,v}(X):=f(Xv),\quad v\in V,\ f\in V^{*}.$ (3.1)

The _quantized coordinate ring_ $\mathcal{O}_{q}[G]$ is the subspace of
$\operatorname{Hom}_{\mathbb{C}(q)}(\mathcal{U}_{q}(\mathfrak{g}),\mathbb{C}(q))$
spanned by the matrix coefficients of finite-dimensional integrable modules.
This is a Hopf algebra with operations obtained from those of
$\mathcal{U}_{q}(\mathfrak{g})$ by duality. Here, we use the algebraists’
convention of a non-skew pairing:

$\displaystyle(\Delta(X),a\otimes b)=(X,ab),$ $\displaystyle(X\otimes
Y,\Delta(a))=(XY,a).$

Explicitly, this corresponds to the matrix coproduct on matrix coefficients:

$\Delta(c^{V}_{f,v})=\sum_{i}c^{V}_{f,v_{i}}\otimes c^{V}_{f^{i},v},$

where $\\{v_{i}\\}$ and $\\{f_{i}\\}$ are any basis and dual basis for $V$ and
$V^{*}$, respectively. We denote the left and right regular representations of
$\mathcal{U}_{q}(\mathfrak{g})$ on $\mathcal{O}_{q}[G]$ by

$\displaystyle X\triangleright c^{V}_{f,v}$ $\displaystyle=c^{V}_{f,Xv},$
$\displaystyle c^{V}_{f,v}\triangleleft X$ $\displaystyle=c^{V}_{f\circ X,v}$
(3.2)

We also have a version of the quantized coordinate ring defined over
$\mathbb{C}$ for a fixed complex parameter $q\in(0,\infty)\setminus\\{1\\}$.
This algebra will be denoted $\mathcal{O}[G_{q}]$, with apologies for the
subtle difference in notation. For every $q\in(0,\infty)\setminus\\{1\\}$ we
have a partially defined specialization map

$\operatorname{ev}_{q}:\mathcal{O}_{q}[G]\to\mathcal{O}[G_{q}],$

induced by evaluation at $q$.

In [Kas93] Kashiwara introduced a crystal lattice for $\mathcal{O}_{q}[G]$
consisting of finite sums of matrix coefficients $c^{V}_{f,v}$ where $v$
belongs to a crystal lattice $\mathcal{L}$ for $V$ and $f$ belongs to
$\mathcal{L}^{*}=\operatorname{Hom}_{\mathbf{A}_{0}}(\mathcal{L},{\mathbf{A}_{0}})\subset
V^{*}$. We denote this ${\mathbf{A}_{0}}$-form by
$\mathcal{O}_{q}^{\mathbf{A}_{0}}[G]$. This algebra was also studied by
Iglesias in [Igl06], where he uses the notation $\widetilde{\mathcal{F}}$.

###  3.2. Star structure and compact integral form

The $*$-structure (2.3) on $\mathcal{U}_{q}(\mathfrak{g})$ induces a
corresponding $*$-structure on $\mathcal{O}_{q}[G]$ and on each
$\mathcal{O}[G_{q}]$ by the formula $(X,a^{*})=\overline{(S(X)^{*},a)}$. When
equipped with this $*$-structure, we denote the $*$-algebras
$\mathcal{O}_{q}[G]$ and $\mathcal{O}[G_{q}]$ by $\mathcal{O}_{q}[K]$ and
$\mathcal{O}[K_{q}]$, respectively. Again, we caution that $q^{*}=q$, and so
the specialization map $\mathcal{O}_{q}[K]\to\mathcal{O}[K_{q}]$ is a
$*$-morphism only when $q$ is real.

As mentioned in the introduction, the ${\mathbf{A}_{0}}$-form
$\mathcal{O}_{q}^{\mathbf{A}_{0}}[G]$ is not stable under the $*$. This can be
seen in the following example.

######  Example 3.1.

Let $V$ be the fundamental representation of
$\mathcal{U}_{q}(\mathfrak{sl}_{2})$ over $\mathbb{C}(q)$. Let $v_{1}$ be the
highest weight vector for $V$ and $v_{2}=Fv_{1}$. Let $\\{f^{1},f^{2}\\}\in
V^{*}$ be the dual basis for $V^{*}$. The subspace of
$\mathcal{O}_{q}^{\mathbf{A}_{0}}[G]$ corresponding to the fundamental
representation is spanned by the matrix coefficients
$u^{V}_{ij}=c^{V}_{f^{i},v_{j}}$ with $i,j\in\\{1,2\\}$.

In this particular case, $\\{v_{1},v_{2}\\}$ is an orthonormal basis for $V$,
so upon specialization at $q\in(0,\infty)\setminus\\{1\\}$, the matrix
coefficients $u^{V}_{ij}$ coincide with the traditional generators for the
operator algebraists’ $*$-algebra $\mathcal{O}[K_{q}]$, which in Woronowicz’s
notation [Wor87b] are denoted

$\displaystyle\alpha$ $\displaystyle=u^{V}_{11},$ $\displaystyle-q\gamma^{*}$
$\displaystyle=u^{V}_{12},$ $\displaystyle\gamma$ $\displaystyle=u^{V}_{21},$
$\displaystyle\alpha^{*}$ $\displaystyle=u^{V}_{22}.$

We see that $u^{V*}_{21}=-q^{-1}u^{V}_{12}$, which does not belong to
$\mathcal{O}_{q}^{\mathbf{A}_{0}}[G]$.

Because of this, we cannot simply equip the algebra
$\mathcal{O}_{q}^{\mathbf{A}_{0}}[G]$ with the $*$ operation. Instead, we
proceed as follows.

For each simple module $V(\lambda)$ fix a lift of the crystal basis to a basis
of weight vectors $\\{v_{i}\\}_{i}\subset\mathcal{L}(\lambda)$, with dual
basis $\\{f^{i}\\}_{i}\subset V(\lambda)^{*}$. We impose that
$v_{1}=v_{\lambda}$ is the highest weight vector of $V(\lambda)$, and so
$f^{1}=f^{-\lambda}$ is the lowest weight element.

With this notation, we define the matrix coefficients

$\mathsf{f}^{\lambda}_{i}:=c^{V(\lambda)}_{f^{i},v_{\lambda}},\quad\mathsf{v}^{\lambda}_{i}:=S(c^{V(\lambda)}_{f^{-\lambda},v_{i}}).$
(3.3)

We also note that
$\mathsf{v}^{\lambda}_{i}=c^{V(\lambda)^{*}}_{\tilde{v}_{i},f^{-\lambda}}$,
where $\\{\tilde{v_{i}}\\}_{i}\subset V(\lambda)^{**}$ is the dual basis to
$\\{f^{i}\\}_{i}$.

Following Joseph [Jos95, §9.1.6], let $\mathcal{O}_{q}[G/N^{+}]$ and
$\mathcal{O}_{q}[G/N^{-}]$ be the $\mathbb{C}(q)$-subalgebras of
$\mathcal{O}_{q}[G]$ generated by the matrix coefficients
$\mathsf{f}^{\lambda}_{i}$ and $\mathsf{v}^{\lambda}_{i}$, respectively (for
all $\lambda\in\mathbf{P}^{+}$ and $i=1,\cdots,\dim V(\lambda)$). Joseph uses
the notation $R_{q}[G/N^{+}]$ and $R_{q}[G/N^{-}]$. The multiplication map
$\mathcal{O}_{q}[G/N^{+}]\otimes_{\mathbb{C}(q)}\mathcal{O}_{q}[G/N^{-}]\to\mathcal{O}_{q}[G]$
is surjective, see [Jos95, Proposition 9.2.2].

We now define ${\mathbf{A}_{0}}$-forms of these subalgebras.

######  Definition 3.2.

We define:

  * •

$\mathcal{O}^{\mathbf{A}_{0}}_{q}[G/N^{+}]$ as the
${\mathbf{A}_{0}}$-subalgebra generated by the elements
$\mathsf{f}^{\lambda}_{i}$,

  * •

$\mathcal{O}^{\mathbf{A}_{0}}_{q}[G/N^{-}]$ as the
${\mathbf{A}_{0}}$-subalgebra generated by the elements
$\mathsf{v}^{\lambda}_{i}$,

  * •

$\mathcal{O}^{\mathbf{A}_{0}}_{q}[K]$ as the ${\mathbf{A}_{0}}$-subalgebra
generated by the elements $\mathsf{f}^{\lambda}_{i}$ and
$\mathsf{v}^{\lambda}_{i}$.

Thus, while $\mathcal{O}_{q}[K]$ and $\mathcal{O}_{q}[G]$ denote the same
$\mathbb{C}(q)$-algebra with or without the $*$-structure, we stress that
$\mathcal{O}_{q}^{\mathbf{A}_{0}}[K]$ and
$\mathcal{O}_{q}^{\mathbf{A}_{0}}[G]$ are not the same
${\mathbf{A}_{0}}$-subalgebra. They represent different $q=0$ limits of the
families of algebras $\mathcal{O}[K_{q}]=\mathcal{O}[G_{q}]$. The notation
$\mathcal{O}^{\mathbf{A}_{0}}_{q}[K]$ suggests that this algebra is closed
under the $*$-structure, which is not immediately obvious from its definition.
However this is true, as shown in the next result.

######  Proposition 3.3.

We have
$(\mathcal{O}^{\mathbf{A}_{0}}_{q}[G/N^{+}])^{*}=\mathcal{O}^{\mathbf{A}_{0}}_{q}[G/N^{-}]$.
Moreover

$\begin{split}(\mathsf{f}^{\lambda}_{i})^{*}&\equiv\mathsf{v}^{\lambda}_{i}\mod
q\mathcal{O}^{\mathbf{A}_{0}}_{q}[G/N^{-}],\\\
(\mathsf{v}^{\lambda}_{i})^{*}&\equiv\mathsf{f}^{\lambda}_{i}\mod
q\mathcal{O}^{\mathbf{A}_{0}}_{q}[G/N^{+}].\end{split}$

###### Proof.

Let $\\{v_{i}\\}_{i}$ be the lift of the crystal basis of $V(\lambda)$ as
above. Let $(\cdot,\cdot)$ be the unique inner product on $V(\lambda)$
invariant under $*$ such that $(v_{1},v_{1})=1$. Let $G^{i}_{j}=(v_{i},v_{j})$
be the Gram matrix for the basis $\\{v_{i}\\}_{i}$. Some linear algebra shows
that we have the identity

$(c^{V(\lambda)}_{f^{i},v_{j}})^{*}=\sum_{k,l}(G^{-1})^{l}_{i}G^{j}_{k}S(c^{V(\lambda)}_{f^{k},v_{l}}).$

Now consider the case of
$\mathsf{f}^{\lambda}_{i}=c^{V(\lambda)}_{f^{i},v_{1}}$. We have
$G^{1}_{k}=(v_{1},v_{k})=\delta_{1k}$, since different weight spaces are
orthogonal under the inner product. Then we get

$(\mathsf{f}^{\lambda}_{i})^{*}=\sum_{l}(G^{-1})^{l}_{i}\mathsf{v}^{\lambda}_{l}.$

Recall from Lemma 2.4 that the basis $\\{v_{i}\\}_{i}$ becomes orthonormal at
$q=0$, which implies that $G^{i}_{j}$ and $(G^{-1})^{i}_{j}$ are equal to
$\delta_{ij}$ modulo terms in $q{\mathbf{A}_{0}}$. Thus we obtain the desired
result for $(\mathsf{f}^{\lambda}_{i})^{*}$. The result for
$(\mathsf{v}^{\lambda}_{i})^{*}$ is proven similarly, and it follows that
$(\mathcal{O}^{\mathbf{A}_{0}}_{q}[G/N^{+}])^{*}=\mathcal{O}^{\mathbf{A}_{0}}_{q}[G/N^{-}]$.
∎

######  Remark 3.4.

It is not clear from the definition above whether
$\mathcal{O}_{q}^{\mathbf{A}_{0}}[G]\subset\mathcal{O}_{q}^{\mathbf{A}_{0}}[K]$,
although we expect this to be the case. Answering this question would require
a more careful study of the crystal lattice structure. Since we don’t need
this property here, we won’t discuss it further.

###  3.3. The gauge action

For the moment, we continue to suppose that $K$ is simply connected, with rank
$r$. That is, we have $T\cong\mathbb{T}^{r}$ for the maximal torus and
$\mathbf{P}\cong\hat{T}\cong\mathbb{Z}^{r}$ for the lattice of integral
weights. The quantized coordinate ring $\mathcal{O}_{q}[K]$ is
$\mathbb{Z}^{r}$-graded as follows.

######  Definition 3.5.

For any $\mu\in\mathbf{P}$, we write $\mathcal{O}_{q}[K]_{\mu}$ for the span
of all matrix coefficients $c^{V}_{f,v}$ where $v\in V$ is a vector of weight
$\mu$, so that
$\mathcal{O}_{q}[K]=\bigoplus_{\mu\in\mathbf{P}}\mathcal{O}_{q}[K]_{\mu}$ is a
$\mathbf{P}$-graded algebra.

The $\mathbf{P}$-grading corresponds to the gauge action of the torus subgroup
$T$,

$z\cdot c^{V}_{f,v}=z^{\mu}c^{V}_{f,v}\quad\text{for
}c^{V}_{f,v}\in\mathcal{O}_{q}[K]_{\mu},$ (3.4)

where $z^{\mu}=e^{(\mu,\log(z))}$ denotes the evaluation at $z\in T$ of the
unitary character associated to $\mu\in\mathbf{P}$. Geometrically, this
corresponds to the realization of $K$ as a principal $T$-bundle over the full
flag variety $X=K/T$, and this language is extended by analogy to the quantum
case. The gauge-invariant subalgebra of $\mathcal{O}_{q}[K]$ is then the
quantized coordinate ring of the flag variety.

###  3.4. Flag manifolds and non-simply connected groups

We finish this preliminary section with the structure theory of the quantized
coordinate rings of general flag varieties. This material could be skipped on
a first reading. We follow the conventions of Stokman [Sto03].

Let $\mathfrak{b}_{+}$ denote the Borel subalgebra of $\mathfrak{g}$ generated
by $\mathfrak{h}$ and the elements $E_{1},\cdots,E_{r}$. The standard
parabolic subalgebras are indexed by subsets of simple roots
$S\subseteq{\bm{\Delta}}$. Specifically, we denote by
$\mathfrak{p}_{S}\supseteq\mathfrak{b}_{+}$ the standard parabolic which
contains $F_{i}$ if and only if $\alpha_{i}\in S$. The Levi factor
$\mathfrak{l}_{S}$ of $\mathfrak{p}_{S}$ is generated by $\mathfrak{h}$ and
those $E_{i},F_{i}$ with $\alpha_{i}\in S$. The associated Lie subgroups of
$G$ will always be denoted by the corresponding uppercase letters. We write
$X_{S}=G/P_{S}$ for the corresponding flag variety.

Passing to the compact form, we have $X_{S}=K/K_{S}$ where
$\mathfrak{k}_{S}=\mathfrak{k}\cap\mathfrak{p}_{S}$. The Lie subalgebra
$\mathfrak{k}_{S}$ decomposes as

$\mathfrak{k}_{S}=\mathfrak{k}^{0}_{S}\oplus\mathfrak{z}_{S},$

where $\mathfrak{z}_{S}$ is the centre of $\mathfrak{k}_{S}$ and
$\mathfrak{k}^{0}_{S}$ is semisimple. Then $Y_{S}=K/K^{0}_{S}$ is a principal
torus bundle over the flag variety $X_{S}$ with fibres
$Z_{S}\cong\mathbb{T}^{N}$, where $N=r-|S|$.

For instance, when $S=\emptyset$ we have $K_{\emptyset}=T$ and
$K^{0}_{\emptyset}=\\{1\\}$, so the torus bundle $Y_{\emptyset}\to
X_{\emptyset}$ is the principal bundle $K\to K/T$ with fibres
$Z_{\emptyset}=T\cong\mathbb{T}^{r}$. More generally, the torus group $Z_{S}$
identifies canonically with the dual torus of the abelian group
$\mathbf{P}_{S^{c}}\subseteq\mathbf{P}$ generated by the fundamental weights
$\varpi_{i}$ with $i\notin S$.

All of the above can be quantized. We shall not spell out all the details
here, but refer directly to [Sto03]. Following our above notation, we write
$\mathcal{O}[X_{S,q}]$ and $\mathcal{O}[Y_{S,q}]$ for what Stokman [Sto03]
calls $\mathbb{C}_{q}[U/K_{S}]$ and $\mathbb{C}_{q}[U/K^{0}_{S}]$ in
Definitions 2.3(a) and Section 4, namely

$\displaystyle\mathcal{O}[Y_{S,q}]$
$\displaystyle=\\{a\in\mathcal{O}[K_{q}]\mid T\triangleright
a=\epsilon(T)a\text{ for all }T\in\mathcal{U}_{q}(\mathfrak{k}^{0}_{S})\\},$
$\displaystyle\mathcal{O}[X_{S,q}]$
$\displaystyle=\\{a\in\mathcal{O}[K_{q}]\mid T\triangleright
a=\epsilon(T)a\text{ for all }T\in\mathcal{U}_{q}(\mathfrak{k}_{S})\\}.$

The following is due to Stokman [Sto03]. We retain the notation
$\mathsf{f}^{\lambda}_{i}=c^{V(\lambda)}_{f,v_{1}^{\lambda}}$,
$\mathsf{v}^{\lambda}_{i}=c^{V(\lambda)^{*}}_{\tilde{v},f^{1}_{\lambda}}$ for
the generators of $\mathcal{O}_{q}[G/N^{+}]$ and $\mathcal{O}_{q}[G/N^{-}]$
from Equation (3.3). We denote the specialized versions of these elements by
the same symbols, for simplicity.

######  Theorem 3.6.

Let $\mathbf{P}_{S}^{+}$ denote the set of dominant weights which are non-
negative integral linear combinations of the fundamental weights $\varpi_{k}$
with $\alpha_{k}\in{\bm{\Delta}}\setminus S$. The quantized coordinate ring
$\mathcal{O}[Y_{S,q}]$ of the torus bundle $Y_{S}\to X_{S}$ is generated as an
algebra by the matrix coefficients

$\left\\{\mathsf{f}^{\lambda}_{i},\mathsf{v}^{\lambda}_{i}\mid\lambda\in\mathbf{P}^{+}_{S}\right\\}.$

This algebra is stable under the gauge action of $T$ defined in Section 3.3,
and the gauge-invariant subalgebra is $\mathcal{O}[X_{S,q}]$.

###### Proof.

In fact, Stokman [Sto03, Theorem 4.1] proves that $\mathcal{O}[Y_{S,q}]$ is
generated by $c^{V(\lambda)}_{f,v_{1}^{\lambda}}$ and
$c^{V(\lambda)^{*}}_{\tilde{v},f^{1}_{\lambda}}$ with $\lambda=\varpi_{k}$ for
$\alpha_{k}\notin S$, so our first claim is in fact weaker. The second claim
follows immediately upon observing that $\mathcal{O}[X_{S,q}]$ consists of
those elements of $\mathcal{O}[Y_{S,q}]$ which are invariant under the action
of $\mathcal{U}_{q}(\mathfrak{h})$. ∎

######  Remark 3.7.

The algebra $\mathcal{O}[Y_{S,q}]$ is already invariant under the action of
$K_{i}$ for $\alpha_{i}\in S$, so the gauge action in Theorem 3.6 is
determined entirely by the action of the generators $K_{i}$ with
$\alpha_{i}\notin S$. Thus the relevant gauge action on $\mathcal{O}[Y_{S,q}]$
reduces to an action of $\mathbb{T}^{N}$ with $N=r-|S|$.

Next we consider the case of a connected but not simply connected compact
semisimple Lie group $K$. Let $\mathfrak{k}$ be the Lie algebra of $K$. Let
$\widetilde{K}$ be the universal cover of $K$, and $\mathbf{Q}$, $\mathbf{P}$
be the root and weight lattices of $\mathfrak{k}$. Then there is a lattice
$\mathbf{P}_{K}\subset\mathfrak{h}^{*}$ with
$\mathbf{Q}\subseteq\mathbf{P}_{K}\subseteq\mathbf{P}$ such that the
irreducible representations of $K$ are precisely those of the simply connected
group $\widetilde{K}$ with highest weights in
$\mathbf{P}_{K}^{+}=\mathbf{P}_{K}\cap\mathbf{P}^{+}$. As a consequence we
have

$\mathcal{O}_{q}[K]=\operatorname{span}\left\\{\left.c^{V(\lambda)}_{f,v}\right|\lambda\in\mathbf{P}_{K}^{+},~{}f\in
V(\lambda)^{*},~{}v\in
V(\lambda)\right\\}\quad\subseteq\mathcal{O}_{q}[\widetilde{K}],$ (3.5)

as a $*$-subalgebra.

We can then define generalized flag varieties $X=K/K_{S}$ and their torus
bundles $Y=K/K^{0}_{S}$ for the non simply connected group $K$. Combining the
above with Stokman’s result, Theorem 3.6, we get the following.

######  Theorem 3.8.

Let $K$ be a compact semisimple Lie group, not necessarily simply connected.
The quantized coordinate ring $\mathcal{O}[Y_{S,q}]$ of the torus bundle
$Y_{S}\to X_{S}$ is generated by the matrix coefficients

$\left\\{\mathsf{f}^{\lambda}_{i},\mathsf{v}^{\lambda}_{i}\mid\lambda\in\mathbf{P}^{+}_{K,S}\right\\}.$

where $\mathbf{P}_{K,S}^{+}:=\mathbf{P}_{K}^{+}\cap\mathbf{P}_{S}^{+}$. The
gauge-invariant subalgebra of $\mathcal{O}[Y_{S,q}]$ is
$\mathcal{O}[X_{S,q}]$.

##  4\. The analytic limit of the quantized coordinate ring

###  4.1. The analytic limit for $SU_{q}(2)$

We identify the integral weights of $\mathfrak{sl}(2)$ with the integers,
$\mathbf{P}=\mathbb{Z}$, so that $V(m)$ is the irreducible integral
representation of dimension $m+1$. Let444Unfortunately, this notation clashes
with the common convention of using $v^{\lambda}_{1}$ for the highest weight
vector in $V(\lambda)$, which we have followed elsewhere in this article when
$K\neq\mathrm{SU}(2)$. $v^{m}_{0}$ denote a highest weight vector of $V(m)$,
and put $v^{m}_{k}=F^{(k)}v^{m}_{0}$. The ${\mathbf{A}_{0}}$-span of these
vectors is a crystal lattice $\mathcal{L}(m)$ and their images
$b^{m}_{k}\in\mathcal{L}(m)/q\mathcal{L}(m)$ define a crystal basis with
crystal $\mathcal{B}(m)=\\{b^{m}_{k}\mid k=0,\cdots,m\\}$.

Let $\\{f_{m}^{k}\mid k=0,\cdots,m\\}$ denote the basis of $V(m)^{*}$ dual to
$\\{v^{m}_{k}\mid k=0,\cdots,m\\}$. We write
$u^{m}_{ij}=c^{V(m)}_{f^{i},v_{j}}$ for the associated matrix coefficients.
For the matrix coefficients of the fundamental representation we will use the
operator algebraists’ notation from [Wor87b], that is

$\displaystyle\alpha$ $\displaystyle=u^{1}_{00},$ $\displaystyle-q\gamma^{*}$
$\displaystyle=u^{1}_{01},$ $\displaystyle\gamma$ $\displaystyle=u^{1}_{10},$
$\displaystyle\alpha^{*}$ $\displaystyle=u^{1}_{11}.$

Note that the basis $\\{v^{m}_{k}\\}$ for $V(m)$ is generally only orthogonal,
but in the case $m=1$ the basis $\\{v^{1}_{0},v^{1}_{1}\\}$ is also
orthonormal, so we do indeed recover Woronowicz’s generators. They satisfy the
following relations

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

######  Definition 4.1 (Woronowicz [Wor87b], Vaksman-Soibelman [VS88]).

Fix $q\in(0,\infty)\setminus\\{1\\}$. The _standard representation_ of the
quantized coordinate ring $\mathcal{O}[\mathrm{SU}_{q}(2)]$ is the
$*$-representation $\tilde{\pi}_{q}$ on
$\ell^{2}(\mathbb{N})=\operatorname{span}\\{e_{0},e_{1},\cdots\\}$ determined
by

$\displaystyle\tilde{\pi}_{q}(\alpha)e_{n}$
$\displaystyle=\sqrt{1-q^{2n}}e_{n-1},$
$\displaystyle\tilde{\pi}_{q}(\gamma)e_{n}$ $\displaystyle=q^{n}e_{n},$
$\displaystyle(n\geq 0).$

We will use the same notation $\tilde{\pi}_{q}$ for the partially defined
representation of the algebraists’ quantized coordinate ring
$\tilde{\pi}_{q}\circ\operatorname{ev}_{q}:\mathcal{O}_{q}[\mathrm{SU}(2)]\to\mathcal{B}(\ell^{2}(\mathbb{N}))$.

If $V$ is any finite-dimensional integrable
$\mathcal{U}_{q}(\mathfrak{g})$-module over the ground field $\mathbb{C}(q)$
then any $v\in V$ can be specialized at all but finitely many
$q\in(0,\infty)$, and likewise for any $f\in V^{*}$. Therefore, for any given
element $u\in\mathcal{O}_{q}[K]$, the representation $\tilde{\pi}_{q}(u)$ is
well-defined for all sufficiently small $q>0$, so it makes sense to ask about
the existence of the limit of $\tilde{\pi}_{q}(u)$ as $q\to 0$. The following
result is essentially known, compare [Wor87b, VS88, HS02].

######  Theorem 4.2.

For any $u\in\mathcal{O}_{q}^{\mathbf{A}_{0}}[\mathrm{SL}(2)]$, the one-
parameter family of operators
$\tilde{\pi}_{q}(u)\in\mathcal{B}(\ell^{2}\mathbb{N})$ admits a norm limit as
$q\to 0$, which we denote by $\tilde{\pi}_{0}(u)$. Explicitly, if
$m\in\mathbb{N}$ and $i,j\in\\{0,\cdots,m\\}$, then

$\displaystyle\tilde{\pi}_{0}(u^{m}_{ij})=\begin{cases}T^{j}P_{0}T^{*m-i},&\text{if
}i>j\\\ T^{i}T^{*m-i},&\text{if }i=j\\\ 0,&\text{if }i<j\end{cases}$

where $T$ is the right-shift operator and $P_{0}$ is the orthogonal projection
onto $e_{0}$.

###### Proof.

For $m=0$, we have $\tilde{\pi}_{0}(u^{0}_{00})=1$, and for $m=1$, the
formulas in Definition 4.1 give the desired limits as $q\to 0$:

$\displaystyle\tilde{\pi}_{0}(u^{1}_{00})$ $\displaystyle=T^{*},$
$\displaystyle\tilde{\pi}_{0}(u^{1}_{01})$ $\displaystyle=0,$
$\displaystyle\tilde{\pi}_{0}(u^{1}_{10})$ $\displaystyle=P_{0},$
$\displaystyle\tilde{\pi}_{0}(u^{1}_{11})$ $\displaystyle=T.$

Since the fundamental matrix coefficients generate
$\mathcal{O}_{q}^{\mathbf{A}_{0}}[\mathrm{SL}(2)]$ as an
$\mathbf{A}_{0}$-algebra, the existence of the limit for all elements of
$\mathcal{O}_{q}^{{\mathbf{A}_{0}}}[\mathrm{SL}(2)]$ follows immediately.

It remains to check the stated formula when $m>1$. There is a unique inclusion
of $\mathcal{U}_{q}(\mathfrak{g})$-modules $\iota_{m}:V(m)\hookrightarrow
V(1)^{\otimes m}$ which sends the highest weight vector $v^{m}_{0}$ to
$v^{1}_{0}\otimes\cdots\otimes v^{1}_{0}$. Equation (2.5) shows that the
Kashiwara operator $\tilde{F}$ acts on $\mathcal{B}(1)^{\otimes m}$ by

$\tilde{F}:(b^{1}_{1})^{\otimes
k}\otimes(b^{1}_{0})^{\otimes(m-k)}\mapsto(b^{1}_{1})^{\otimes(k+1)}\otimes(b^{1}_{0})^{\otimes(m-k-1)},$

for $k=0,\cdots,m-1$, so it follows that

$\iota_{m}(v^{m}_{k})=(v^{1}_{1})^{\otimes
k}\otimes(v^{1}_{0})^{\otimes(m-k)}\mod q\mathcal{L}(1)^{\otimes m}.$

Note that for every $i,j\in\\{0,\ldots,m\\}$ and every $a,b\in\mathbb{N}$ we
have

$\big{(}(f^{1}_{1})^{\otimes i}\otimes(f^{1}_{0})^{\otimes
m-i},\tilde{E}^{a}\tilde{F}^{b}(v^{1}_{1})^{\otimes
j}\otimes(v^{1}_{0})^{\otimes(m-j)}\big{)}=\big{(}f^{i}_{m},\tilde{E}^{a}\tilde{F}^{b}v^{m}_{j})\mod
q{\mathbf{A}_{0}},$

which implies that

$\displaystyle u^{m}_{ij}$
$\displaystyle=\begin{cases}(u^{1}_{11})^{j}(u^{1}_{10})^{i-j}(u^{1}_{00})^{m-i},&\text{if
}i>j\\\ (u^{1}_{11})^{i}(u^{1}_{00})^{m-i},&\text{if }i=j\\\
(u^{1}_{11})^{i}(u^{1}_{01})^{j-i}(u^{1}_{00})^{m-j},&\text{if
}i<j\end{cases}\quad\mod q\mathcal{O}_{q}^{\mathbf{A}_{0}}[\mathrm{SL}(2)].$

Now we can use the result for $m=1$ to deduce the general formula. ∎

This result immediately extends to our compact ${\mathbf{A}_{0}}$-form
$\mathcal{O}_{q}^{\mathbf{A}_{0}}[\mathrm{SU}(2)]$ from Definition 3.2, as
follows.

######  Corollary 4.3.

For any $u\in\mathcal{O}_{q}^{\mathbf{A}_{0}}[\mathrm{SU}(2)]$, the one-
parameter family of operators
$\tilde{\pi}_{q}(u)\in\mathcal{B}(\ell^{2}\mathbb{N})$ admits a norm limit as
$q\to 0$, which we denote by $\tilde{\pi}_{0}(u)$.

###### Proof.

By Theorem 4.2, for every generator $\mathsf{f}^{\lambda}_{i}$ of
$\mathcal{O}_{q}^{\mathbf{A}_{0}}[\mathrm{SL}(2)/N^{+}]$, the operators
$\tilde{\pi}_{q}(\mathsf{v}^{\lambda}_{i})$ admit a norm-limit as $q\to 0$.
Since the $\tilde{\pi}_{q}$ are $*$-representations, Proposition 3.3 shows
that the same is true of the generators $\mathsf{v}^{\lambda}_{i}$ of
$\mathcal{O}_{q}^{\mathbf{A}_{0}}[\mathrm{SL}(2)/N^{-}]$. The result follows.
∎

Note, for instance, that the matrix coefficient
$\gamma^{*}=(u^{1}_{10})^{*}=-q^{-1}u^{1}_{01}$ does not belong to the
${\mathbf{A}_{0}}$-form
$\widetilde{\mathcal{F}}=\mathcal{O}_{q}^{\mathbf{A}_{0}}[\mathrm{SL}(2)]$,
but nonetheless it does admit an analytic limit at $q=0$, namely $\lim_{q\to
0}\tilde{\pi}_{q}(\gamma^{*})=P_{0}$.

###  4.2. The analytic limit in higher rank

Next, we generalize the previous result to higher rank. Let $K$ be a
connected, simply connected compact semisimple Lie group. Let $W$ denote its
Weyl group.

For every $\alpha_{i}\in{\bm{\Delta}}$ there is a Hopf $*$-morphism
$\iota_{i}:\mathcal{U}_{q_{i}}(\mathfrak{su}(2))\hookrightarrow\mathcal{U}_{q}(\mathfrak{k})$
sending $E$ to $E_{i}$. The dual map
$\operatorname{Res}_{S_{i}}^{K_{q}}:=\iota_{i}^{*}:\mathcal{O}_{q}[K]\twoheadrightarrow\mathcal{O}_{q_{i}}[\mathrm{SU}(2)]$
is called the _restriction to the quantum subgroup
$S_{i}\cong\mathrm{SU}_{q_{i}}(2)$_ associated to the simple root
$\alpha_{i}$. It specializes at $q\in(0,\infty)\setminus\\{1\\}$ to a
$*$-morphism
$\operatorname{Res}_{S_{i}}^{K_{q}}:\mathcal{O}[K_{q}]\twoheadrightarrow\mathcal{O}[\mathrm{SU}_{q_{i}}(2)]$.
This allows us to define a $*$-representation for each
$\alpha_{i}\in{\bm{\Delta}}$ and each $q\in(0,\infty)\setminus\\{1\\}$ by

$\tilde{\pi}_{q,i}:=\tilde{\pi}_{q}\circ\operatorname{Res}_{S_{i}}^{K_{q}}:\mathcal{O}[K_{q}]\to\mathcal{B}(\ell^{2}(\mathbb{N})).$
(4.1)

As previously, we will use the same notation for the partially defined
representation of the $\mathbb{C}(q)$-algebra $\mathcal{O}_{q}[K]$ obtained by
first specializing at $q$.

We also have a restriction morphism to the maximal torus subgroup
$\operatorname{res}^{K_{q}}_{T}:\mathcal{O}[K_{q}]\to\mathcal{O}[T]$ for each
$q\in(0,\infty)\setminus\\{1\\}$. Explicitly,

$\operatorname{res}^{K_{q}}_{T}:c^{V}_{f^{i},v_{j}}\mapsto\delta_{ij}c^{V}_{f^{i},v_{i}},$

where $\\{v_{i}\\}$ is a basis of weight vectors, $\\{f^{i}\\}$ a dual basis,
and the right-hand side is understood as a matrix coefficient for $T\subset
K$. Composing this with the multiplication representation
$\mathsf{M}:\mathcal{O}[T]\to\mathcal{B}(L^{2}(T))$ yields the
$*$-representation

$\chi=\mathsf{M}\circ\operatorname{res}^{K_{q}}_{T}:\mathcal{O}[K_{q}]\to\mathcal{B}(L^{2}(T)),$
(4.2)

as well as its partially defined analogue on $\mathcal{O}_{q}[K]$.

######  Definition 4.4 (Soibelman [Soĭ91]).

Fix a decomposition $w_{0}=s_{i_{1}}\cdots s_{i_{l}}$ of the longest word of
$W$. We put $\mathsf{H}=\ell^{2}(\mathbb{N})^{\otimes l}\otimes L^{2}(T)$. For
$q\in(0,\infty)\setminus\\{1\\}$, we define the _Soibelman representation_ to
be

$\displaystyle\pi_{q}:=(\tilde{\pi}_{i_{1},q}\otimes\tilde{\pi}_{i_{2},q}\otimes\cdots\otimes\tilde{\pi}_{i_{l},q}\otimes\chi)\circ\Delta^{(l)}:\mathcal{O}[K_{q}]$
$\displaystyle\to\mathcal{B}(\mathsf{H}).$

Again, we use the same notation for the partially defined representation of
$\mathcal{O}_{q}[K]$.

######  Remark 4.5.

Soibelman [Soĭ91] defined a family of irreducible $*$-representations indexed
by the symplectic leaves of $K$. The representation in Definition 4.4
corresponds to the direct integral over all symplectic leaves of maximal
dimension. It is faithful but not irreducible. One can obtain an irreducible
representation by replacing the multiplication representation of the torus
$\chi$ by any character
$\chi_{t}=\operatorname{ev}_{t}\circ\operatorname{res}^{K_{q}}_{T}$ where
$\operatorname{ev}_{t}$ denotes evaluation at a point $t\in T$. For a nice
account of Soibelman’s work, and generalizations, see [NT12].

######  Theorem 4.6.

For any $u\in\mathcal{O}_{q}^{\mathbf{A}_{0}}[G]$, the one-parameter family of
operators $\pi_{q}(u)$ admits a norm-limit as $q\to 0$, which we denote by
$\pi_{0}(u)$.

###### Proof.

Firstly, observe that
$\Delta:\mathcal{O}_{q}^{\mathbf{A}_{0}}[G]\to\mathcal{O}_{q}^{\mathbf{A}_{0}}[G]\otimes\mathcal{O}_{q}^{\mathbf{A}_{0}}[G]$.
Next, note that a crystal lattice $\mathcal{L}$ for a
$\mathcal{U}_{q}(\mathfrak{k})$-module $V$ is also a
$\mathcal{U}_{q}(\mathfrak{su}_{2})$-crystal lattice for the restriction of
the module $V$ to $S_{i}$ for every $i=1,\cdots,r$. Therefore the maps
$\operatorname{res}^{K_{q}}_{S_{i}}:\mathcal{O}_{q}[K]\to\mathcal{O}_{q_{i}}[\mathrm{SU}(2)]$
restrict to morphisms
$\mathcal{O}_{q}^{\mathbf{A}_{0}}[G]\to\mathcal{O}_{q_{i}}^{\mathbf{A}_{0}}[\mathrm{SL}(2)]$.
The result now follows from Theorem 4.2 and the definition of the Soibelman
representation. ∎

By the same argument as in Corollary 4.3, we obtain limits for our compact
${\mathbf{A}_{0}}$-form $\mathcal{O}_{q}^{\mathbf{A}_{0}}[K]$.

######  Corollary 4.7.

For any $u\in\mathcal{O}_{q}^{\mathbf{A}_{0}}[K]$, the one-parameter family of
operators $\pi_{q}(u)$ admits a norm-limit as $q\to 0$, which we denote by
$\pi_{0}(u)$.

######  Definition 4.8.

We put $\mathcal{O}[K_{0}]:=\pi_{0}(\mathcal{O}_{q}^{\mathbf{A}_{0}}[K])$ and
refer to this $*$-algebra as the crystal limit of $\mathcal{O}[K_{q}]$.

We make a similar definition for the $q=0$ limit of the quantized coordinate
rings of flag varieties, following Theorem 3.8. Note that $\mathcal{O}[K_{0}]$
inherits a gauge action of the torus $T\cong\mathbb{T}^{r}$ from that of
$\mathcal{O}_{q}[K]$.

######  Definition 4.9.

With notation as in Section 3.4, let $K$ be a compact connected semisimple Lie
group, not necessarily simply connected, let $S\subset{\bm{\Delta}}$ be a set
of simple roots, and let $Y_{S}$ be the corresponding principal torus bundle
over the flag variety $X_{S}$. We define $\mathcal{O}[Y_{S,0}]$ to be the
subalgebra of $\mathcal{O}[K_{0}]$ generated by the elements
$\tilde{\pi}_{0}(\mathsf{f}^{\lambda}_{i})$ and
$\tilde{\pi}_{0}(\mathsf{v}^{\lambda}_{i})$ with
$\lambda\in\mathbf{P}^{+}_{K,S}$. The gauge-invariant subalgebra is denoted by
$\mathcal{O}[X_{S,0}]$.

The remainder of the paper will be dedicated to describing the structure of
the crystal limit $\mathcal{O}[K_{0}]$, as well as the subalgebras
$\mathcal{O}[Y_{S,0}]\subset\mathcal{O}[K_{0}]$ as above.

##  5\. The Cartan braiding and the crystal limit

###  5.1. Braiding and commutation relations

At this point, we will be obliged to enlarge our base field slightly in order
to work with expressions of the form $q^{(\lambda,\lambda^{\prime})}$ where
$\lambda,\lambda^{\prime}\in\mathbf{P}$ are any integral weights. Let
$L\in\mathbb{N}$ be the smallest positive integer such that
$(\mathbf{P},\mathbf{P})\subseteq\frac{1}{L}\mathbb{Z}$. Then we work over
$\mathbb{C}(s)$ and put $q=s^{L}$. Likewise we redefine the rings
${\mathbf{A}_{0}}$, $\mathbf{A}_{\infty}$ and $\mathbf{A}$ with the parameter
$s$ in place of $q$. This has essentially no effect on the crystal theory.
Rather than rewrite expressions in terms of $s$, we will write $q^{k}$ where
$k\in\frac{1}{L}\mathbb{Z}$.

We denote by $\hat{R}_{V,W}:V\otimes W\to W\otimes V$ the braiding
corresponding to the finite-dimensional integrable
$\mathcal{U}_{q}(\mathfrak{g})$-modules $V$ and $W$. It is a
$\mathcal{U}_{q}(\mathfrak{g})$-module isomorphism which generalizes the
classical flip map. The braiding is not quite unique; the choice we make can
be characterized uniquely by

$\begin{split}\hat{R}_{V,W}(v\otimes
w)&=q^{-(\operatorname{wt}(v),\operatorname{wt}(w))}w\otimes
v+\sum_{i}w_{i}\otimes v_{i},\\\ &\text{ where
$\operatorname{wt}(w_{i})<\operatorname{wt}(w)$ and
$\operatorname{wt}(v_{i})>\operatorname{wt}(v)$ for every $i$. }\end{split}$
(5.1)

Here, $\operatorname{wt}$ denotes the weight of a weight vector. In
particular, in the case $V=V(\lambda)$ and with $v_{\lambda}$ a highest weight
vector, we get $\hat{R}_{V,V}(v_{\lambda}\otimes
v_{\lambda})=q^{-(\lambda,\lambda)}v_{\lambda}\otimes v_{\lambda}$.

The braiding can be used to give commutation relations for the elements of
$\mathcal{O}_{q}[G]$. Let $V$ and $W$ be
$\mathcal{U}_{q}(\mathfrak{g})$-modules with bases $\\{v_{i}\\}_{i}$ and
$\\{w_{i}\\}_{i}$ and denote by $\\{f^{i}\\}_{i}$ and $\\{g^{i}\\}_{i}$ the
dual bases. Let us introduce the coefficients of $\hat{R}_{V,W}$ by

$\hat{R}_{V,W}(v_{i}\otimes
w_{j})=\sum_{k,l}(\hat{R}_{V,W})_{ij}^{kl}w_{k}\otimes v_{l}.$

Then, using the fact that the braiding satisfies
$\hat{R}_{V,W}\Delta(X)=\Delta(X)\hat{R}_{V,W}$ for any
$X\in\mathcal{U}_{q}(\mathfrak{g})$, it is easy to derive the relations

$\begin{split}c^{V}_{f^{i},v_{k}}c^{W}_{g^{j},w_{l}}&=\sum_{a,b,c,d}(\hat{R}_{V,W}^{-1})^{ij}_{ab}(\hat{R}_{V,W})^{cd}_{kl}c^{W}_{g^{a},w_{c}}c^{V}_{f^{b},v_{d}}\\\
&=\sum_{a,b,c,d}(\hat{R}_{W,V})^{ij}_{ab}(\hat{R}_{W,V}^{-1})^{cd}_{kl}c^{W}_{g^{a},w_{c}}c^{V}_{f^{b},v_{d}}.\end{split}$
(5.2)

###  5.2. The Cartan braiding

Before discussing the crystal limit of the braiding operators, let us make
some observations about tensor products of irreducible crystals.

######  Definition 5.1.

A finite-dimensional integrable $\mathcal{U}_{q}(\mathfrak{g})$-module $V$
will be called a _product of irreducibles_ if it is of the form
$V(\lambda_{1})\otimes\cdots\otimes V(\lambda_{n})$ for some
$\lambda_{1},\cdots,\lambda_{n}\in\mathbf{P}^{+}$. Such a module contains a
unique irreducible submodule of highest weight $\lambda=\sum_{i}\lambda_{i}$,
which we call the _Cartan component_. We refer to $\lambda$ as the _highest
weight_ of $V$ (although strictly speaking it is the largest among the highest
weights of all irreducible submodules of $V$).

The same terminology will be used to refer to the associated _Cartan
component_ of a product of irreducible crystals. If
$\mathcal{B}=\mathcal{B}(\lambda_{1})\otimes\cdots\otimes\mathcal{B}(\lambda_{n})$
is a product of irreducible crystals, we denote by
$\eta:\mathcal{B}\sqcup\\{0\\}\to\\{0,1\\}$ the indicator function of its
Cartan component:

$\eta(b)=\begin{cases}1&\text{if $b$ is in the Cartan component},\\\
0&\text{otherwise}.\end{cases}$

The following well-known property of the Cartan component will occasionally be
useful (we include the easy proof for completeness).

######  Lemma 5.2.

Let $b_{\lambda}$ and $b_{w_{0}\lambda}$ denote the highest and lowest weight
elements, respectively, of the irreducible crystal $\mathcal{B}(\lambda)$.
Then the tensor $c\otimes b_{\lambda^{\prime}}$ belongs to the Cartan
component of $\mathcal{B}(\lambda)\otimes\mathcal{B}(\lambda^{\prime})$ for
every $c\in\mathcal{B}(\lambda)$. Similarly, $b_{w_{0}\lambda}\otimes
c^{\prime}$ belongs to the Cartan component for every
$c^{\prime}\in\mathcal{B}(\lambda^{\prime})$.

###### Proof.

The first statement follows by repeatedly applying the Kashiwara operators
$\tilde{F}_{i}$ to the highest weight element $b_{\lambda}\otimes
b_{\lambda^{\prime}}$ and using the tensor product formula (2.5). Similarly,
the second follows by applying the Kashiwara operators $\tilde{E}_{i}$ to the
lowest weight element $b_{w_{0}\lambda}\otimes b_{w_{0}\lambda^{\prime}}$. ∎

We now return to the braiding. Consider two simple modules $V(\lambda)$ and
$V(\lambda^{\prime})$. We have crystal lattices of $V(\lambda)\otimes
V(\lambda^{\prime})$ and $V(\lambda^{\prime})\otimes V(\lambda)$ given by

$\mathcal{L}_{V(\lambda)\otimes
V(\lambda^{\prime})}:=\mathcal{L}(\lambda)\otimes_{{\mathbf{A}_{0}}}\mathcal{L}(\lambda^{\prime}),\quad\mathcal{L}_{V(\lambda^{\prime})\otimes
V(\lambda)}:=\mathcal{L}(\lambda^{\prime})\otimes_{{\mathbf{A}_{0}}}\mathcal{L}(\lambda).$

The braiding $\hat{R}_{V(\lambda),V(\lambda^{\prime})}$ does not typically map
$\mathcal{L}_{V(\lambda)\otimes V(\lambda^{\prime})}$ into
$\mathcal{L}_{V(\lambda^{\prime})\otimes V(\lambda)}$ because of the negative
powers of $q$ in (5.1). This deficiency can be remedied by multiplying by an
appropriate power of $q$, as we show in the next theorem.

######  Theorem 5.3.

Given any simple modules $V(\lambda)$ and $V(\lambda^{\prime})$ we have

$q^{(\lambda,\lambda^{\prime})}\hat{R}_{V(\lambda),V(\lambda^{\prime})}(\mathcal{L}_{V(\lambda)\otimes
V(\lambda^{\prime})})\subseteq\mathcal{L}_{V(\lambda^{\prime})\otimes
V(\lambda)}.$

Moreover, the map
$q^{(\lambda,\lambda^{\prime})}\hat{R}_{V(\lambda),V(\lambda^{\prime})}$
induces a morphism of crystals

$\sigma_{\mathcal{B}(\lambda),\mathcal{B}(\lambda^{\prime})}:\mathcal{B}(\lambda)\otimes\mathcal{B}(\lambda^{\prime})\to\mathcal{B}(\lambda^{\prime})\otimes\mathcal{B}(\lambda).$

It is an isomorphism between the Cartan components of the two crystals, and is
zero on all other components.

###### Proof.

It is a consequence of the tensor product rule (2.4) for $\tilde{E}_{i}$ that
every highest weight element of the crystal
$\mathcal{B}(\lambda)\otimes\mathcal{B}(\lambda^{\prime})$ is of the form
$b_{\lambda}\otimes b^{\prime}$, where $b_{\lambda}$ is the highest weight
element of $\mathcal{B}(\lambda)$, and $b^{\prime}$ is some element of
$\mathcal{B}(\lambda^{\prime})$. Let $v_{\lambda}\in\mathcal{L}(\lambda)$ and
$w\in\mathcal{L}(\lambda^{\prime})$ be lifts of $b_{\lambda}$ and
$b^{\prime}$, respectively, to weight vectors. Write $\mu$ for the weight of
$w$.

Let us write $V=V(\lambda)$ and $W=V(\lambda^{\prime})$ for convenience. By
the nature of the braiding described in (5.1), we have

$q^{(\lambda,\lambda^{\prime})}\hat{R}_{V,W}(v_{\lambda}\otimes
w)=q^{(\lambda,\lambda^{\prime})-(\lambda,\mu)}w\otimes v_{\lambda}.$ (5.3)

Since $\lambda\in\mathbf{P}^{+}$ and
$\lambda^{\prime}-\mu\in\mathbf{Q}^{+}=\mathbb{N}\cdot{\bm{\Delta}}$, we have
$(\lambda,\lambda^{\prime}-\mu)\geq 0$. Therefore, (5.3) is in
$\mathcal{L}_{W\otimes V}$, and moreover is in $q\mathcal{L}_{W\otimes V}$
unless $w$ is the highest weight vector of $W$.

The images of the various highest weight elements $v_{\lambda}\otimes w$
above, under repeated action of the Kashiwara operators $\tilde{F}_{i}$,
generate the crystal lattice $\mathcal{L}_{V,W}$ as an
${\mathbf{A}_{0}}$-module. Since $\hat{R}_{V,W}$ is a morphism of
$\mathcal{U}_{q}(\mathfrak{g})$-modules, we get

$q^{(\lambda,\lambda^{\prime})}\hat{R}_{V,W}(\tilde{F}(v_{\lambda}\otimes
w))=q^{(\lambda,\lambda^{\prime})-(\lambda,\mu)}\tilde{F}(w\otimes
v_{\lambda})\in\mathcal{L}_{W\otimes V}$

for any word $\tilde{F}=\tilde{F}_{i_{1}}\cdots\tilde{F}_{i_{n}}$ in the
Kashiwara operators $\tilde{F}_{i}$. This proves that the map
$q^{(\lambda,\lambda^{\prime})}\hat{R}_{V,W}$ respects the crystal lattices
and induces a map of crystals.

For the final claim, note that crystal element $b^{\prime}\otimes b_{\lambda}$
associated to the right hand side of (5.3) always belongs to the Cartan
component by Lemma 5.2. Since the left-hand side is a highest-weight element
by assumption, it belongs to the Cartan component only if $w$ is of weight
$\lambda^{\prime}$. It follows that the crystal morphism
$\sigma_{\mathcal{B}(\lambda),\mathcal{B}(\lambda^{\prime})}$ must vanish on
all non-Cartan components. On the Cartan component, (5.3) gives
$q^{(\lambda,\lambda^{\prime})}\hat{R}_{V,W}(v_{\lambda}\otimes
v_{\lambda^{\prime}})=v_{\lambda^{\prime}}\otimes v_{\lambda}$, from which we
conclude that $\sigma_{\mathcal{B}(\lambda),\mathcal{B}(\lambda^{\prime})}$ is
an isomorphism here. ∎

######  Definition 5.4.

Let $V$ and $V^{\prime}$ be products of irreducible modules with highest
weights $\lambda$ and $\lambda^{\prime}$, respectively, and let
$(\mathcal{L},\mathcal{B})$ and $(\mathcal{L}^{\prime},\mathcal{B}^{\prime})$
be their crystal bases. The _Cartan braiding_ is the morphism of crystals

$\sigma=\sigma_{\mathcal{B},\mathcal{B}^{\prime}}:\mathcal{B}\otimes\mathcal{B}^{\prime}\to\mathcal{B}^{\prime}\otimes\mathcal{B}$

induced by the morphism of crystal lattices
$q^{(\lambda,\lambda^{\prime})}\hat{R}_{V,V^{\prime}}:\mathcal{L}\otimes_{\mathbf{A}_{0}}\mathcal{L}^{\prime}\to\mathcal{L}^{\prime}\otimes_{\mathbf{A}_{0}}\mathcal{L}$.

The use of the word braiding here is a bit misleading. Firstly, braidings are
supposed to be isomorphisms, while our Cartan braiding is not, see Theorem
5.3. More importantly, the Cartan braiding
$\sigma_{\mathcal{B},\mathcal{B}^{\prime}}$ is only defined when $\mathcal{B}$
and $\mathcal{B}^{\prime}$ products of irreducible modules, not on general
direct sums of irreducibles. This is because we need a well-defined notion of
highest weights $\lambda$ and $\lambda^{\prime}$ for $\mathcal{B}$ and
$\mathcal{B}^{\prime}$ in order to define it, see below.

If $V$ and $V^{\prime}$ are irreducible modules, then the Cartan braiding is
the crystal morphism described in Theorem 5.3, that is, it is an isomorphism
between the Cartan components of $\mathcal{B}\otimes\mathcal{B}^{\prime}$ and
$\mathcal{B}^{\prime}\otimes\mathcal{B}$, and zero on all other irreducible
components.

If $V$ and $V^{\prime}$ are merely products of irreducibles, then we need a
few remarks to justify that the Cartan braiding is well-defined. If
$\mu,\mu^{\prime}\in\mathbf{P}^{+}$ are the highest weights of any components
of $V$ and $V^{\prime}$ respectively, then we have $\lambda-\mu$ and
$\lambda-\mu^{\prime}$ in $\mathbb{N}\cdot{\bm{\Delta}}$. Therefore

$(\lambda,\lambda^{\prime})=(\lambda-\mu,\lambda^{\prime})+(\mu,\lambda^{\prime}-\mu^{\prime})+(\mu,\mu^{\prime})\geq(\mu,\mu^{\prime}),$

so
$q^{(\lambda,\lambda^{\prime})}\hat{R}_{V(\mu),V(\mu^{\prime})}=q^{k}q^{(\mu,\mu^{\prime})}\hat{R}_{V(\mu),V(\mu^{\prime})}$
for some $k\geq 0$. Now Theorem 5.3 shows that
$q^{(\lambda,\lambda^{\prime})}\hat{R}_{V,V^{\prime}}:\mathcal{L}\otimes_{\mathbf{A}_{0}}\mathcal{L}^{\prime}\to\mathcal{L}^{\prime}\otimes_{\mathbf{A}_{0}}\mathcal{L}$
is well defined and descends to a morphism of crystals. Note though that in
this case, it is possible for $\sigma_{\mathcal{B},\mathcal{B}^{\prime}}$ to
be non-zero on components other than the Cartan component, as the following
example shows.

######  Example 5.5.

Let $\mathfrak{g}=\mathfrak{sl}_{4}$, and consider
$\mathcal{B}=\mathcal{B}(\varpi_{1})$ and
$\mathcal{B}^{\prime}=\mathcal{B}(\varpi_{2})\otimes\mathcal{B}(\varpi_{3})$,
where $\varpi_{1},\varpi_{2},\varpi_{3}$ are the fundamental weights. Then
$\mathcal{B}^{\prime}\cong\mathcal{B}(\varpi_{2}+\varpi_{3})\oplus\mathcal{B}(\varpi_{1})$.
One can calculate

$(\varpi_{1},\varpi_{2}+\varpi_{3})={\frac{3}{4}}=(\varpi_{1},\varpi_{1}).$

It follows that $\sigma_{\mathcal{B},\mathcal{B}^{\prime}}$ is non-zero on two
distinct components in the triple product
$\mathcal{B}(\varpi_{1})\otimes\mathcal{B}(\varpi_{2})\otimes\mathcal{B}(\varpi_{3})$.

On the other hand, if we consider the map
$\operatorname{id}_{\mathcal{B}(\varpi_{1})}\otimes\sigma_{\mathcal{B}(\varpi_{2}),\mathcal{B}(\varpi_{3})}$,
this will kill
$\mathcal{B}(\varpi_{1})\otimes\mathcal{B}(\varpi_{1})\subset\mathcal{B}(\varpi_{1})\otimes(\mathcal{B}(\varpi_{2})\otimes\mathcal{B}(\varpi_{3}))$.
In fact, we will show that the Cartan component of any tensor product of
irreducibles is characterized by the non-vanishing of every possible action of
the Cartan braidings. See Theorem 6.3 below for a precise statement.

The Cartan braiding is very easy to describe in one particular situation.

######  Example 5.6.

When $V=V^{\prime}=V(\lambda)$, the Cartan braiding is the “projection” onto
the component of highest weight $2\lambda$, _i.e._ ,

$\sigma_{\mathcal{B}(\lambda),\mathcal{B}(\lambda)}(b\otimes
b^{\prime})=\begin{cases}b\otimes b^{\prime}&\text{if }\eta(b\otimes
b^{\prime})=1,\\\ 0&\text{otherwise.}\end{cases}$

Throughout this paper we will consider the running example of
$K=\mathrm{SU}(3)$, with the aim of rederiving the result of Giselsson
mentioned in the introduction. We begin by determining the Cartan braiding in
this case.

######  Example 5.7.

For $\mathfrak{g}=\mathfrak{sl}_{3}$, we denote the crystals graphs of
$\mathcal{B}(\varpi_{1})$ and $\mathcal{B}(\varpi_{2})$ by

${a_{1}}$${a_{2}}$${a_{3}}$${b_{1}}$${b_{2}}$${b_{3}}$$\scriptstyle{1}$$\scriptstyle{2}$$\scriptstyle{2}$$\scriptstyle{1}$

Using the tensor product rule for the Kashiwara operators, we find that the
crystal graphs of $\mathcal{B}(\varpi_{1})\otimes\mathcal{B}(\varpi_{1})$ and
$\mathcal{B}(\varpi_{2})\otimes\mathcal{B}(\varpi_{2})$ are given by

${a_{1}\otimes a_{1}}$${a_{2}\otimes a_{1}}$${a_{3}\otimes
a_{1}}$${b_{1}\otimes b_{1}}$${b_{2}\otimes b_{1}}$${b_{3}\otimes
b_{1}}$${a_{1}\otimes a_{2}}$${a_{2}\otimes a_{2}}$${a_{3}\otimes
a_{2}}$${b_{1}\otimes b_{2}}$${b_{2}\otimes b_{2}}$${b_{3}\otimes
b_{2}}$${a_{1}\otimes a_{3}}$${a_{2}\otimes a_{3}}$${a_{3}\otimes
a_{3}}$${b_{1}\otimes b_{3}}$${b_{2}\otimes b_{3}}$${b_{3}\otimes
b_{3}}$$\scriptstyle{1}$$\scriptstyle{2}$$\scriptstyle{1}$$\scriptstyle{1}$$\scriptstyle{2}$$\scriptstyle{1}$$\scriptstyle{2}$$\scriptstyle{2}$$\scriptstyle{2}$$\scriptstyle{2}$$\scriptstyle{2}$$\scriptstyle{1}$$\scriptstyle{1}$$\scriptstyle{1}$$\scriptstyle{1}$$\scriptstyle{2}$

Similarly, for the products
$\mathcal{B}(\varpi_{1})\otimes\mathcal{B}(\varpi_{2})$ and
$\mathcal{B}(\varpi_{2})\otimes\mathcal{B}(\varpi_{1})$ we have

${a_{1}\otimes b_{1}}$${a_{2}\otimes b_{1}}$${a_{3}\otimes
b_{1}}$${b_{1}\otimes a_{1}}$${b_{2}\otimes a_{1}}$${b_{3}\otimes
a_{1}}$${a_{1}\otimes b_{2}}$${a_{2}\otimes b_{2}}$${a_{3}\otimes
b_{2}}$${b_{1}\otimes a_{2}}$${b_{2}\otimes a_{2}}$${b_{3}\otimes
a_{2}}$${a_{1}\otimes b_{3}}$${a_{2}\otimes b_{3}}$${a_{3}\otimes
b_{3}}$${b_{1}\otimes a_{3}}$${b_{2}\otimes a_{3}}$${b_{3}\otimes
a_{3}}$$\scriptstyle{1}$$\scriptstyle{2}$$\scriptstyle{2}$$\scriptstyle{2}$$\scriptstyle{2}$$\scriptstyle{1}$$\scriptstyle{1}$$\scriptstyle{1}$$\scriptstyle{1}$$\scriptstyle{1}$$\scriptstyle{1}$$\scriptstyle{2}$$\scriptstyle{2}$$\scriptstyle{2}$$\scriptstyle{2}$$\scriptstyle{1}$

Using Theorem 5.3, we deduce from the latter two diagrams that the Cartan
braiding $\sigma=\sigma_{\mathcal{B}(\varpi_{1}),\mathcal{B}(\varpi_{2})}$ is
given by

$\begin{array}[]{lll}\sigma(a_{1}\otimes b_{1})=b_{1}\otimes
a_{1},&\sigma(a_{2}\otimes b_{1})=b_{1}\otimes a_{2},&\sigma(a_{3}\otimes
b_{1})=b_{2}\otimes a_{2},\\\ \sigma(a_{1}\otimes b_{2})=b_{2}\otimes
a_{1},&\sigma(a_{2}\otimes b_{2})=b_{3}\otimes a_{1},&\sigma(a_{3}\otimes
b_{2})=b_{2}\otimes a_{3},\\\ \sigma(a_{1}\otimes
b_{3})=0,&\sigma(a_{2}\otimes b_{3})=b_{3}\otimes a_{2},&\sigma(a_{3}\otimes
b_{3})=b_{3}\otimes a_{3}.\end{array}$ (5.4)

Note that this is different from the flip map, even on the Cartan component.

###  5.3. Crystal limit of the commutation relations

As previously, we fix a weight basis
$\\{v^{\lambda}_{i}\\}_{i}\subset\mathcal{L}(\lambda)$ lifting the crystal
basis $\\{b^{\lambda}_{i}\\}_{i}$ of $\mathcal{B}(\lambda)$, and let
$\\{f^{i}_{\lambda}\\}_{i}\subset V(\lambda)^{*}$ be the dual basis. We will
usually suppress the highest weight $\lambda$ in the notation and write
$v_{i}$ and $f^{i}$. We make the convention that $v^{\lambda}_{1}$ is the
highest weight vector, and hence $f_{\lambda}^{1}$ is the lowest weight
vector. We also denote by $\\{\tilde{v}^{\lambda}_{i}\\}_{i}\subset
V(\lambda)^{**}$ the dual basis to $\\{f_{\lambda}^{i}\\}_{i}$.

We recall the notation for the generators of
$\mathcal{O}_{q}^{\mathbf{A}_{0}}[K]$ from Section 3.2,

$\mathsf{f}^{\lambda}_{i}:=c^{V(\lambda)}_{f^{i},v_{1}},\quad\mathsf{v}^{\lambda}_{i}:=c^{V(\lambda)^{*}}_{\tilde{v}_{i},f^{1}}.$
(5.5)

######  Proposition 5.8.

Let $\lambda,\lambda^{\prime}\in\mathbf{P}^{+}$ and let
$\phi_{\lambda,\lambda^{\prime}}:\mathcal{B}(\lambda)\otimes\mathcal{B}(\lambda^{\prime})\to\mathcal{B}(\lambda+\lambda^{\prime})$
be the unique non-trivial crystal morphism. If $b^{\lambda}_{i}\otimes
b^{\lambda^{\prime}}_{j}$ is in the Cartan component, with
$\phi_{\lambda,\lambda^{\prime}}(b^{\lambda}_{i}\otimes
b^{\lambda^{\prime}}_{j})=b^{\lambda+\lambda^{\prime}}_{m}$, then we have

$\displaystyle\mathsf{f}^{\lambda}_{i}\mathsf{f}^{\lambda^{\prime}}_{j}$
$\displaystyle\equiv\mathsf{f}^{\lambda+\lambda^{\prime}}_{m}\mod{q\mathcal{O}_{q}^{\mathbf{A}_{0}}[K]},$
(5.6) $\displaystyle\mathsf{v}^{\lambda^{\prime}}_{j}\mathsf{v}^{\lambda}_{i}$
$\displaystyle\equiv\mathsf{v}^{\lambda+\lambda^{\prime}}_{m}\mod{q\mathcal{O}_{q}^{\mathbf{A}_{0}}[K]}.$
(5.7)

Otherwise $\mathsf{f}^{\lambda}_{i}\mathsf{f}^{\lambda^{\prime}}_{j}\equiv 0$
and $\mathsf{v}^{\lambda^{\prime}}_{j}\mathsf{v}^{\lambda}_{i}\equiv
0\pmod{q\mathcal{O}_{q}^{\mathbf{A}_{0}}[K]}$.

###### Proof.

We have
$\mathsf{f}^{\lambda}_{i}\mathsf{f}^{\lambda^{\prime}}_{j}=c^{V(\lambda)\otimes
V(\lambda^{\prime})}_{f^{i}_{\lambda}\otimes
f^{j}_{\lambda^{\prime}},v^{\lambda}_{1}\otimes v^{\lambda^{\prime}}_{1}}$.
Let us write
$\mathcal{B}(\lambda)\otimes\mathcal{B}(\lambda^{\prime})=\bigsqcup_{n}\mathcal{B}_{n}$
for the decomposition of the tensor product crystal into irreducible
components, with $\mathcal{B}_{1}$ being the Cartan component. Let $V_{n}$ and
$\mathcal{L}_{n}$ be the corresponding irreducible components of
$V(\lambda)\otimes V(\lambda^{\prime})$ and
$\mathcal{L}(\lambda)\otimes_{\mathbf{A}_{0}}\mathcal{L}(\lambda^{\prime})$.

The vector $v^{\lambda}_{1}\otimes v^{\lambda^{\prime}}_{1}$ is the highest
weight vector for the Cartan component $\mathcal{B}_{1}$. If
$b^{\lambda}_{i}\otimes b^{\lambda^{\prime}}_{j}$ is contained in some
component $\mathcal{B}_{n}$ other than the Cartan component, then
$v^{\lambda}_{i}\otimes
v^{\lambda^{\prime}}_{j}\in\mathcal{L}_{n}+q(\mathcal{L}(\lambda)\otimes_{{\mathbf{A}_{0}}}\mathcal{L}(\lambda^{\prime}))$,
and likewise $f^{i}_{\lambda}\otimes
f^{j}_{\lambda^{\prime}}\in\mathcal{L}_{n}^{*}+q(\mathcal{L}(\lambda)^{*}\otimes_{{\mathbf{A}_{0}}}\mathcal{L}(\lambda^{\prime})^{*})$.
Therefore $\mathsf{f}^{\lambda}_{i}\mathsf{f}^{\lambda^{\prime}}_{j}\in
q\mathcal{O}_{q}^{\mathbf{A}_{0}}[K]$.

On the other hand, if $b^{\lambda}_{i}\otimes b^{\lambda^{\prime}}_{j}$ is
contained in the Cartan component, then $f^{\lambda}_{i}\otimes
f^{\lambda^{\prime}}_{j}\equiv
f^{\lambda+\lambda^{\prime}}_{m}\pmod{q(\mathcal{L}(\lambda)\otimes_{{\mathbf{A}_{0}}}\mathcal{L}(\lambda^{\prime}))}$,
where we are identifying $f^{\lambda+\lambda^{\prime}}_{m}$ with its image in
$\mathcal{L}_{1}^{*}\cong\mathcal{L}(\lambda+\lambda^{\prime})^{*}$. The
relation (5.6) follows, and by applying the involution $*$ and Proposition 3.3
we obtain (5.7). ∎

We note that Proposition 5.8 implies the exchange relation
$\mathsf{f}^{\lambda}_{i}\mathsf{f}^{\lambda^{\prime}}_{j}\equiv\mathsf{f}^{\lambda^{\prime}}_{k}\mathsf{f}^{\lambda}_{l}$
modulo $q\mathcal{O}_{q}^{\mathbf{A}_{0}}[K]$ when $b^{\lambda}_{i}\otimes
b^{\lambda^{\prime}}_{j}$ is in the Cartan component and
$\sigma_{\mathcal{B}(\lambda),\mathcal{B}(\lambda^{\prime})}(b^{\lambda}_{i}\otimes
b^{\lambda^{\prime}}_{j})=b^{\lambda^{\prime}}_{k}\otimes b^{\lambda}_{l}$.
The following proposition gives more precision, as well as giving exchange
relations for the $\mathsf{f}^{\lambda}_{i}$ with the
$\mathsf{v}^{\lambda^{\prime}}_{j}$.

######  Proposition 5.9.

We have the commutation relations

$\begin{split}\mathsf{f}^{\lambda}_{i}\mathsf{f}^{\lambda^{\prime}}_{j}&=\sum_{k,l}q^{(\lambda,\lambda^{\prime})}\left(\hat{R}_{V(\lambda^{\prime}),V(\lambda)}\right)^{ij}_{kl}\mathsf{f}^{\lambda^{\prime}}_{k}\mathsf{f}^{\lambda}_{l},\\\
\mathsf{v}^{\lambda}_{i}\mathsf{v}^{\lambda^{\prime}}_{j}&=\sum_{k,l}q^{(\lambda,\lambda^{\prime})}\left(\hat{R}_{V(\lambda^{\prime}),V(\lambda)}\right)^{lk}_{ji}\mathsf{v}^{\lambda^{\prime}}_{k}\mathsf{v}^{\lambda}_{l},\\\
\mathsf{f}^{\lambda}_{i}\mathsf{v}^{\lambda^{\prime}}_{j}&=\sum_{k,l}q^{(\lambda,\lambda^{\prime})}\left(\hat{R}_{V(\lambda),V(\lambda^{\prime})}\right)^{ki}_{lj}\mathsf{v}^{\lambda^{\prime}}_{k}\mathsf{f}^{\lambda}_{l}.\end{split}$

Moreover we have $\sum_{i}\mathsf{v}^{\lambda}_{i}\mathsf{f}^{\lambda}_{i}=1$.

###### Proof.

These can be derived from the commutation relations (5.2) which hold in
$\mathcal{O}_{q}[G]$. Since $v_{1}$ is the highest weight vector we have
$(\hat{R}_{V(\lambda^{\prime}),V(\lambda)}^{-1})^{cd}_{11}=q^{(\lambda,\lambda^{\prime})}\delta^{c}_{1}\delta^{d}_{1}$,
which gives the relation for
$\mathsf{f}^{\lambda}_{i}\mathsf{f}^{\lambda^{\prime}}_{j}$ in the form

$c^{V(\lambda)}_{f^{i},v_{1}}c^{V(\lambda^{\prime})}_{f^{j},v_{1}}=\sum_{a,b}q^{(\lambda,\lambda^{\prime})}(\hat{R}_{V(\lambda^{\prime}),V(\lambda)})^{ij}_{ab}c^{V(\lambda^{\prime})}_{f^{a},v_{1}}c^{V(\lambda)}_{f^{b},v_{1}}.$

Similarly, since $f^{1}$ is the lowest weight vector we have
$(\hat{R}_{V(\lambda^{\prime})^{*},V(\lambda)^{*}}^{-1})^{cd}_{11}=q^{(\lambda,\lambda^{\prime})}\delta^{c}_{1}\delta^{d}_{1}$,
which gives the relation for
$\mathsf{v}^{\lambda}_{i}\mathsf{v}^{\lambda^{\prime}}_{j}$ in the form

$c^{V(\lambda)^{*}}_{\tilde{v}_{i},f^{1}}c^{V(\lambda^{\prime})^{*}}_{\tilde{v}_{j},f^{1}}=\sum_{a,b}q^{(\lambda,\lambda^{\prime})}(\hat{R}_{V(\lambda^{\prime})^{*},V(\lambda)^{*}})^{ij}_{ab}c^{V(\lambda^{\prime})^{*}}_{\tilde{v}_{a},f^{1}}c^{V(\lambda)^{*}}_{\tilde{v}_{b},f^{1}}.$

It can be rewritten in terms of $\hat{R}_{V(\lambda^{\prime}),V(\lambda)}$ by
using the identities

$(\hat{R}_{V^{*},W})_{ij}^{kl}=(\hat{R}_{V,W}^{-1})_{jl}^{ik},\quad(\hat{R}_{V,W^{*}}^{-1})_{ij}^{kl}=(\hat{R}_{V,W})_{jl}^{ik}.$
(5.8)

These are valid for any $\mathcal{U}_{q}(\mathfrak{g})$-modules $V,W$ (with
dual bases for $V^{*},W^{*}$) and follow from the fact that the finite-
dimensional $\mathcal{U}_{q}(\mathfrak{g})$-modules form a braided monoidal
category (see [Mat19, Lemma A.1] for a proof). They imply that

$(\hat{R}_{V(\lambda^{\prime})^{*},V(\lambda)^{*}})^{ij}_{ab}=(\hat{R}_{V(\lambda^{\prime}),V(\lambda)})^{ba}_{ji},$

which gives the relation for
$\mathsf{v}^{\lambda}_{i}\mathsf{v}^{\lambda^{\prime}}_{j}$ as stated.

Proceeding as above, we have
$(\hat{R}_{V(\lambda),V(\lambda^{\prime})})^{cd}_{11}=q^{(\lambda,\lambda^{\prime})}\delta^{c}_{1}\delta^{d}_{1}$
and this gives the relation for
$\mathsf{f}^{\lambda}_{i}\mathsf{v}^{\lambda^{\prime}}_{j}$ in the form

$c^{V(\lambda)}_{f^{i},v_{1}}c^{V(\lambda^{\prime})^{*}}_{\tilde{v}_{j},f^{1}}=\sum_{a,b}q^{(\lambda,\lambda^{\prime})}(\hat{R}_{V(\lambda),V(\lambda^{\prime})^{*}}^{-1})^{ij}_{ab}c^{V(\lambda^{\prime})^{*}}_{\tilde{v}_{a},f^{1}}c^{V(\lambda)}_{f^{b},v_{1}}.$

It can be rewritten using the identity
$(\hat{R}_{V(\lambda),V(\lambda^{\prime})^{*}}^{-1})^{ij}_{ab}=(\hat{R}_{V(\lambda),V(\lambda^{\prime})})^{ai}_{bj}$
from (5.8), which gives the relation as stated.

Finally we want to show that
$\sum_{i}\mathsf{v}^{\lambda}_{i}\mathsf{f}^{\lambda}_{i}=1$. It is not
difficult show that $\sum_{i}\mathsf{v}^{\lambda}_{i}\mathsf{f}^{\lambda}_{i}$
is invariant under the right action of $\mathcal{U}_{q}(\mathfrak{g})$, which
means that $\sum_{i}\mathsf{v}^{\lambda}_{i}\mathsf{f}^{\lambda}_{i}=c\cdot 1$
for some $c\in\mathbb{C}$. Applying the counit to the left-hand side we have

$\sum_{i}\varepsilon(\mathsf{v}^{\lambda}_{i}\mathsf{f}^{\lambda}_{i})=\sum_{i}\varepsilon(c^{V(\lambda)^{*}}_{\tilde{v}_{i},f^{1}})\varepsilon(c^{V(\lambda)}_{f^{i},v_{1}})=\sum_{i}\tilde{v}_{i}(f^{1})f^{i}(v_{1})=1.$

Therefore $c=1$, which gives the claimed relation. ∎

Note that the third exchange relations from Proposition 5.9 yields the
following factorization for our compact ${\mathbf{A}_{0}}$-form
$\mathcal{O}^{\mathbf{A}_{0}}_{q}[K]$.

######  Corollary 5.10.

We have
$\mathcal{O}^{\mathbf{A}_{0}}_{q}[K]=\mathcal{O}^{\mathbf{A}_{0}}_{q}[G/N^{-}]\cdot\mathcal{O}^{\mathbf{A}_{0}}_{q}[G/N^{+}]$.

The key consequence of Proposition 5.9 is that, thanks to Theorem 5.3, the
stated commutation relations also make sense when we specialize at $q=0$. This
allows us to deduce a set of commutation relations for the analytic limits
$\pi_{0}(\mathsf{f}^{\lambda}_{i})$ and $\pi_{0}(\mathsf{v}^{\lambda}_{i})$ in
$\mathcal{O}[K_{0}]$. Combining all the above results, we get relations at
$q=0$ as follows.

######  Theorem 5.11.

The following relations hold in
$\mathcal{O}[K_{0}]=\pi_{0}(\mathcal{O}_{q}^{\mathbf{A}_{0}}[K])$.

  1. 1)

For any $\lambda,\lambda^{\prime}\in\mathbf{P}^{+}$ and any $i,j$ we have

$\displaystyle\pi_{0}(\mathsf{f}^{\lambda}_{i})\pi_{0}(\mathsf{f}^{\lambda^{\prime}}_{j})$
$\displaystyle=\eta(b^{\lambda}_{i}\otimes
b^{\lambda^{\prime}}_{j})\pi_{0}(\mathsf{f}^{\lambda+\lambda^{\prime}}_{m}),$
(5.9)
$\displaystyle\pi_{0}(\mathsf{v}^{\lambda^{\prime}}_{j})\pi_{0}(\mathsf{v}^{\lambda}_{i})$
$\displaystyle=\eta(b^{\lambda}_{i}\otimes
b^{\lambda^{\prime}}_{j})\pi_{0}(\mathsf{v}^{\lambda+\lambda^{\prime}}_{m}),$
(5.10)

where we write $b^{\lambda}_{i}\otimes b^{\lambda^{\prime}}_{j}\mapsto
b^{\lambda+\lambda^{\prime}}_{m}$ under the unique surjective crystal morphism
$\mathcal{B}(\lambda)\otimes\mathcal{B}(\lambda^{\prime})\to\mathcal{B}(\lambda+\lambda^{\prime})$.

  2. 2)

For any $\lambda,\lambda^{\prime}\in\mathbf{P}^{+}$ and any $i,j$ we have

$\displaystyle\pi_{0}(\mathsf{f}^{\lambda}_{i})\pi_{0}(\mathsf{v}^{\lambda^{\prime}}_{j})$
$\displaystyle=\sum_{k,l}\pi_{0}(\mathsf{v}^{\lambda^{\prime}}_{k})\pi_{0}(\mathsf{f}^{\lambda}_{l}),$
(5.11)

where the sum is over all pairs $(k,l)$ such that
$\sigma_{\mathcal{B}(\lambda),\mathcal{B}(\lambda^{\prime})}(b^{\lambda}_{l}\otimes
b^{\lambda^{\prime}}_{j})=b^{\lambda^{\prime}}_{k}\otimes b^{\lambda}_{i}$.

  3. 3)

For any $\lambda\in\mathbf{P}^{+}$ we have

$\sum_{i}\pi_{0}(\mathsf{v}^{\lambda}_{i})\pi_{0}(\mathsf{f}^{\lambda}_{i})=1.$
(5.12)

  4. 4)

For any $\lambda\in\mathbf{P}^{+}$ and any $i$ we have
$\pi_{0}(\mathsf{f}^{\lambda}_{i})^{*}=\pi_{0}(\mathsf{v}^{\lambda}_{i})$.

###### Proof.

1) and 4) follow from Proposition 5.8 and Proposition 3.3, while 3) follows
from Proposition 5.9. The relation 2) follows from the third relation of
Proposition 5.9 using the following consequence of Theorem 5.3:

$\lim_{q\to
0}q^{(\lambda,\lambda^{\prime})}\left(\hat{R}_{V(\lambda),V(\lambda^{\prime})}\right)_{ij}^{kl}=\begin{cases}1&\text{if
}\sigma_{\mathcal{B}(\lambda),\mathcal{B}(\lambda^{\prime})}(b^{\lambda}_{i}\otimes
b^{\lambda^{\prime}}_{j})=b^{\lambda^{\prime}}_{k}\otimes
b^{\lambda^{\prime}}_{l},\\\ 0&\text{if }\eta(b^{\lambda}_{i}\otimes
b^{\lambda^{\prime}}_{j})=0.\end{cases}\qed$

Ultimately, we will show that Theorem 5.11 gives a complete set of generators
and relations for $\mathcal{O}[K_{0}]$.

######  Remark 5.12.

Note that relations (5.9) and (5.10) imply the exchange relations

$\displaystyle\pi_{0}(\mathsf{f}^{\lambda}_{i})\pi_{0}(\mathsf{f}^{\lambda^{\prime}}_{j})$
$\displaystyle=\pi_{0}(\mathsf{f}^{\lambda^{\prime}}_{k})\pi_{0}(\mathsf{f}^{\lambda}_{l}),$
(5.13)
$\displaystyle\pi_{0}(\mathsf{v}^{\lambda^{\prime}}_{j})\pi_{0}(\mathsf{v}^{\lambda}_{i})$
$\displaystyle=\pi_{0}(\mathsf{v}^{\lambda}_{l})\pi_{0}(\mathsf{v}^{\lambda^{\prime}}_{k}),$
(5.14)

whenever
$\sigma_{\mathcal{B}(\lambda),\mathcal{B}(\lambda^{\prime})}(b^{\lambda}_{i}\otimes
b^{\lambda^{\prime}}_{j})=b^{\lambda^{\prime}}_{k}\otimes b^{\lambda}_{l}$.
These also follow from the $q\to 0$ limit of the relations in Proposition 5.9.

##  6\. Properties of the Cartan braiding

###  6.1. Hexagon and braid relations

The Cartan braiding of Definition 5.4 inherits properties from the braiding
operators $\hat{R}_{V,V^{\prime}}$ as follows.

######  Theorem 6.1.

Let $V$, $V^{\prime}$ and $V^{\prime\prime}$ be products of irreducibles with
crystal bases $(\mathcal{L},\mathcal{B})$,
$(\mathcal{L}^{\prime},\mathcal{B}^{\prime})$ and
$(\mathcal{L}^{\prime\prime},\mathcal{B}^{\prime\prime})$, respectively. Then
the crystal braiding maps satisfy the hexagon relations

$\displaystyle\sigma_{\mathcal{B},\mathcal{B}^{\prime}\otimes\mathcal{B}^{\prime\prime}}$
$\displaystyle=(\operatorname{id}_{\mathcal{B}^{\prime}}\otimes\sigma_{\mathcal{B},\mathcal{B}^{\prime\prime}})(\sigma_{\mathcal{B},\mathcal{B}^{\prime}}\otimes\operatorname{id}_{\mathcal{B}^{\prime\prime}}),$
(6.1)
$\displaystyle\sigma_{\mathcal{B}\otimes\mathcal{B}^{\prime},\mathcal{B}^{\prime\prime}}$
$\displaystyle=(\sigma_{\mathcal{B},\mathcal{B}^{\prime\prime}}\otimes\operatorname{id}_{\mathcal{B}^{\prime}})(\operatorname{id}_{\mathcal{B}}\otimes\sigma_{\mathcal{B}^{\prime},\mathcal{B}^{\prime\prime}}),$
(6.2)

and the braid relation

$\displaystyle(\sigma_{\mathcal{B}^{\prime},\mathcal{B}^{\prime\prime}}\otimes\operatorname{id}_{\mathcal{B}})$
$\displaystyle(\operatorname{id}_{\mathcal{B}}^{\prime}\otimes\sigma_{\mathcal{B},\mathcal{B}^{\prime\prime}})(\sigma_{\mathcal{B},\mathcal{B}^{\prime}}\otimes\operatorname{id}_{\mathcal{B}^{\prime\prime}})$
$\displaystyle=(\operatorname{id}_{\mathcal{B}^{\prime\prime}}\otimes\sigma_{\mathcal{B},\mathcal{B}^{\prime}})(\sigma_{\mathcal{B},\mathcal{B}^{\prime\prime}}\otimes\operatorname{id}_{\mathcal{B}^{\prime}})(\operatorname{id}_{\mathcal{B}}\otimes\sigma_{\mathcal{B}^{\prime},\mathcal{B}^{\prime\prime}}).$
(6.3)

###### Proof.

Let $\lambda$, $\lambda^{\prime}$ and $\lambda^{\prime\prime}$ be the highest
weights of $V$, $V^{\prime}$ and $V^{\prime\prime}$, with crystals
$\mathcal{B}$, $\mathcal{B}^{\prime}$ and $\mathcal{B}^{\prime\prime}$,
respectively. Then the highest weight of
$\mathcal{B}^{\prime}\otimes\mathcal{B}^{\prime\prime}$ is
$\lambda^{\prime}+\lambda^{\prime\prime}$. Using the hexagon relation for
$\hat{R}$, we get

$\displaystyle
q^{(\lambda,\lambda^{\prime}+\lambda^{\prime\prime})}\hat{R}_{V,V^{\prime}\otimes
V^{\prime\prime}}$
$\displaystyle=q^{(\lambda,\lambda^{\prime\prime})}q^{(\lambda,\lambda^{\prime})}(\operatorname{id}_{V^{\prime}}\otimes\hat{R}_{V,V^{\prime\prime}})(\hat{R}_{V,V^{\prime}}\otimes\operatorname{id}_{V^{\prime\prime}}).$

This specializes at $q=0$ by Theorem 5.3, and the specialization induces the
hexagon relation (6.1). The other hexagon relation is proven similarly, and
the braid relation follows by an analogous calculation. ∎

###  6.2. Partial action of the symmetric group

Let $\mathcal{B}_{1},\cdots,\mathcal{B}_{n}$ be products of irreducible
crystals, see Definition 5.1. For long tensor products of the form
$\mathcal{B}_{1}\otimes\cdots\otimes\mathcal{B}_{n}$, we will denote the
action of the crystal braiding on two successive terms by
$\sigma_{i}=\operatorname{id}_{\mathcal{B}_{1}}\otimes\cdots\otimes\sigma_{\mathcal{B}_{i},\mathcal{B}_{i+1}}\otimes\cdots\otimes\operatorname{id}_{\mathcal{B}_{n}}$,
so that

$\sigma_{i}:\mathcal{B}_{1}\otimes\cdots\mathcal{B}_{i}\otimes\mathcal{B}_{i+1}\otimes\cdots\otimes\mathcal{B}_{n}\to\mathcal{B}_{1}\otimes\cdots\mathcal{B}_{i+1}\otimes\mathcal{B}_{i}\otimes\cdots\otimes\mathcal{B}_{n}.$

######  Proposition 6.2.

Let $\mathcal{B}_{1},\cdots,\mathcal{B}_{n}$ be irreducible crystals. Let
$s\in S_{n}$ be a permutation and let $s=s_{i_{1}}\cdots s_{i_{k}}$ be a
realization of $s$ as a reduced word in the transpositions
$s_{i}=(i\;\;i\\!+\\!1)$. Then the crystal morphism

$\sigma_{s}:=\sigma_{i_{1}}\circ\cdots\circ\sigma_{i_{n}}:\mathcal{B}_{1}\otimes\cdots\mathcal{B}_{n}\to\mathcal{B}_{s(1)}\otimes\cdots\otimes\mathcal{B}_{s(n)}$

is independent of the choice of reduced word representing $s$.

###### Proof.

Thanks to Theorem 5.3, this follows from the well-known fact that any two
reduced words can be linked by repeated application of the braid relations
$s_{i}s_{i+1}s_{i}=s_{i+1}s_{i}s_{i+1}$. ∎

Note that the relation $s_{i}^{2}=\operatorname{id}$ is not necessary in the
above proof. This is important since $\sigma_{i}^{2}\neq\operatorname{id}$,
although $\sigma_{i}^{2}$ is the identity on the Cartan component. As a
consequence, the map $s\mapsto\sigma_{s}$ doesn’t define an action of $S_{n}$,
but it does define an action between the Cartan components, thanks to the
following theorem.

######  Theorem 6.3.

Let $\mathcal{B}_{1},\cdots,\mathcal{B}_{n}$ be irreducible crystals and let
$b_{i}\in\mathcal{B}_{i}$ for each $i$. The following are equivalent:

  1. (i)

$b_{1}\otimes\cdots\otimes b_{n}$ is in the Cartan component of
$\mathcal{B}_{1}\otimes\cdots\otimes\mathcal{B}_{n}$,

  2. (ii)

$\sigma_{s}(b_{1}\otimes\cdots\otimes b_{n})\neq 0$ for every $s\in S_{n}$,

  3. (iii)

$\sigma_{s_{0}}(b_{1}\otimes\cdots\otimes b_{n})\neq 0$, where
$s_{0}=\scriptstyle\begin{pmatrix}1&2&\cdots&n\\\ n&n-1&\cdots&1\end{pmatrix}$
is the longest permutation.

If $\mathcal{B}_{1},\cdots,\mathcal{B}_{n}$ are products of irreducibles, the
equivalence holds if we add to (ii) and (iii) the condition that each $b_{i}$
be in the Cartan component of $\mathcal{B}_{i}$.

###### Proof.

To begin with, assume that all the $\mathcal{B}_{i}$ are irreducible.

For (i) $\Rightarrow$ (ii), note that if $b_{1}\otimes\cdots\otimes b_{n}$ is
in the Cartan component of
$\mathcal{B}_{1}\otimes\cdots\otimes\mathcal{B}_{n}$, then necessarily
$b_{i}\otimes b_{i+1}$ is in the Cartan component of
$\mathcal{B}_{i}\otimes\mathcal{B}_{i+1}$ for every $i$. Therefore
$\sigma_{\mathcal{B}_{i},\mathcal{B}_{i+1}}(b_{i}\otimes b_{i+1})$ is non-zero
and, since it corresponds to the same element of the Cartan component as
$b_{i}\otimes b_{i+1}$, we have that $\sigma_{i}(b_{1}\otimes\cdots\otimes
b_{n})$ again belongs to the Cartan component. Inductively, any repeated
composition of the $\sigma_{i}$ is non-zero.

The implication (ii) $\Rightarrow$ (iii) is obvious. To prove (iii)
$\Rightarrow$ (i), we work inductively on $n$. When $n=1$ or $2$, the claim is
immediate. Suppose it is true for $n-1$. If
$\sigma_{s_{0}}(b_{1}\otimes\cdots\otimes b_{n})\neq 0$, write

$s_{0}=s_{0}^{\prime}(s_{n-1}s_{n-2}\cdots s_{1})=(s_{n-1}s_{n-2}\cdots
s_{1})s_{0}^{\prime\prime},$ (6.4)

where

$s_{0}^{\prime}=\scriptstyle\begin{pmatrix}1&2&\cdots&n-1&n\\\
n-1&n-2&\cdots&1&n\end{pmatrix},\quad
s_{0}^{\prime\prime}=\scriptstyle\begin{pmatrix}1&2&\cdots&n-1&n\\\
1&n&\cdots&3&2\end{pmatrix}$

are the permutations reversing the first $n-1$ and last $n-1$ entries,
respectively. Upon writing $s_{0}^{\prime}$ and $s_{0}^{\prime\prime}$ as
reduced words, both expressions in (6.4) give a reduced expression for
$s_{0}$, so we have

$\sigma_{s_{0}^{\prime}}(\sigma_{n-1}\circ\cdots\circ\sigma_{1}(b_{1}\otimes\cdots\otimes
b_{n}))\\\
=\sigma_{n-1}\circ\cdots\circ\sigma_{1}(\sigma_{s_{0}^{\prime\prime}}(b_{1}\otimes\cdots\otimes
b_{n}))=\sigma_{s_{0}}(b_{1}\otimes\cdots\otimes b_{n})\neq 0.$

In particular, $\sigma_{s_{0}^{\prime\prime}}(b_{1}\otimes\cdots\otimes
b_{n})\neq 0$, so by the inductive hypothesis, $b_{2}\otimes\cdots\otimes
b_{n}$ is in the Cartan component of
$\mathcal{B}_{2}\otimes\cdots\otimes\mathcal{B}_{n}$. But also
$\sigma_{n-1}\circ\cdots\circ\sigma_{1}(b_{1}\otimes\cdots\otimes b_{n})\neq
0$, and repeated application of the hexagon relation (6.1) shows that

$\sigma_{n-1}\circ\sigma_{n-2}\circ\cdots\circ\sigma_{1}=\sigma_{\mathcal{B}_{1},(\mathcal{B}_{2}\otimes\cdots\otimes\mathcal{B}_{n})},$

so $b_{1}\otimes(b_{2}\otimes\cdots\otimes b_{n})$ is necessarily in the
Cartan component of
$\mathcal{B}_{1}\otimes(\mathcal{B}_{2}\otimes\cdots\otimes\mathcal{B}_{n})$.

This completes the proof for $\mathcal{B}_{i}$ irreducible. The general case
where the $\mathcal{B}_{i}$ are products of irreducibles follows readily from
this by naturality of the braiding, which implies that we can restrict our
attention to the Cartan components in each $\mathcal{B}_{i}$. ∎

###  6.3. Left and right ends

We now make a definition that is going to play a major role in the rest of
this work, namely the notion of right end of a crystal element.

######  Definition 6.4.

Let $\lambda,\mu\in\mathbf{P}^{+}$ with $\lambda\geq\mu$. Considering the
crystal $\mathcal{B}(\lambda)$, the _left end_ and _right end_ with respect to
$\mu$ are the set-theoretic maps

$\displaystyle\mathsf{L}_{\mu}:\mathcal{B}(\lambda)\to\mathcal{B}(\mu),$
$\displaystyle\mathsf{R}_{\mu}:\mathcal{B}(\lambda)\to\mathcal{B}(\mu),$

defined as follows. There exist unique injective morphisms of crystals

$\displaystyle\mathcal{B}(\lambda)\to\mathcal{B}(\mu)\otimes\mathcal{B}(\lambda-\mu),$
$\displaystyle\mathcal{B}(\lambda)\to\mathcal{B}(\lambda-\mu)\otimes\mathcal{B}(\mu),$

and we define $\mathsf{L}_{\mu}(b)$ and $\mathsf{R}_{\mu}(b)$ to be the unique
elements of $\mathcal{B}(\mu)$ such that

$\displaystyle b\mapsto\mathsf{L}_{\mu}(b)\otimes b^{\prime},$ $\displaystyle
b\mapsto b^{\prime\prime}\otimes\mathsf{R}_{\mu}(b),$

for some $b^{\prime},b^{\prime\prime}\in\mathcal{B}(\lambda-\mu)$.

More generally, let
$\mathcal{B}=\mathcal{B}(\lambda_{1})\otimes\cdots\otimes\mathcal{B}(\lambda_{n})$
be a product of irreducibles of highest weight $\lambda=\sum_{i}\lambda_{i}$.
If $b\in\mathcal{B}$ is in the Cartan component we define
$\mathsf{L}_{\mu}(b)$ and $\mathsf{R}_{\mu}(b)$ as above, under the
identification of the Cartan component of $\mathcal{B}$ with
$\mathcal{B}(\lambda)$. Otherwise $\mathsf{L}_{\mu}(b)=\mathsf{R}_{\mu}(b)=0$.

We will also consider the right ends with respect to a _family_ of dominant
weights, and in particular for the family of fundamental weights
${\bm{\Pi}}=(\varpi_{1},\cdots,\varpi_{r})$.

######  Definition 6.5.

Let $\mathsf{S}=(\mu_{1},\cdots,\mu_{n})$ be an $n$-tuple of dominant weights.
Let $\mathcal{B}$ be a product of irreducible crystals of highest weight
$\lambda$ with $\lambda\geq\mu_{i}$ for all $i$. Then the _$\mathsf{S}$ -left
end_ and _$\mathsf{S}$ -right end_ maps are the set-theoretic maps

$\mathsf{L}_{\mathsf{S}},\mathsf{R}_{\mathsf{S}}:\mathcal{B}\to\mathcal{B}(\mu_{1})\times\cdots\times\mathcal{B}(\mu_{n})\sqcup\\{0\\}$

defined by

$\displaystyle\mathsf{L}_{\mathsf{S}}(b):=\left(\mathsf{L}_{\mu_{1}}(b),\cdots,\mathsf{L}_{\mu_{n}}(b)\right),$
$\displaystyle\mathsf{R}_{\mathsf{S}}(b):=\left(\mathsf{R}_{\mu_{1}}(b),\cdots,\mathsf{R}_{\mu_{n}}(b)\right).$

######  Remark 6.6.

According to the usual conventions on crystal bases, the notations
$\mathcal{B}(\mu_{1})\otimes\cdots\otimes\mathcal{B}(\mu_{n})$ and
$\mathcal{B}(\mu_{1})\times\cdots\times\mathcal{B}(\mu_{n})$ denote the same
object. We use the former when we take into account the additional crystal
structure, but the latter when regarded simply as a set. This is the notation
we use for left and right ends, since $\mathsf{L}_{\mathsf{S}}$ and
$\mathsf{R}_{\mathsf{S}}$ are only set-theoretic maps.

The rest of this section is devoted to the properties of the right end maps.
We focus on the right ends since these play the main role in our setup, but
similar results can of course be obtained for the left end maps.

######  Proposition 6.7.

Let $\lambda_{1},\cdots,\lambda_{n}\in\mathbf{P}^{+}$ be dominant weights. An
element $b_{1}\otimes\cdots\otimes b_{n}$ belongs to the Cartan component of
$\mathcal{B}(\lambda_{1})\otimes\cdots\otimes\mathcal{B}(\lambda_{n})$ if and
only if we have

$\sigma_{n-1}\circ\cdots\circ\sigma_{k}(b_{1}\otimes\cdots\otimes b_{n})\neq
0$ (6.5)

for every $k\in\\{1,\cdots,n-1\\}$. In this case, for each
$k\in\\{1,\cdots,n\\}$, the right end
$\mathsf{R}_{\lambda_{k}}(b_{1}\otimes\cdots\otimes b_{n})$ is equal to the
rightmost tensor factor of (6.5).

###### Proof.

For $n=1$ the statement is trivial, so suppose it is true for tensor products
of length $n-1$. The necessity of (6.5) is obvious from Theorem 6.3, so it
remains to check sufficiency.

By the inductive hypothesis, $b_{2}\otimes\cdots\otimes b_{n}$ is in the
Cartan component of $\mathcal{B}(\lambda_{2})\otimes\mathcal{B}(\lambda_{n})$,
so by Theorem 6.3 we have $s_{0}^{\prime}(b_{2}\otimes\cdots\otimes b_{n})\neq
0$ where $s_{0}^{\prime}$ denotes the longest word of $S_{n-1}$. But now the
longest word $s_{0}$ of $S_{n}$ satisfies

$s_{0}\circ(\operatorname{id}\otimes s_{0}^{\prime})(b_{1}\otimes\cdots\otimes
b_{n})=\sigma_{n-1}\circ\cdots\circ\sigma_{1}(b_{1}\otimes\cdots\otimes
b_{n})\neq 0$

by (6.5). Therefore, by Theorem 6.3, $(\operatorname{id}\otimes
s_{0}^{\prime})(b_{1}\otimes\cdots\otimes b_{n})$ is in the Cartan component
of $\mathcal{B}(\lambda_{1})\otimes\cdots\otimes\mathcal{B}(\lambda_{n})$, and
so $b_{1}\otimes\cdots\otimes b_{n}$ as well. This proves the first statement.

Now the map from (6.5) gives a morphism of crystals

$\sigma_{n-1}\circ\cdots\circ\sigma_{k}:\mathcal{B}(\lambda_{1})\otimes\cdots\otimes\mathcal{B}(\lambda_{n})\to\mathcal{B}(\lambda_{1})\otimes\cdots\otimes\widehat{\mathcal{B}(\lambda_{k})}\otimes\cdots\otimes\mathcal{B}(\lambda_{n})\otimes\mathcal{B}(\lambda_{k})$

with the hat denoting omission, which gives the second statement. ∎

######  Remark 6.8.

We briefly mention how this connects to the theory of set-theoretic solutions
to the Yang-Baxter equations, as in [ESS99]. Writing the Cartan braiding as
$\sigma(b\otimes
b^{\prime})=\mathsf{f}_{b}(b^{\prime})\otimes\mathsf{g}_{b^{\prime}}(b)$,
Proposition 6.7 can be recast in the form

$\mathsf{R}_{\lambda_{k}}(b_{1}\otimes\cdots\otimes
b_{n})=\mathsf{g}_{b_{n}}\circ\cdots\circ\mathsf{g}_{b_{k+1}}(b_{k}),\quad
k=1,\cdots,n.$

This gives the components of the map $J_{n}$ introduced in [ESS99, Proposition
1.7] (if we work with left ends instead of right ends).

Next, we discuss how right ends behave with respect to tensor products.

######  Proposition 6.9.

Let $\lambda,\lambda^{\prime},\mu\in\mathbf{P}^{+}$ and let $b\otimes
b^{\prime}$ be in the Cartan component of
$\mathcal{B}(\lambda)\otimes\mathcal{B}(\lambda^{\prime})$.

  1. 1)

If $\lambda^{\prime}\geq\mu$ then $\mathsf{R}_{\mu}(b\otimes
b^{\prime})=\mathsf{R}_{\mu}(b^{\prime})$.

  2. 2)

If $\lambda\geq\mu$ then $\mathsf{R}_{\mu}(b\otimes
b^{\prime})=\mathsf{R}_{\mu}(\mathsf{R}_{\mu}(b)\otimes b^{\prime})$.
Consequently, if $\lambda^{\prime\prime}\geq\mu$ and
$c\in\mathcal{B}(\lambda^{\prime\prime})$ has the same $\mu$-right end as
$b\in\mathcal{B}(\lambda)$, then $\mathsf{R}_{\mu}(b\otimes
b^{\prime})=\mathsf{R}_{\mu}(c\otimes b^{\prime})$.

###### Proof.

1) is almost immediate from the definitions. For 2), consider the inclusion of
the Cartan component
$\iota:\mathcal{B}(\lambda)\to\mathcal{B}(\lambda-\mu)\otimes\mathcal{B}(\mu)$.
We have

$\textstyle{b\otimes
b^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota\otimes\operatorname{id}}$$\textstyle{b^{\prime\prime}\otimes\mathsf{R}_{\mu}(b)\otimes
b^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{2}}$$\textstyle{b^{\prime\prime}\otimes
b^{\prime\prime\prime}\otimes\mathsf{R}_{\mu}(\mathsf{R}_{\mu}(b)\otimes
b^{\prime}),}$

for some $b^{\prime\prime}\in\mathcal{B}(\lambda-\mu)$ and
$b^{\prime\prime\prime}\in\mathcal{B}(\lambda^{\prime})$. The result then
follows from Proposition 6.7. The final statement is now immediate. ∎

The next result shows that we obtain the same set of non-zero
$\mathsf{S}$-right ends from any appropriate crystal.

######  Proposition 6.10.

Let $\mathsf{S}=(\mu_{1},\cdots,\mu_{n})$ be a family of dominant weights. Let
$\mathcal{B}$ be a product of irreducible crystals. Then the set of (non-zero)
$\mathsf{S}$-right ends of elements of $\mathcal{B}$, namely

$\\{\mathsf{R}_{\mathsf{S}}(b)\mid
b\in\mathcal{B}\\}\setminus\\{0\\}\quad\subseteq\prod_{i=1}^{N}\mathcal{B}(\mu_{i}),$
(6.6)

is independent of the choice of $\mathcal{B}$, as long as the highest weight
$\lambda$ of $\mathcal{B}$ satisfies $\lambda\geq\mu_{i}$ for every $i$.

###### Proof.

We may assume without loss of generality that
$\mathcal{B}=\mathcal{B}(\lambda)$ is irredudible, with $\lambda\geq\mu_{i}$
for all $i$, since if $\mathcal{B}$ has highest weight $\lambda$ then by
definition the right ends of $\mathcal{B}$ are the same as those of
$\mathcal{B}(\lambda)$.

Now suppose $\lambda^{\prime}\geq\lambda$. Proposition 6.9(1) shows that the
$\mu_{i}$-right ends of $\mathcal{B}(\lambda)$ are the same as the
$\mu_{i}$-right ends of
$\mathcal{B}(\lambda^{\prime}-\lambda)\otimes\mathcal{B}(\lambda)$, which by
definition are the same as the $\mu_{i}$-right ends of
$\mathcal{B}(\lambda^{\prime})$. To complete the proof, just note that for any
two $\lambda_{1},\lambda_{2}$ there is $\lambda^{\prime}$ which is greater
than both of them. ∎

##  7\. The higher-rank graph algebra

###  7.1. Higher-rank graphs

We begin by recalling the definition of a $k$-graph from [KP00]. For this, we
need to regard the abelian monoid $\mathbb{N}^{k}$ as a category with one
object, in which composition is given by addition.

######  Definition 7.1.

A _higher-rank graph of rank $k$_ (or $k$-graph) is a small category $\Lambda$
equipped with a functor $\mathsf{d}:\Lambda\to\mathbb{N}^{k}$ satisfying the
_factorization property_ : for every $e\in\Lambda$, and any pair of multi-
indices $m,n\in\mathbb{N}^{k}$ with $\mathsf{d}(e)=m+n$, there are unique
elements $e_{1},e_{2}\in\Lambda$ with $\mathsf{d}(e_{1})=m$,
$\mathsf{d}(e_{2})=n$ such that $e=e_{1}\cdot e_{2}$.

The elements of $\Lambda$ are called _paths_ , the indices $\\{1,\cdots,k\\}$
are referred to as _colours_ , and $\mathsf{d}(e)$ is called the _degree_ or
_coloured length_ of $e$. We write $\Lambda^{n}$ for the set of paths of
length $n\in\mathbb{N}^{k}$. Paths of length $0$ are called _vertices_ ,
naturally enough. Paths of length $\delta_{i}=(0,\cdots,1,\cdots,0)$, with $1$
in the $i$th slot, are called _edges of colour $i$_. We use the notation
$\Lambda^{\neq 0}$ for the set of paths of non-zero length.

The factorization property implies that for every path $e\in\Lambda$, there
are unique vertices $\mathsf{r}(e)$ and $\mathsf{s}(e)$ in $\Lambda^{0}$, with
the property that

$e=\mathsf{r}(e)\cdot e\cdot\mathsf{s}(e).$

This defines the _range_ and _source_ maps
$\mathsf{r},\mathsf{s}:\Lambda\to\Lambda^{0}$. Given a vertex
$v\in\Lambda^{0}$ and a multi-index $n\in\mathbb{N}^{k}$, we write

$\displaystyle v\Lambda^{n}:=\\{e\in\Lambda^{n}:\mathsf{r}(e)=v\\},$
$\displaystyle\Lambda^{n}v:=\\{e\in\Lambda^{n}:\mathsf{s}(e)=v\\}.$

It is common to impose the following additional hypotheses on a higher-rank
graph, which will apply in our case, see Section 7.4.

######  Definition 7.2.

A $k$-graph $\Lambda$ is _row-finite_ if $v\Lambda^{n}$ is finite for every
$v\in\Lambda^{0}$ and $n\in\mathbb{N}^{k}$. It has _no sources_ if
$v\Lambda^{n}\neq\emptyset$ for every $v\in\Lambda^{0}$ and
$n\in\mathbb{N}^{k}$, and _no sinks_ if $\Lambda^{n}v\neq\emptyset$ for every
$v\in\Lambda^{0}$ and $n\in\mathbb{N}^{k}$.

###  7.2. Higher-rank graph algebras

Higher-rank graph algebras were first defined as $C^{*}$-algebras, by Kumjian
and Pask [KP00]. We will work initially with the algebraic version, which is
due to Aranda Pino, Clark, an Huef and Raeburn [APCaHR13], although we will
modify their definition slightly to take advantage of the fact that we only
work over the field $\mathbb{C}$ rather than an arbitrary commutative ring
with unit.

######  Definition 7.3.

Let $\Lambda$ be a row-finite $k$-graph without sources. The _Kumjian-Pask
algebra_ $\mathrm{KP}_{\mathbb{C}}(\Lambda)$ is the $*$-algebra over
$\mathbb{C}$ generated by the elements $\\{p_{v}\\}_{v\in\Lambda^{0}}$ and
$\\{s_{e}\\}_{e\in\Lambda^{\neq 0}}$ with the following relations:

  1. (KP1)

$\\{p_{v}\\}_{v\in\Lambda^{0}}$ is a set of mutually orthogonal projections,
that is $p_{v}^{*}=p_{v}=p_{v}^{2}$ and
$p_{v}p_{v^{\prime}}=\delta_{v,v^{\prime}}p_{v}$,

  2. (KP2)

for all $e,e^{\prime}\in\Lambda^{\neq 0}$ with
$\mathsf{s}(e)=\mathsf{r}(e^{\prime})$ we have

$s_{e}s_{e^{\prime}}=s_{e\cdot e^{\prime}},\quad
p_{\mathsf{r}(e)}s_{e}=s_{e}=s_{e}p_{\mathsf{s}(e)},$

  3. (KP3)

for all $e,e^{\prime}\in\Lambda^{\neq 0}$ with
$\mathsf{d}(e)=\mathsf{d}(e^{\prime})$ we have

$s_{e}^{*}s_{e^{\prime}}=\delta_{e,e^{\prime}}p_{\mathsf{s}(e)},$

  4. (KP4)

for all $v\in\Lambda^{0}$ and $n\in\mathbb{N}^{k}\backslash\\{0\\}$ we have

$p_{v}=\sum_{e\in v\Lambda^{n}}s_{e}s_{e}^{*}.$

######  Remark 7.4.

The definition in [APCaHR13] is given in terms of paths $e\in\Lambda^{\neq 0}$
and ghost paths $e^{*}\in G(\Lambda^{\neq 0})$, corresponding to generators
$s_{e}$ and $s_{e^{*}}$. When working over $\mathbb{C}$ there is a
$*$-structure uniquely defined by $p_{v}^{*}=p_{v}$ and $s_{e}^{*}=s_{e^{*}}$,
so we prefer to give the definition in this way.

For the definition of the $C^{*}$-algebra of a higher-rank graph, see [KP00].

###  7.3. Higher-rank graphs associated to quantum groups

Let $K$ be a connected, simply connected compact semisimple Lie group of rank
$r$ with Lie algebra $\mathfrak{k}$ and complexification
$\mathfrak{g}=\mathfrak{k}_{\mathbb{C}}$. Our goal in this section is to
define a higher-rank graph $\Lambda_{\mathfrak{g}}$ of rank $r$ associated to
$\mathfrak{g}$. In fact, we will make a more general construction, in
anticipation of the results required for torus bundles over flag varieties,
compare Section 3.4 and in particular Theorem 3.8.

Consider any $N$-tuple $\mathsf{C}=(\vartheta_{1},\cdots,\vartheta_{N})$ of
linearly independent dominant weights. We refer to $\mathsf{C}$ as the set of
_colours_ and we blur the distinction between $\mathsf{C}$ and the index set
$\\{1,\cdots,N\\}$. Then to any such choice of colours we will associate a
higher-rank graph $\Lambda_{\mathfrak{g},\mathsf{C}}$ of rank $N$. The basic
case $\Lambda_{\mathfrak{g}}$ is obtained when
$\mathsf{C}={\bm{\Pi}}=(\varpi_{1},\cdots,\varpi_{r})$ is the $r$-tuple
fundamental weights.

Write $\mathbf{P}^{+}_{\mathsf{C}}:=\mathbb{N}\cdot\mathsf{C}$ for the
submonoid of $\mathbf{P}^{+}$ generated by the colours $\mathsf{C}$. We
identify $\mathbf{P}^{+}_{\mathsf{C}}$ with the monoid $\mathbb{N}^{N}$ via
the map

$\mathbb{N}^{N}\to\mathbf{P}^{+}_{\mathsf{C}},\qquad(n_{1},\cdots,n_{N})\mapsto\sum_{i=1}^{N}n_{i}\vartheta_{i}.$
(7.1)

In this way, the degree of a path in our higher rank graph will be given by a
weight in $\mathbf{P}^{+}_{\mathsf{C}}$. We also fix the dominant weight
$\displaystyle\rho_{\mathsf{C}}:=\sum_{i=1}^{N}\vartheta_{i}\in\mathbf{P}^{+}_{\mathsf{C}}.$

######  Definition 7.5.

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and fix a set of
colours $\mathsf{C}=(\vartheta_{1},\cdots,\vartheta_{N})$ as above. We define
the pair
$(\Lambda_{\mathfrak{g},\mathsf{C}}^{0},\Lambda_{\mathfrak{g},\mathsf{C}})$ as
follows.

  * •

The _set of vertices_ consists of the $\mathsf{C}$-right ends of
$\mathcal{B}(\rho_{\mathsf{C}})$, that is

$\Lambda_{\mathfrak{g},\mathsf{C}}^{0}:=\left\\{\mathsf{R}_{\mathsf{C}}(b)\mid
b\in\mathcal{B}(\rho_{\mathsf{C}})\right\\}\subset\mathcal{B}(\vartheta_{1})\times\cdots\times\mathcal{B}(\vartheta_{N}).$

  * •

The _set of paths_ $\Lambda_{\mathfrak{g},\mathsf{C}}$ is the set of pairs
$(v,b)$, where
$v=(c_{1},\cdots,c_{N})\in\Lambda_{\mathfrak{g},\mathsf{C}}^{0}$ and
$b\in\mathcal{B}(\lambda)$ for some $\lambda\in\mathbf{P}^{+}_{\mathsf{C}}$
such that $c_{i}\otimes b$ is in the Cartan component of
$\mathcal{B}(\vartheta_{i})\otimes\mathcal{B}(\lambda)$ for every colour $i$.

We define the _degree_ of the path $e=(v,b)$ above to be
$\mathsf{d}(e)=\lambda$, where we use the identification
$\mathbf{P}^{+}_{\mathsf{C}}\cong\mathbb{N}^{N}$ from (7.1).

Some remarks are in order. Firstly, we can identify the vertex set
$\Lambda_{\mathfrak{g},\mathsf{C}}^{0}$ with the set
$\Lambda_{\mathfrak{g},\mathsf{C}}^{0}$ of paths of degree $0$ via the map
$v\mapsto(v,b_{0})$, where $b_{0}\in\mathcal{B}(0)$ is the unique element of
the trivial crystal. Secondly, note that in the definition of the vertex set
$\Lambda_{\mathfrak{g},\mathsf{C}}^{0}$, we could equally well use the set of
$\mathsf{C}$-right ends of any crystal $\mathcal{B}(\mu)$ with highest weight
$\mu\geq\rho_{\mathsf{C}}$, thanks to Proposition 6.10.

######  Lemma 7.6.

Let $v\in\Lambda_{\mathfrak{g},\mathsf{C}}^{0}$ and fix $c\in\mathcal{B}(\mu)$
with $\mathsf{R}_{\mathsf{C}}(c)=(c_{1},\cdots,c_{N})=v$, for some
$\mu\in\mathbf{P}^{+}_{\mathsf{C}}$ with $\mu\geq\rho_{\mathsf{C}}$. Let also
$b\in\mathcal{B}(\lambda)$ for some $\lambda\in\mathbf{P}^{+}_{\mathsf{C}}$.
Then $(v,b)\in\Lambda_{\mathfrak{g},\mathsf{C}}$ if and only if $c\otimes b$
is in the Cartan component of $\mathcal{B}(\mu)\otimes\mathcal{B}(\lambda)$.
In this case, for every $\vartheta_{i}\in\mathsf{C}$, we have
$\mathsf{R}_{\vartheta_{i}}(c_{i}\otimes
b)=\mathsf{R}_{\vartheta_{i}}(c\otimes b)$.

###### Proof.

First, suppose that $c\otimes b$ is in the Cartan component of
$\mathcal{B}(\mu)\otimes\mathcal{B}(\lambda)$. Then Proposition 6.9(2) shows
that $\mathsf{R}_{\vartheta_{i}}(c_{i}\otimes
b)=\mathsf{R}_{\vartheta_{i}}(c\otimes b)\neq 0$ for all $i$, so
$(v,b)\in\Lambda_{\mathfrak{g},\mathsf{C}}$.

Conversely, suppose that $(v,b)\in\Lambda_{\mathfrak{g},\mathsf{C}}$, so that
$c_{i}\otimes b$ is in the Cartan component of
$\mathcal{B}(\vartheta_{i})\otimes\mathcal{B}(\lambda)$ for every $i$. We can
write $\mu=\sum_{i=1}^{N}n_{i}\vartheta_{i}$ with $n_{i}>0$ for every $i$, and
we put $|\mu|=\sum_{i}n_{i}$. We have an inclusion of crystals

$\mathcal{B}(\mu)\hookrightarrow\mathcal{B}(\vartheta_{1})^{\otimes
n_{1}}\otimes\cdots\otimes\mathcal{B}(\vartheta_{N})^{\otimes n_{N}}$

as the Cartan component on the right-hand side. Let us identify $c$ with its
image $c\mapsto a_{1}\otimes\cdots\otimes a_{|\mu|}$ in the Cartan component.
We can then identify $c\otimes b$ with its image $a_{1}\otimes\cdots\otimes
a_{|\mu|}\otimes b$ in $\mathcal{B}(\vartheta_{1})^{\otimes
n_{1}}\otimes\cdots\otimes\mathcal{B}(\vartheta_{N})^{\otimes
n_{N}}\otimes\mathcal{B}(\lambda)$. For each $k\in\\{1,\cdots,|\mu|\\}$, if
the $k$-th factor in the tensor product above is $\mathcal{B}(\vartheta_{i})$,
then using Proposition 6.7 we have

$\displaystyle\sigma_{|\mu|}\circ$
$\displaystyle\cdots\circ\sigma_{k}(a_{1}\otimes\cdots\otimes a_{|\mu|}\otimes
b)$ $\displaystyle=\sigma_{|\mu|}(a_{1}\otimes\cdots a_{k-1}\otimes
a_{k}^{\prime}\otimes\cdots\otimes a^{\prime}_{|\mu|-1}\otimes c_{i}\otimes
b)$ $\displaystyle=a_{1}\otimes\cdots a_{k-1}\otimes
a_{k}^{\prime}\otimes\cdots\otimes a^{\prime}_{|\mu|-1}\otimes
b^{\prime}\otimes\mathsf{R}_{\vartheta_{i}}(c_{i}\otimes b)\neq 0$

for some $a^{\prime}_{k},\cdots,a^{\prime}_{|\mu|-1}$ and some $b^{\prime}$.
By Proposition 6.7 again, we deduce that $c\otimes b$ is in the Cartan
component. Moreover, $\mathsf{R}_{\vartheta_{i}}(c\otimes
b)=\mathsf{R}_{\vartheta_{i}}(c_{i}\otimes b)$, which proves the final
statement. ∎

In particular, Lemma 7.6 shows that $\mathsf{R}_{\mathsf{C}}(c\otimes b)$
depends only on the right end
$v=\mathsf{R}_{\mathsf{C}}(c)\in\Lambda_{\mathfrak{g},\mathsf{C}}^{0}$ of $c$,
and not on the choice of the element $c$ representing it. Therefore, the
following definition makes sense.

######  Definition 7.7.

We define the _source_ and _range_ maps
$\mathsf{s},\mathsf{r}:\Lambda_{\mathfrak{g},\mathsf{C}}\to\Lambda_{\mathfrak{g},\mathsf{C}}^{0}$
as follows. Let $e=(v,b)$ be a path with $v=(c_{1},\cdots,c_{N})$. Then we
define

$\mathsf{s}(e):=v,\qquad\mathsf{r}(e):=\left(\mathsf{R}_{\vartheta_{1}}(c_{1}\otimes
b),\cdots,\mathsf{R}_{\vartheta_{N}}(c_{N}\otimes b)\right).$

Equivalently, choosing $c\in\mathcal{B}(\rho_{\mathsf{C}})$ such that
$\mathsf{R}_{\mathsf{C}}(c)=v$, we have
$\mathsf{r}(e)=\mathsf{R}_{\mathsf{C}}(c\otimes b)$.

As before, in the equality $\mathsf{r}(e)=\mathsf{R}_{\mathsf{C}}(c\otimes b)$
we can replace $c\in\mathcal{B}(\rho_{\mathsf{C}})$ with any crystal element
$c\in\mathcal{B}(\mu)$, where $\mu\in\mathbf{P}^{+}_{\mathsf{C}}$ and
$\mu\geq\rho_{\mathsf{C}}$, such that $\mathsf{R}_{\mathsf{C}}(c)=v$.

######  Theorem 7.8.

The set $\Lambda_{\mathfrak{g},\mathsf{C}}$ becomes a higher-rank graph of
rank $N$ with:

  * •

degree map $\mathsf{d}$ as in Definition 7.5,

  * •

source and range maps as in Definition 7.7,

  * •

composition of paths
$(v,b),(v^{\prime},b^{\prime})\in\Lambda_{\mathfrak{g},\mathsf{C}}$ with
$\mathsf{r}(v,b)=\mathsf{s}(v^{\prime},b^{\prime})$ defined by

$(v^{\prime},b^{\prime})\cdot(v,b)=(v,\phi(b\otimes b^{\prime})),$ (7.2)

where
$\phi:\mathcal{B}(\lambda)\otimes\mathcal{B}(\lambda^{\prime})\to\mathcal{B}(\lambda+\lambda^{\prime})$
denotes the projection onto the Cartan component.

###### Proof.

First we need to show that the composition law is well-defined. Let $(v,b)$
and $(v^{\prime},b^{\prime})\in\Lambda_{\mathfrak{g},\mathsf{C}}$ with
$\mathsf{r}(v,b)=\mathsf{s}(v^{\prime},b^{\prime})$. By Lemma 7.6, this means
that if we fix any $c\in\mathcal{B}(\rho_{\mathsf{C}})$ with
$\mathsf{R}_{\mathsf{C}}(c)=v$, then $c\otimes b$ is in the Cartan component
of $\mathcal{B}(\rho_{\mathsf{C}})\otimes\mathcal{B}(\lambda)$ and
$\mathsf{R}_{\mathsf{C}}(c\otimes b)=v^{\prime}$. Again using Lemma 7.6, we
get that $c\otimes b\otimes b^{\prime}$ is in the Cartan component of
$\mathcal{B}(\rho_{\mathsf{C}})\otimes\mathcal{B}(\lambda)\otimes\mathcal{B}(\lambda^{\prime})$,
and hence $c\otimes\phi(b\otimes b^{\prime})$ is in the Cartan component of
$\mathcal{B}(\rho_{\mathsf{C}})\otimes\mathcal{B}(\lambda+\lambda^{\prime})$.
This proves that $(v,\phi(b\otimes
b^{\prime}))\in\Lambda_{\mathfrak{g},\mathsf{C}}$.

Associativity follows from the associativity of the tensor product of
crystals. The identity arrows are the paths of length $0$, see the remarks
immediately after Definition 7.5. Thus $\Lambda_{\mathfrak{g},\mathsf{C}}$ is
a (small) category.

If $\mathsf{d}(v,b)=\lambda$ and
$\mathsf{d}(v^{\prime},b^{\prime})=\lambda^{\prime}$ then clearly
$\mathsf{d}(v,\phi(b\otimes b^{\prime}))=\lambda+\lambda^{\prime}$.
Conversely, if $\mathsf{d}(v,b^{\prime\prime})=\lambda+\lambda^{\prime}$, then
$b^{\prime\prime}\in\mathcal{B}(\lambda\otimes\lambda^{\prime\prime})$ and
there is a unique element $b\otimes b^{\prime}$ in the Cartan component of
$\mathcal{B}(\lambda)\otimes\mathcal{B}(\lambda^{\prime})$ with $\phi(b\otimes
b^{\prime})=b^{\prime\prime}$. Fix any $c\in\mathcal{B}(\rho_{\mathsf{C}})$
with $\mathsf{R}_{\mathsf{C}}(c)=v$. Since
$(v,b^{\prime\prime})\in\Lambda_{\mathfrak{g},\mathsf{C}}$, we have that
$c\otimes b\otimes b^{\prime}$ is in the Cartan component of
$\mathcal{B}(\rho_{\mathsf{C}})\otimes\mathcal{B}(\lambda)\otimes\mathcal{B}(\lambda^{\prime})$,
hence both $(\mathsf{R}_{\mathsf{C}}(c),b)=(v,b)$ and
$(\mathsf{R}_{\mathsf{C}}(c\otimes b),b^{\prime})$ are in
$\Lambda_{\mathfrak{g},\mathsf{C}}$. If we put
$v^{\prime}=\mathsf{R}_{\mathsf{C}}(c\otimes b)$ then
$\mathsf{r}(v,b)=v^{\prime}$, and we get a factorization
$(v,b^{\prime\prime})=(v^{\prime},b^{\prime})\cdot(v,b)$. This factorization
is unique, by the uniqueness of $b$ and $b^{\prime}$. ∎

We describe this structure in some detail in the case of our running example.

######  Example 7.9.

Consider $\mathfrak{g}=\mathfrak{sl}_{3}$ with the colours
$\mathsf{C}=(\varpi_{1},\varpi_{2})$. We want to determine the $2$-graph
corresponding to this case.

We can identify the crystal
$\mathcal{B}(\rho)=\mathcal{B}(\varpi_{1}+\varpi_{2})$ with the Cartan
component of the product
$\mathcal{B}(\varpi_{1})\otimes\mathcal{B}(\varpi_{2})$, which consists of $8$
elements. Then using the Cartan braiding (5.4) from Example 5.7, Proposition
6.7 gives

$\begin{array}[]{lll}\mathsf{R}_{\mathsf{C}}(a_{1}\otimes
b_{1})=(a_{1},b_{1}),&\mathsf{R}_{\mathsf{C}}(a_{2}\otimes
b_{1})=(a_{2},b_{1}),&\mathsf{R}_{\mathsf{C}}(a_{3}\otimes
b_{1})=(a_{2},b_{1}),\\\ \mathsf{R}_{\mathsf{C}}(a_{1}\otimes
b_{2})=(a_{1},b_{2}),&\mathsf{R}_{\mathsf{C}}(a_{2}\otimes
b_{2})=(a_{1},b_{2}),&\mathsf{R}_{\mathsf{C}}(a_{3}\otimes
b_{2})=(a_{3},b_{2}),\\\ &\mathsf{R}_{\mathsf{C}}(a_{2}\otimes
b_{3})=(a_{2},b_{3}),&\mathsf{R}_{\mathsf{C}}(a_{3}\otimes
b_{3})=(a_{3},b_{3}).\end{array}$

From these $8$ elements we obtain $6$ distinct right ends, namely

$\displaystyle v_{1}$ $\displaystyle=(a_{1},b_{1}),$ $\displaystyle v_{2}$
$\displaystyle=(a_{1},b_{2}),$ $\displaystyle v_{3}$
$\displaystyle=(a_{2},b_{1}),$ $\displaystyle v_{4}$
$\displaystyle=(a_{2},b_{3}),$ $\displaystyle v_{5}$
$\displaystyle=(a_{3},b_{2}),$ $\displaystyle v_{6}$
$\displaystyle=(a_{3},b_{3}).$

Next, we determine the edges (paths of length $1$) of colour $\varpi_{1}$.
Recall that these consists of pairs $((a_{i},b_{j}),a_{k})$ such that
$a_{i}\otimes a_{k}$ and $b_{j}\otimes a_{k}$ are in their respective Cartan
components. Using the graphs from Example 5.7 we find

$\displaystyle e_{1}$ $\displaystyle=(v_{1},a_{1}),$ $\displaystyle e_{2}$
$\displaystyle=(v_{2},a_{1}),$ $\displaystyle e_{3}$
$\displaystyle=(v_{3},a_{1}),$ $\displaystyle e_{4}$
$\displaystyle=(v_{3},a_{2}),$ $\displaystyle e_{5}$
$\displaystyle=(v_{4},a_{1}),$ $\displaystyle e_{6}$
$\displaystyle=(v_{4},a_{2}),$ $\displaystyle e_{7}$
$\displaystyle=(v_{5},a_{1}),$ $\displaystyle e_{8}$
$\displaystyle=(v_{5},a_{2}),$ $\displaystyle e_{9}$
$\displaystyle=(v_{5},a_{3}),$ $\displaystyle e_{10}$
$\displaystyle=(v_{6},a_{1}),$ $\displaystyle e_{11}$
$\displaystyle=(v_{6},a_{2}),$ $\displaystyle e_{12}$
$\displaystyle=(v_{6},a_{3}).$

The sources and ranges $(\mathsf{s}(e_{i}),\mathsf{r}(e_{i}))$ of these edges
are given by

$\displaystyle e_{1}$ $\displaystyle:(v_{1},v_{1}),$ $\displaystyle e_{2}$
$\displaystyle:(v_{2},v_{2}),$ $\displaystyle e_{3}$
$\displaystyle:(v_{3},v_{1}),$ $\displaystyle e_{4}$
$\displaystyle:(v_{3},v_{3}),$ $\displaystyle e_{5}$
$\displaystyle:(v_{4},v_{2}),$ $\displaystyle e_{6}$
$\displaystyle:(v_{4},v_{4}),$ $\displaystyle e_{7}$
$\displaystyle:(v_{5},v_{2}),$ $\displaystyle e_{8}$
$\displaystyle:(v_{5},v_{3}),$ $\displaystyle e_{9}$
$\displaystyle:(v_{5},v_{5}),$ $\displaystyle e_{10}$
$\displaystyle:(v_{6},v_{2}),$ $\displaystyle e_{11}$
$\displaystyle:(v_{6},v_{4}),$ $\displaystyle e_{12}$
$\displaystyle:(v_{6},v_{6}).$

This gives the following portion of the graph, which we depict in red

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{6}$

Similar computations can be made for the colour $\varpi_{2}$, or alternatively
we can employ the obvious symmetry coming from the Dynkin diagram
automorphism. This leads to the $2$-graph presented in the introduction.

###  7.4. Properties of the higher-rank graph

We now prove that our graphs satisfy the properties from Definition 7.2.

######  Proposition 7.10.

The $N$-graph $\Lambda_{\mathfrak{g},\mathsf{C}}$ is row-finite and has no
sources and sinks.

###### Proof.

Row-finiteness is clear, because
$v\Lambda_{\mathfrak{g},\mathsf{C}}^{\lambda}\subseteq\\{v\\}\otimes\mathcal{B}(\lambda)$,
which is finite.

Now let $v\in\Lambda_{\mathfrak{g},\mathsf{C}}^{0}$ and
$\lambda\in\mathbf{P}^{+}_{\mathsf{C}}$. Pick
$c\in\mathcal{B}(\rho_{\mathsf{C}})$ with $\mathsf{R}_{\mathsf{C}}(c)=v$. Let
$b_{\lambda}$ and $b_{w_{0}\lambda}$ denote the highest and lowest weight
elements of $\mathcal{B}(\lambda)$, respectively. By Lemma 5.2, $c\otimes
b_{\lambda}$ belongs to the Cartan component of
$\mathcal{B}(\rho_{\mathsf{C}})\otimes\mathcal{B}(\lambda)$ and so
$(v,b_{\lambda})\in\Lambda_{\mathfrak{g},\mathsf{C}}^{\lambda}v$.

Likewise, $b_{w_{0}\lambda}\otimes c$ is in the Cartan component of
$\mathcal{B}(\lambda)\otimes\mathcal{B}(\rho_{\mathsf{C}})$. Let
$c^{\prime}\otimes
b=\sigma_{\mathcal{B}(\lambda),\mathcal{B}(\rho_{\mathsf{C}})}(b_{w_{0}\lambda}\otimes
c)$. Then $\mathsf{R}_{\mathsf{C}}(c^{\prime}\otimes
b)=\mathsf{R}_{\mathsf{C}}(b_{w_{0}\lambda}\otimes
c)=\mathsf{R}_{\mathsf{C}}(c)=v$ by Proposition 6.9, so putting
$v^{\prime}=\mathsf{R}_{\mathsf{C}}(c^{\prime})$ we have $(v^{\prime},b)\in
v\Lambda_{\mathfrak{g},\mathsf{C}}^{\lambda}$. ∎

The higher rank graph $\Lambda_{\mathfrak{g},\mathsf{C}}$ is compatible with a
partial ordering on the vertices, as we now describe. Firstly, every
irreducible crystal $\mathcal{B}(\lambda)$ admits a partial ordering by
declaring that $b\leq b^{\prime}$ if and only if $b=\tilde{F}b^{\prime}$ where
$\tilde{F}=\tilde{F}_{i_{1}}\cdots\tilde{F}_{i_{m}}$ is some product of the
Kashiwara operators $\tilde{F}_{i}$. Then for $N$-tuples
$v=(b_{1},\cdots,b_{N})$ and
$v^{\prime}=(b_{1}^{\prime},\cdots,b_{N}^{\prime})$ in
$\Lambda_{\mathfrak{g},\mathsf{C}}^{0}$, we write $v\leq v^{\prime}$ if and
only if $b_{i}\leq b_{i}^{\prime}$ for every $i$.

######  Proposition 7.11.

Consider the partial order $\leq$ on $\Lambda_{\mathfrak{g},\mathsf{C}}^{0}$
defined above. Then:

  1. 1)

for any $e\in\Lambda_{\mathfrak{g},\mathsf{C}}$ we have
$\mathsf{s}(e)\leq\mathsf{r}(e)$,

  2. 2)

there are unique maximal and minimal elements in
$\Lambda_{\mathfrak{g},\mathsf{C}}^{0}$, namely

$\displaystyle v_{\mathrm{max}}=(b_{\vartheta_{1}},\cdots,b_{\vartheta_{N}}),$
$\displaystyle
v_{\mathrm{min}}=(b_{w_{0}\vartheta_{1}},\cdots,b_{w_{0}\vartheta_{N}}),$

where $b_{\lambda}$ and $b_{w_{0}\lambda}$ denote the highest and lowest
weight elements of $\mathcal{B}(\lambda)$, respectively.

###### Proof.

1) Consider a path $e=(v,b)$ with $b\in\mathcal{B}(\lambda)$. Write
$v=(b_{1},\cdots,b_{N})$ for $\mathsf{s}(e)$ and
$v^{\prime}=(b_{1}^{\prime},\cdots,b_{N}^{\prime})$ for $\mathsf{r}(e)$.
Fixing $i$, we have $b^{\prime}_{i}=\mathsf{R}_{\vartheta_{i}}(b_{i}\otimes
b)$, so by Proposition 6.7 we have
$\sigma_{\mathcal{B}(\vartheta_{i}),\mathcal{B}(\lambda)}:b_{i}\otimes
b\mapsto c\otimes b^{\prime}_{i}$ for some $c\in\mathcal{B}(\lambda)$.
Considering the structure of the braiding operators
$\hat{R}_{\mathcal{B}(\vartheta_{i}),\mathcal{B}(\lambda)}$ fixed in (5.1), we
obtain that $b_{i}\leq b_{i}^{\prime}$. The result follows.

2) Consider any $\lambda,\mu\in\mathbf{P}^{+}$ with $\lambda\geq\mu$. The
inclusion
$\mathcal{B}(\lambda)\hookrightarrow\mathcal{B}(\lambda-\mu)\otimes\mathcal{B}(\mu)$
maps $b_{\lambda}$ to $b_{\lambda-\mu}\otimes b_{\mu}$, from which we see that
$\mathsf{R}_{\mu}(b_{\lambda})=b_{\mu}$. It follows that
$\mathsf{R}_{\mathsf{C}}(b_{\rho_{\mathsf{C}}})=(b_{\vartheta_{1}},\cdots,b_{\vartheta_{N}})\in\Lambda_{\mathfrak{g},\mathsf{C}}^{0}$,
and this element is clearly maximal. The result for $v_{\mathrm{min}}$ is
obtained in a similar way. ∎

##  8\. The crystal algebra

###  8.1. Definition of the algebra

In this section we introduce an abstract $*$-algebra
$\mathcal{A}_{\mathfrak{k}}$ which is universal for the generators and
relations for the crystal limit $\mathcal{O}[K_{0}]$ given in Theorem 5.11, so
that we get a surjective $*$-homomorphism
$\mathcal{A}_{\mathfrak{k}}\to\mathcal{O}[K_{0}]$. It provides a convenient
bridge between $\mathcal{O}[K_{0}]$ and the higher-rank graph algebra
$\mathrm{KP}_{\mathbb{C}}(\Lambda_{\mathfrak{g}})$. Ultimately, we are going
to prove that these three algebras are all $*$-isomorphic.