#  Categorified Crystal Structure on Localized Quantum Coordinate Rings

Toshiki Nakashima

###### Abstract.

For the quiver Hecke algebra $R$ associated with a simple Lie algebra, let
$R$-gmod be the category of finite-dimensional graded $R$-modules. It is well-
known that it categorifies the unipotent quantum coordinate ring
${\mathcal{A}}_{q}(\mathfrak{n})$. The localization of $R$-gmod has been
defined in [12], which will be denoted by $\widetilde{R}\hbox{-gmod}$. Its
Grothendieck ring ${\mathcal{K}}(\widetilde{R}\hbox{-gmod})$ defines the
localized (unipotent) quantum coordinate ring
$\widetilde{\mathcal{A}}_{q}(\mathfrak{n})$. We shall give a certain crystal
structure on the localized quantum coordinate ring by regarding the set of
self-dual simple objects $\mathbb{B}(\widetilde{R}\hbox{-gmod})$ in
$\widetilde{R}\hbox{-gmod}$. We also give the isomorphism of crystals from
$\mathbb{B}(\widetilde{R}\hbox{-gmod})$ to the cellular crystal
$\mathbb{B}_{\mathbf{i}}$ for an arbitrary reduced word ${\mathbf{i}}$ of the
longest Weyl group element. This result can be seen as a localized version for
the categorification of the crystal $B(\infty)$ by Lauda-Vazirani [19] since
the crystal $B(\infty)$ is realized in the cellular crystal
$\mathbb{B}_{\mathbf{i}}$.

Division of Mathematics, Sophia University, Kioicho 7-1, Chiyoda-ku, Tokyo
102-8554, Japan<EMAIL_ADDRESS>supported in part by JSPS Grants
in Aid for Scientific Research $\\#$20K03564

##  1\. Introduction

The quiver Hecke algebra has been introduced by Khovanov-Lauda([10, 11]) and
Rouquier([28]) independently, which is a family of $\mathbb{Z}$-graded ${\bf
k}$-algebras $R(\beta)$ defined for $\beta=\sum_{i}m_{i}\alpha_{i}\in Q_{+}$,
where $Q_{+}$ is the positive root lattice and ${\bf k}$ is a field. For a
symmetrizable Kac-Moody Lie algebra $\mathfrak{g}$, the remarkable property on
the algebra $R:=\bigoplus_{\beta\in Q_{+}}R(\beta)$ is that it categorifies
the nilpotent half of the quantum algebra $U^{-}_{q}(\mathfrak{g})$ and the
unipotent quantum coordinate ring ${\mathcal{A}}_{q}(\mathfrak{n})$, that is,
there exist isomorphisms of algebras:

${\mathcal{K}}(R\hbox{\rm-proj})\cong
U^{-}_{q}(\mathfrak{g})_{\mathbb{Z}[q,q^{-1}]}\qquad\qquad{\mathcal{K}}(\hbox{$R$-\hbox{gmod}})\cong{\mathcal{A}}(\mathfrak{n})_{\mathbb{Z}[q,q^{-1}]},$

where ${\mathcal{K}}(R\hbox{\rm-proj})$ (resp.
${\mathcal{K}}(\hbox{$R$-\hbox{gmod}})$) is the Grothendieck ring of
$R\hbox{\rm-proj}$ (resp. $R$-gmod) the category of graded projective(resp.
finite-dimensional) $R$-modules, which is equipped with a multiplication by
the convolution product on modules. In particular, in [29, 30] for a symmetric
$\mathfrak{g}$ this isomorphism yields one to one correspondence between the
set of indecomposable projective $R$-modules (resp. finite-dimensional simple
$R$-modules) and the set of the lower global bases/canonical bases (resp.
upper global bases/dual canonical bases). Those bases have been introduced by
Kashiwara [7] and Lusztig [27] independently.

The theory of crystal bases has been introduced by Kashiwara and it involved
various applications to many areas, such as, mathematical physics [4, 5],
combinatorics [18, 21, 22], cluster algebras [3], modular representations and
LLT conjecture [1, 20], etc. In particular, in [1] Ariki has just open the
door to the method of categorification, which has become the standard in
representation theory nowadays. The categorification as above also played a
significant role in realizing the cluster algebra on the unipotent quantum
coordinate rings, which is a kind of monoidal categorification.

Theory of crystal bases would be integrated into crystals, which is a certain
combinatorial object holding the similar properties of crystal bases and
consists of 6-tuple
$(B,\mathrm{wt},\\{\varepsilon_{i}\\},\\{\varphi_{i}\\},\\{\tilde{e}_{i}\\},\\{\tilde{f}_{i}\\})_{i\in
I}$ with a set $B$, the maps $\tilde{e}_{i},\tilde{f}_{i}:B\sqcup\\{0\\}\to
B\sqcup\\{0\\}$ and the functions
$\varepsilon_{i},\varphi_{i}:B\to\mathbb{Z}\sqcup\\{-\infty\\}$ and
$\mathrm{wt}:B\to P$ satisfy certain conditions (see Definition 3.1). Lauda
and Vazirani [19] introduced the crystal structure on the set of finite
dimensional simple $R$-modules and obtained the outstanding result that this
crystal is isomorphic to the crystal $B(\infty)$ of the nilpotent half of the
quantum algebra $U^{-}_{q}(\mathfrak{g})$, that is, it can be said that they
succeeded in categorifying the crystal $B(\infty)$ in terms of the
$R$-modules.

Theory of crystal base gave rise to another important object ”geometric
crystal”[2], which is a kind of geometric lifting of crystal base and is
defined by replacing the components in the 6-tuple of crystal such as the set
$B$ by a variety $X$, the integer-function $\varepsilon_{i}$ by rational
function $\tilde{\varepsilon}_{i}$ and the map $\tilde{e}_{i}$ (or
$\tilde{f}_{i}$) by some unital rational $\mathbb{C}^{\times}$\- action
$e_{i}^{c}$ on $X$ ($c\in\mathbb{C}^{\times}$) satisfying the specific
conditions. One of the most remarkable feature of geometric crystals is that
some geometric crystals possess ”positive structure” and it yields
”tropicalization functor”, by which one has a Langlands dual crystal. Indeed,
in [25] we considered the geometric crystals on the Schubert cell $X_{w}$
associated with a Weyl group element $w$ and applying the
tropicalization/ultra-discretization functor we obtained a Langlands dual
crystal ${}^{L}\mathbb{B}_{{\mathbf{i}}}$, where ${\mathbf{i}}=i_{1}\cdots
i_{k}$ is a reduced word for $w$ and this crystal called a ”cellular crystal”
associated with a reduced word ${\mathbf{i}}$, which is defined as follows:
Let $B_{i}=\\{(n)_{i}\mid n\in\mathbb{Z}\\}$ ($i\in I$) be the crystal as in
Example 3.4 and then define $\mathbb{B}_{i_{1}i_{2}\cdots
i_{k}}:=B_{i_{1}}\otimes B_{i_{2}}\otimes\cdots\otimes B_{i_{k}}$. Indeed,
since the tensor product of crystals mean a direct product as sets (see
Proposition 3.3), one finds that $\mathbb{B}_{i_{1}i_{2}\cdots i_{k}}$ can be
identified with $\mathbb{Z}^{k}$ as a set. Thus, its crystal structure is
rather simple, but it enables us to realize the crystal base $B(\infty)$ of
the nilpotent half of the quantum algebra $U^{-}_{q}(\mathfrak{g})$ inside the
crystal $\mathbb{B}_{i_{1}i_{2}\cdots i_{k}}$ (see [24, 26]) if $\mathfrak{g}$
is of finite type. The crystal $B_{i}$ and $\mathbb{B}_{\mathbf{i}}$ are
sometimes called virtual crystals, which means that they are not obtained by
crystallization process from any representation-theoretical object, such as,
$U^{-}_{q}(\mathfrak{g})$ or irreducible highest weight modules $V(\lambda)$
but defined as abstract combinatorial objects. Nevertheless, the
tropicalization functor involved a certain representation-theoretical
background on such ”virtual crystals”.

In [12] (see also [6]), Kashiwara et al has invented the ”localization”
functor on graded monoidal categories. For a graded monoidal category
$({\mathcal{T}}=\oplus_{\lambda\in\Lambda}{\mathcal{T}}_{\lambda},\otimes)$,
where $\Lambda$ is a $\mathbb{Z}$-lattice, to construct the localization
functor it is required the ingredients, called a real commuting family of
graded braiders $\\{(C_{i},R_{C_{i}},\phi_{i})\\}_{i\in I}$, which consists of
objects $C_{i}$ in ${\mathcal{T}}$, morphisms
$R_{C_{i}}:(C_{i}\otimes-)\to(-\otimes C_{i})$ and a $\mathbb{Z}$-valued
function $\phi_{i}$ on $\Lambda$ satisfying the conditions as in Definition
5.2. Here note that this real commuting family of graded braiders
$\\{(C_{i},R_{C_{i}},\phi_{i})\\}_{i\in I}$ plays the similar role to a
multiplicative set for the localization of commutative ring theory. The
resultant localized category $\widetilde{\mathcal{T}}$ holds several nice
properties as in Theorem 5.6 and Proposition 5.7.

Applying this localization method to the subcategory ${\mathscr{C}}_{w}$ of
$R$-gmod for a Weyl group element $w\in W$ one has the localized category
$\widetilde{\mathscr{C}}_{w}$. In particular, since for finite type
$\mathfrak{g}$ and the longest element $w=w_{0}$ in $W$,
${\mathscr{C}}_{w_{0}}$ coincides with the category $R$-gmod, then one can say
that through the isomorphism
${\mathcal{K}}(\hbox{$R$-\hbox{gmod}})\cong{\mathcal{A}}(\mathfrak{n})_{\mathbb{Z}[q,q^{-1}]}$
the localized quantum coordinate ring
${\widetilde{\mathcal{A}}_{q}(\mathfrak{n})}_{\mathbb{Z}[q,q^{-1}]}$ has been
defined as ${\mathcal{K}}(\widetilde{R}\hbox{-gmod})$.

The localized category $\widetilde{R}\hbox{-gmod}$ possess several interesting
properties, such as, rigidity defined as in Definition 9.7 and the operation
$\tilde{\mathfrak{a}}$, which is a sort of generalization of the anti-
automorphism $\star$ on $U_{q}(\mathfrak{g})=\langle
e_{i},f_{i},q^{h}\rangle_{i\in I,h\in P^{*}}$ defined by
$e_{i}^{\star}=e_{i}$, $f_{i}^{\star}=f_{i}$ and $(q^{h})^{\star}=q^{-h}$.
There exists the automorphism $\mathfrak{a}$ on the quiver Hecke algebra
$R(\beta)$ as in Sect.9 and then it induces the functor
$\tilde{\mathfrak{a}}:\widetilde{R}\hbox{-gmod}\to\widetilde{R}\hbox{-gmod}$
which satisfies $\tilde{\mathfrak{a}}^{2}\cong{\rm id}$ and
$\tilde{\mathfrak{a}}(X\circ
Y)\cong\tilde{\mathfrak{a}}(Y)\circ\tilde{\mathfrak{a}}(X)$ for
$X,Y\in\widetilde{R}\hbox{-gmod}$. it follows from the fact
${\mathcal{K}}(\widetilde{R}\hbox{-gmod})\cong\widetilde{\mathcal{A}}_{q}(\mathfrak{n})_{\mathbb{Z}[q,q^{-1}]}$
that the functor $\tilde{\mathfrak{a}}$ defines an anti-automorphism on
$\widetilde{\mathcal{A}}_{q}(\mathfrak{n})$ and it can be seen as a
generalization of the anti-automorphism $\star$.

Furthermore, it is remarkable that for any simple object
$M\in\widetilde{R}\hbox{-gmod}$ there exist
$\lambda=\sum_{i}m_{i}\Lambda_{i}\in P$ and a simple
$S\in\hbox{$R$-\hbox{gmod}}$ such that $M\cong C_{\lambda}\circ S$, where
$C_{\lambda}\cong C_{1}^{\circ m_{1}}\circ\cdots\circ C_{n}^{\circ m_{n}}$ up
to grading shift.

In the middle of study for the connectedness of the cellular crystal
$\mathbb{B}_{\mathbf{i}}$ ([17]), we found the following equality under some
condition in [17] and without any condition in Proposition 3.11 below:

$\mathbb{B}_{\mathbf{i}}=\bigcup_{h\in{\mathcal{H}}_{\mathbf{i}}}B^{h}(\infty)_{\mathbf{i}},$

where ${\mathcal{H}}_{\mathbf{i}}$ is a sub $\mathbb{Z}$-lattice of
$\mathbb{B}_{\mathbf{i}}$ given as the solution space of some system of linear
equations (3.24) and

$B^{h}(\infty)_{\mathbf{i}}:=\\{x+h\in\mathbb{Z}^{N}(=\mathbb{B}_{\mathbf{i}})\,|\,x\in
B(\infty)_{\mathbf{i}}\\}\subset\mathbb{B}_{\mathbf{i}},$

which is a subset of $\mathbb{B}_{\mathbf{i}}$ obtained by shifting
$B(\infty)$ by $h\in{\mathcal{H}}_{\mathbf{i}}$. Note that
${\mathcal{H}}_{\mathbf{i}}$ has a natural $\mathbb{Z}$-basis
$\\{{\mathbf{h}}_{i}\mid i\in I\\}$ as in Proposition 3.8 and then any
$h\in{\mathcal{H}}_{\mathbf{i}}$ can be expressed as
$h=\sum_{i}m_{i}{\mathbf{h}}_{i}$. By Lauda-Vazirni results, any simple module
in $R$-gmod corresponds to unique crystal base element in $B(\infty)$ up to
grading shift, where we denote it by $\Psi:\\{\hbox{simple
}S\in\hbox{$R$-\hbox{gmod}}\\}\quad\smash{\mathop{\longrightarrow}\limits^{\sim}}\quad
B(\infty)$. Here, we observe the following correspondence between the set of
simple modules in $\widetilde{R}\hbox{-gmod}$ and $\mathbb{B}_{\mathbf{i}}$ as

(1.1) $\begin{array}[]{ccc}\\{\hbox{self-dual simple
}M\in\widetilde{R}\hbox{-gmod}\\}&\longleftrightarrow&\mathbb{B}_{\mathbf{i}}=\bigcup_{h\in{\mathcal{H}}_{\mathbf{i}}}B^{h}(\infty)_{\mathbf{i}}\\\
\rotatebox{90.0}{$\in$}&&\rotatebox{90.0}{$\in$}\\\
M=C_{\sum_{i}m_{i}\Lambda_{i}}\circ
S&\longleftrightarrow&(\sum_{i}m_{i}{\mathbf{h}}_{i})+\Psi(S).\end{array}$

One of the main objectives of this article is to realize this correspondence
as an isomorphism of crystals. In order to pursue it, first we equip the
crystal structure with the set $\mathbb{B}(\widetilde{R}\hbox{-gmod})=\\{S\mid
S\hbox{ is self-dual simple module}\in\widetilde{R}\hbox{-gmod}\\}$ (Theorem
7.4). To define the crystal structure on
$\mathbb{B}(\widetilde{R}\hbox{-gmod})$, we shall take into account the
construction by Lauda-Vazirani[19] and consider how we shall generalize their
construction to the localized category. The most crucial step to complete the
task is to define the action $\widetilde{E}_{i}$ on
$\mathbb{B}(\widetilde{R}\hbox{-gmod})$ as in (7.6). It is also essential task
to show the well-definedness of all operators and functions on
$\mathbb{B}(\widetilde{R}\hbox{-gmod})$. Then, finally, we shall construct the
isomorphism of crystals
$\widetilde{\Psi}:\mathbb{B}(\widetilde{R}\hbox{-gmod})\smash{\mathop{\longrightarrow}\limits^{\sim}}\mathbb{B}_{\mathbf{i}}$
(Theorem 8.1), which seems to be a localized version of Lauda-Vazirani’s
isomorphism from the set of simple objects of $R$-gmod to
$B(\infty)\subset\mathbb{B}_{\mathbf{i}}$ as mentioned above.

As has been seen, there exists the functor
$\tilde{\mathfrak{a}}:\widetilde{R}\hbox{-gmod}\to\widetilde{R}\hbox{-gmod}$.
We shall show the stability of $\mathbb{B}(\widetilde{R}\hbox{-gmod})$ by
$\tilde{\mathfrak{a}}$, that is,

$\tilde{\mathfrak{a}}(\mathbb{B}(\widetilde{R}\hbox{-gmod}))=\mathbb{B}(\widetilde{R}\hbox{-gmod}),$

which seems to be a generalization of the stability
$B(\infty)^{\star}=B(\infty)$ ([8]).

The organization of the article is as follows. In Sect.2, we fix the setting
of the article and prepare several notation and definitions which will be
needed later parts. In Sect.3, we shall review basic definition and properties
of crystal bases, crystals and the cellular crystal $\mathbb{B}_{\mathbf{i}}$
associated with reduced word of Weyl group element ${\mathbf{i}}=i_{1}\cdots
i_{l}$. Several crucial properties of cellular crystals will be shown. In
Sect.4, we shall review basic definitions and several results on quiver Hecke
algebra and its modules. We also review the results by Lauda-Vazirani on
categorification of crystal $B(\infty)$. Sect.5 is devoted to review
localization method on monoidal categories following [12]. In Sect.6, the
localization method will be applied to the category $R$-gmod using the
derterminantial modules $\\{{\bf M}(w_{0}\Lambda_{i},\Lambda_{i})\mid i\in
I\\}$. Sect.7 introduces one of the main results of this paper that the
localized monoidal category $\widetilde{R}\hbox{-gmod}$ possesses the crystal
$\mathbb{B}(\widetilde{R}\hbox{-gmod})$. In Sect.8, we shall mention another
main results of this article that the crystal
$\mathbb{B}(\widetilde{R}\hbox{-gmod})$ is isomorphic to the cellular crystal
associated with the longest Weyl group element. In the last section, we shall
see several miscellaneous results as applications of the crystal structure on
$\mathbb{B}(\widetilde{R}\hbox{-gmod})$. It will be shown the stability
$\tilde{\mathfrak{a}}(\mathbb{B}(\widetilde{R}\hbox{-gmod}))=\mathbb{B}(\widetilde{R}\hbox{-gmod})$,
which is a generalization of the stability $B(\infty)^{\star}=B(\infty)$ [8].
We also define a certain additive structure on
$\mathbb{B}(\widetilde{R}\hbox{-gmod})$ induced by the one of
$\mathbb{B}_{\mathbf{i}}$ as a free $\mathbb{Z}$-lattice. Finally, several
problems will be raised, which ask how to translate categorical properties of
$\mathbb{B}(\widetilde{R}\hbox{-gmod})$ to combinatorial ones of
$\mathbb{B}_{\mathbf{i}}$, and vice versa. Some of them would be discussed
elsewhere soon.

## Acknowledgement

The author would like to thank Masaki Kashiwara from the bottom of his heart
for valuable discussions and helpful advices. He also thanks Yoshiyuki Kimura
and Hironori Oya for their suggestions.

##  2\. Preliminaries

Let $\mathfrak{g}=\mathfrak{n}\oplus\mathfrak{t}\oplus\mathfrak{n}_{-}=\langle
e_{i},h_{i},f_{i}\rangle_{i\in I:=\\{1,2,\cdots,n\\}}$ be a simple Lie algebra
associated with a Cartan matrix $A=(a_{ij})_{i,j\in I}$ where
$\\{e_{i},\,f_{i},\,h_{i}\\}_{i\in I}$ are the standard Chevalley generators
and $\mathfrak{n}=\langle e_{i}\rangle_{i\in I}$ (resp. $\mathfrak{t}=\langle
h_{i}\rangle_{i\in I}$, $\mathfrak{n}_{-}=\langle f_{i}\rangle_{i\in I}$) is
the positive nilpotent subalgebra (resp. the Cartan subalgebra, the negative
nilpotent subalgebra).

Let $\\{\alpha_{i}\\}_{i\in I}$ be the set of simple roots of $\mathfrak{g}$
and $\langle\quad,\quad\rangle$ a pairing on
$\mathfrak{t}\times\mathfrak{t}^{*}$ satisfying $a_{ij}=(\langle
h_{i},\alpha_{j}\rangle)_{i,j\in I}$. We also define a symmetric bilinear form
$(\,,\,)$ on $\mathfrak{t}^{*}$ such that $(\alpha_{i},\alpha_{i})\in
2\mathbb{Z}_{>0}$ and $\langle
h_{i},\lambda\rangle={{2(\alpha_{i},\lambda)}\over{(\alpha_{i},\alpha_{i})}}$
for $\lambda\in\mathfrak{t}^{*}$.

Let $P:=\\{\lambda\in\mathfrak{t}^{*}\,|\,\langle
h_{i},\lambda\rangle\in\mathbb{Z}\,\,\hbox{for any }i\in I\\}$ be the weight
lattice and $P_{+}:=\\{\lambda\in P\mid\langle h_{i},\lambda\rangle\geq
0\hbox{ for any }i\in I\\}$ the set of dominant weights. Set $Q:=\oplus_{i\in
I}\mathbb{Z}\alpha_{i}$ (resp. $Q_{+}:=\sum_{i\in I}\mathbb{Z}_{\geq
0}\alpha_{i}$), which is called the root lattice (resp. positive root
lattice). For an element $\beta=\sum_{i}m_{i}\alpha_{i}\in Q_{+}$ define
$|\beta|=\sum_{i}m_{i}$, which is called the height of $\beta$. Let $W=\langle
s_{i}\,|\,s_{i}\rangle_{i\in I}$ be the Weyl group associated with $P$, where
$s_{i}$ is the simple reflection defined by $s_{i}(\lambda)=\lambda-\langle
h_{i},\lambda\rangle\alpha_{i}$ $(\lambda\in P)$.

We denote the dual weight lattice of $P$ by
$P^{*}:=\\{h\in\mathfrak{t}\,|\,\langle h,P\rangle\subset\mathbb{Z}\\}$. Let
$U_{q}(\mathfrak{g}):=\langle e_{i},\,f_{i},\,q^{h}\rangle_{i\in I,h\in
P^{*}}$ be the quantum algebra associated with $\mathfrak{g}$ with the
defining relations (see e.g.,[7, 8]) and $U^{-}_{q}(\mathfrak{g}):=\langle
f_{i}\rangle_{i\in I}$ (resp. $U^{+}_{q}(\mathfrak{g}):=\langle
e_{i}\rangle_{i\in I}$) the negative (resp. positive) nilpotent subalgebras of
$U_{q}(\mathfrak{g})$. We also define the $\mathbb{Z}$-form
$U^{-}_{\mathbb{Z}[q,q^{-1}]}(\mathfrak{g})$ of $U^{-}_{q}(\mathfrak{g})$ as
in [12].

Now, let us define the (unipotent) quantum coordinate ring
${\mathcal{A}}_{q}(\mathfrak{n})$ by

${\mathcal{A}}_{q}(\mathfrak{n})=\bigoplus_{\beta\in
Q_{-}}{\mathcal{A}}_{q}(\mathfrak{n})_{\beta}\qquad{\mathcal{A}}_{q}(\mathfrak{n})_{\beta}:={\rm
Hom}_{\mathbb{Q}(q)}(U^{+}_{q}(\mathfrak{g})_{-\beta},\mathbb{Q}(q))$

Note that $U^{-}_{q}(\mathfrak{g})\cong{\mathcal{A}}_{q}(\mathfrak{n})$ as a
$\mathbb{Q}(q)$-algebra. The $\mathbb{Z}$-form
${\mathcal{A}}(\mathfrak{n})_{\mathbb{Z}[q,q^{-1}]}$ is defined as in [12].

##  3\. Crystal Bases and Crystals

###  3.1. Crystal Base of
$U^{-}_{q}(\mathfrak{g})\cong{\mathcal{A}}_{q}(\mathfrak{n})$

As in [7], there exists the crystal base $(L(\infty),B(\infty))$ of
$U^{-}_{q}(\mathfrak{g})$ defined by

$\displaystyle L(\infty):=\sum_{k\geq 0,i_{1},\cdots,i_{k}\in
I}\mathbb{A}\tilde{f}_{i_{1}}\cdots\tilde{f}_{i_{k}}u_{\infty},\qquad
B(\infty)=\\{\tilde{f}_{i_{1}}\cdots\tilde{f}_{i_{k}}u_{\infty}\,{\rm
mod}\,qL(\infty)\,|\,k\geq 0,i_{1},\cdots,i_{k}\in I\\}\setminus\\{0\\},$
$\displaystyle\varepsilon_{i}(b)={\rm max}\\{k:\tilde{e}_{i}^{k}b\neq
0\\},\quad\varphi_{i}(b)=\varepsilon_{i}(b)+\langle
h_{i},\mathrm{wt}(b)\rangle,$

where $u_{\infty}=1\in U_{q}(\mathfrak{g})$, $\tilde{e}_{i}$ and
$\tilde{f}_{i}\in{\rm End}_{\mathbb{Q}(q)}(U_{q}(\mathfrak{g}))$ are the
Kashiwara operators ([7]) and $\mathbb{A}\subset\mathbb{Q}(q)$ is the local
subring at $q=0$.

###  3.2. Crystals

We define the notion of crystal as in [8], which is an combinatorial object
abstracting the properties of crystal bases:

######  Definition 3.1 ([8]).

A 6-tuple
$(B,\mathrm{wt},\\{\varepsilon_{i}\\},\\{\varphi_{i}\\},\\{\tilde{e}_{i}\\},\\{\tilde{f}_{i}\\})_{i\in
I}$ is a crystal if $B$ is a set and there exists a certain special element
$0$ outside of $B$ and maps:

(3.1) $\displaystyle{\rm wt}:B\to
P,\quad\varepsilon_{i}:B\to\mathbb{Z}\sqcup\\{-\infty\\},\quad\varphi_{i}:B\to\mathbb{Z}\sqcup\\{-\infty\\}\quad\,(i\in
I),$ (3.2) $\displaystyle\tilde{e}_{i}:B\sqcup\\{0\\}\to
B\sqcup\\{0\\},\quad\tilde{f}_{i}:B\sqcup\\{0\\}\to B\sqcup\\{0\\}\,\,(i\in
I),$

satisfying :

  1. (1)

$\varphi_{i}(b)=\varepsilon_{i}(b)+\langle h_{i},\mathrm{wt}(b)\rangle$.

  2. (2)

If $b,\tilde{e}_{i}b\in B$, then
$\mathrm{wt}(\tilde{e}_{i}b)=\mathrm{wt}(b)+\alpha_{i}$,
$\varepsilon_{i}(\tilde{e}_{i}b)=\varepsilon_{i}(b)-1$,
$\varphi_{i}(\tilde{e}_{i}b)=\varphi_{i}(b)+1$.

  3. (3)

If $b,\tilde{f}_{i}b\in B$, then
$\mathrm{wt}(\tilde{f}_{i}b)=\mathrm{wt}(b)-\alpha_{i}$,
$\varepsilon_{i}(\tilde{f}_{i}b)=\varepsilon_{i}(b)+1$,
$\varphi_{i}(\tilde{f}_{i}b)=\varphi_{i}(b)-1$.

  4. (4)

For $b,b^{\prime}\in B$ and $i\in I$, one has $\tilde{f}_{i}b=b^{\prime}$ iff
$b=\tilde{e}_{i}b^{\prime}.$

  5. (5)

If $\varphi_{i}(b)=-\infty$ for $b\in B$, then
$\tilde{e}_{i}b=\tilde{f}_{i}b=0$ and $\tilde{e}_{i}(0)=\tilde{f}_{i}(0)=0$.

Here, a crystal graph of crystal $B$ is a $I$-colored oriented graph defined
by
$b\smash{\mathop{\longrightarrow}\limits^{i}}b^{\prime}\Leftrightarrow\tilde{f}_{i}(b)=b^{\prime}$
for $b,b^{\prime}\in B$.

######  Definition 3.2 ([8]).

For crystals $B_{1}$ and $B_{2}$, $\Psi$ is a strict embedding (resp.
isomorphism) from $B_{1}$ to $B_{2}$ if $\Psi:B_{1}\sqcup\\{0\\}\to
B_{2}\sqcup\\{0\\}$ is an injective (resp. bijective) map satisfying that
$\Psi(0)=0$, $\mathrm{wt}(\Psi(b))=\mathrm{wt}(b)$,
$\varepsilon_{i}(\Psi(b))=\varepsilon_{i}(b)$ and
$\varphi_{i}(\Psi(b))=\varphi_{i}(b)$ for any $b\in B_{1}$ and $\Psi$ commutes
with all $\tilde{e}_{i}$’s and $\tilde{f}_{i}$’s,.

We obtain the tensor structure of crystals as follows([7, 8]):

######  Proposition 3.3.

For crystals $B_{1}$ and $B_{2}$, set

$B_{1}\otimes B_{2}=\\{b_{1}\otimes b_{2}:=(b_{1},b_{2})\mid b_{1}\in
B_{1},\,b_{2}\in B_{2}\\}(=B_{1}\times B_{2}).$

Then, $B_{1}\otimes B_{2}$ becomes a crystal by defining:

(3.3) $\displaystyle wt(b_{1}\otimes b_{2})=wt(b_{1})+wt(b_{2}),$ (3.4)
$\displaystyle\varepsilon_{i}(b_{1}\otimes
b_{2})={\hbox{max}}(\varepsilon_{i}(b_{1}),\varepsilon_{i}(b_{2})-\langle
h_{i},wt(b_{1})\rangle),$ (3.5) $\displaystyle\varphi_{i}(b_{1}\otimes
b_{2})={\hbox{max}}(\varphi_{i}(b_{2}),\varphi_{i}(b_{1})+\langle
h_{i},wt(b_{2})\rangle),$ (3.8) $\displaystyle\tilde{e}_{i}(b_{1}\otimes
b_{2})=\left\\{\begin{array}[]{ll}\tilde{e}_{i}b_{1}\otimes b_{2}&{\mbox{ if
}}\varphi_{i}(b_{1})\geq\varepsilon_{i}(b_{2})\\\
b_{1}\otimes\tilde{e}_{i}b_{2}&{\mbox{ if
}}\varphi_{i}(b_{1})<\varepsilon_{i}(b_{2}),\end{array}\right.$ (3.11)
$\displaystyle\tilde{f}_{i}(b_{1}\otimes
b_{2})=\left\\{\begin{array}[]{ll}\tilde{f}_{i}b_{1}\otimes b_{2}&{\mbox{ if
}}\varphi_{i}(b_{1})>\varepsilon_{i}(b_{2})\\\
b_{1}\otimes\tilde{f}_{i}b_{2}&{\mbox{ if
}}\varphi_{i}(b_{1})\leq\varepsilon_{i}(b_{2}).\end{array}\right.$

######  Example 3.4.

For $i\in I$, set $B_{i}:=\\{(n)_{i}\,|\,n\in\mathbb{Z}\\}$ and

$\displaystyle\mathrm{wt}((n)_{i})=n\alpha_{i},\,\,\varepsilon_{i}((n)_{i})=-n,\,\,\varphi_{i}((n)_{i})=n,\,\,$
$\displaystyle\varepsilon_{j}((n)_{i})=\varphi_{j}((n)_{i})=-\infty\,\,\,(i\neq
j),$
$\displaystyle\tilde{e}_{i}((n)_{i})=(n+1)_{i},\quad\tilde{f}_{i}((n)_{i})=(n-1)_{i},$
$\displaystyle\tilde{e}_{j}((n)_{i})=\tilde{f}_{j}((n)_{i})=0\,\,\,(i\neq j).$

Then $B_{i}$ ($i\in I$) possesses a crystal structure. Note that as a set the
crystal $B_{i}$ can be identified with the set of integers $\mathbb{Z}$.

###  3.3. Explicit structure of the crystal $B_{i_{1}}\otimes\cdots\otimes
B_{i_{m}}$

In this subsection, we shall investigate explicit structure of a tensor
product of the crystal $B_{i}$’s.

Fix a sequence of indices ${\mathbf{i}}=(i_{1},\cdots,i_{m})\in I^{m}$ and
write

$(x_{1},\cdots,x_{m}):=\tilde{f}_{i_{1}}^{x_{1}}(0)_{i_{1}}\otimes\cdots\otimes\tilde{f}_{i_{m}}^{x_{m}}(0)_{i_{m}}=(-x_{1})_{i_{1}}\otimes\cdots\otimes(-x_{m})_{i_{m}},$

where if $n<0$, then $\tilde{f}_{i}^{n}(0)_{i}$ means
$\tilde{e}_{i}^{-n}(0)_{i}$. Note that here we do not necessarily assume that
${\mathbf{i}}$ is a reduced word though later we will take ${\mathbf{i}}$ to
be a reduced longest word. By the tensor structure of crystals in Proposition
3.3, for the sequence ${\mathbf{i}}$ as above, we can describe the explicit
crystal structure on $\mathbb{B}_{\mathbf{i}}:=B_{i_{1}}\otimes\cdots\otimes
B_{i_{m}}$ as follows: For $x=(x_{1},\cdots,x_{m})\in\mathbb{B}_{\mathbf{i}}$,
define

$\sigma_{k}(x):=x_{k}+\sum_{j<k}\langle h_{i_{k}},\alpha_{i_{j}}\rangle x_{j}$

and for $i\in I$ define

$\displaystyle\widetilde{\sigma}^{(i)}(x):={\rm max}\\{\sigma_{k}(x)\,|\,1\leq
k\leq m\,{\rm and}\,i_{k}=i\\},$
$\displaystyle\widetilde{M}^{(i)}=\widetilde{M}^{(i)}(x):=\\{k\,|\,1\leq k\leq
m,\,i_{k}=i,\,\sigma_{k}(x)=\widetilde{\sigma}^{(i)}(x)\\},$
$\displaystyle\widetilde{m}^{(i)}_{f}=\widetilde{m}^{(i)}_{f}(x):={\rm
max}\,\widetilde{M}^{(i)}(x),\quad\widetilde{m}^{(i)}_{e}=\widetilde{m}^{(i)}_{e}(x):=\min\,\widetilde{M}^{(i)}(x).$

Now, the actions of the Kashiwara operators $\tilde{e}_{i},\tilde{f}_{i}$ and
the functions $\varepsilon_{i},\varphi_{i}$ and $\mathrm{wt}$ are written
explicitly:

(3.12)
$\displaystyle\tilde{f}_{i}(x)_{k}:=x_{k}+\delta_{k,\widetilde{m}^{(i)}_{f}},\qquad\qquad\tilde{e}_{i}(x)_{k}:=x_{k}-\delta_{k,\widetilde{m}^{(i)}_{e}},$
(3.13)
$\displaystyle\mathrm{wt}(x):=-\sum_{k=1}^{m}x_{k}\alpha_{i_{k}},\quad\varepsilon_{i}(x):=\widetilde{\sigma}^{(i)}(x),\quad\varphi_{i}(x):=\langle
h_{i},\mathrm{wt}(x)\rangle+\varepsilon_{i}(x).$

Define the function $\beta^{({\mathbf{i}})}_{k}$ on $\mathbb{B}_{\mathbf{i}}$
by :

(3.14)
$\beta^{({\mathbf{i}})}_{k}(x):=\sigma_{k^{+}}(x)-\sigma_{k}(x)=x_{k}+\sum_{k<j<k^{+}}\langle
h_{i},\alpha_{i_{j}}\rangle x_{j}+x_{k^{+}},$

for $x=(x_{1},\cdots,x_{m})\in\mathbb{B}_{\mathbf{i}}$, where for $k\in[1,N]$,
$k^{+}$ (resp. $k^{-}$) is the minimum (resp. maximum) number $j\in[1,N]$ such
that $k<j$ (resp. $l<k$) and $i_{k}=i_{j}$ if it exists, otherwise $N+1$
(resp. 0). Here one knows that $\widetilde{m}^{(i)}_{f}(x)$ and
$\widetilde{m}^{(i)}_{e}(x)$ are determined by
$\\{\beta^{({\mathbf{i}})}_{k}(x)\,|1\leq k\leq N,\,\,i_{k}=i\\}$.

###  3.4. Braid-type isomorphism

Set $c_{ij}:=\langle h_{i},\alpha_{j}\rangle\langle h_{j},\alpha_{i}\rangle$,
$c_{1}:=-\langle h_{i},\alpha_{j}\rangle$ and $c_{2}:=-\langle
h_{j},\alpha_{i}\rangle$. In the sequel, for $x\in\mathbb{Z}$, put

$x_{+}:=\begin{cases}x&\hbox{ if }x\geq 0,\\\ 0&\hbox{ if }x<0.\end{cases}$

######  Proposition 3.5 ([23]).

There exist the following isomorphisms of crystals $\phi^{(k)}_{ij}$
($k=0,1,2,3$)

  1. (1)

If $c_{ij}=0$,

(3.15) $\phi^{(0)}_{ij}:B_{i}\otimes
B_{i}\smash{\mathop{\longrightarrow}\limits^{\sim}}B_{j}\otimes B_{i},$

where $\phi^{(0)}_{ij}((x)_{i}\otimes(y)_{j})=(y)_{j}\otimes(x)_{i}$.

  2. (2)

If $c_{ij}=1$,

(3.16) $\phi^{(1)}_{ij}:B_{i}\otimes B_{j}\otimes
B_{i}\smash{\mathop{\longrightarrow}\limits^{\sim}}B_{j}\otimes B_{i}\otimes
B_{j},$

where

$\phi^{(1)}_{ij}((x)_{i}\otimes(y)_{j}\otimes(z)_{i})=(z+(-x+y-z)_{+})_{j}\otimes(x+z)_{i}\otimes(y-z-(-x+y-z)_{+})_{j}.$

  3. (3)

If $c_{ij}=2$,

(3.17) $\phi^{(2)}_{ij}:B_{i}\otimes B_{j}\otimes B_{i}\otimes
B_{j}\smash{\mathop{\longrightarrow}\limits^{\sim}}B_{j}\otimes B_{i}\otimes
B_{j}\otimes B_{i},$

where $\phi^{(2)}_{ij}$ is given by the following: for
$(x)_{i}\otimes(y)_{j}\otimes(z)_{i}\otimes(w)_{j}$ we set
$(X)_{j}\otimes(Y)_{i}\otimes(Z)_{j}\otimes(W)_{i}:=\phi^{(2)}_{ij}((x)_{i}\otimes(y)_{j}\otimes(z)_{i}\otimes(w)_{j})$.

(3.18) $\displaystyle X$ $\displaystyle=$ $\displaystyle
w+(-c_{2}x+y-w+c_{2}(x-c_{1}y+z)_{+})_{+},$ (3.19) $\displaystyle Y$
$\displaystyle=$ $\displaystyle x+c_{1}w+(-x+z-c_{1}w+(x-c_{1}y+z)_{+})_{+},$
(3.20) $\displaystyle Z$ $\displaystyle=$ $\displaystyle
y-(-c_{2}x+y-w+c_{2}(x-c_{1}y+z)_{+})_{+},$ (3.21) $\displaystyle W$
$\displaystyle=$ $\displaystyle z-c_{1}w-(-x+z-c_{1}w+(x-c_{1}y+z)_{+})_{+}.$

  4. (4)

If $c_{ij}=3$, the map

(3.22) $\phi^{(3)}_{ij}:B_{i}\otimes B_{j}\otimes B_{i}\otimes B_{j}\otimes
B_{i}\otimes B_{j}\smash{\mathop{\longrightarrow}\limits^{\sim}}B_{j}\otimes
B_{i}\otimes B_{j}\otimes B_{i}\otimes B_{j}\otimes B_{i},$

is defined by the following: for
$(x)_{i}\otimes(y)_{j}\otimes(z)_{i}\otimes(u)_{j}\otimes(v)_{i}\otimes(w)_{j}$
we set $A:=-x+c_{1}y-z$, $B:=-y+c_{2}z-u$, $C:=-z+c_{1}u-v$ and
$D:=-u+c_{2}v-w$. Then
$(X)_{j}\otimes(Y)_{i}\otimes(Z)_{j}\otimes(U)_{i}\otimes(V)_{j}\otimes(W)_{i}:=\phi^{(3)}_{ij}((x)_{i}\otimes(y)_{j}\otimes(z)_{i}\otimes(u)_{j}\otimes(v)_{i}\otimes(w)_{j})$
is given by

$\displaystyle X$ $\displaystyle=$ $\displaystyle
w+(D+(c_{2}C+(2B+A_{+})_{+})_{+})_{+},$ $\displaystyle Y$ $\displaystyle=$
$\displaystyle x+c_{1}w+(c_{1}D+(3C+(2c_{1}B+2A_{+})_{+})_{+})_{+},$
$\displaystyle Z$ $\displaystyle=$ $\displaystyle y+u+w-X-V,$ $\displaystyle
U$ $\displaystyle=$ $\displaystyle x+z+v-Y-W,$ $\displaystyle V$
$\displaystyle=$ $\displaystyle
u-w-(2D+(2c_{2}C+(3B+c_{2}A_{+})_{+})_{+})_{+},$ $\displaystyle W$
$\displaystyle=$ $\displaystyle
v-c_{1}w-(c_{1}D+(2C+(c_{1}B+A_{+})_{+})_{+})_{+}.$

They also satisfy $\phi_{ij}^{(k)}\circ\phi_{ji}^{(k)}={\rm id}$.

We call such isomorphisms of crystals braid-type isomorphisms.

We also define a braid-move on the set of reduced words of $w\in W$ to be a
composition of the following transformations induced from braid relations:

$\displaystyle\cdots ij\cdots\to\cdots ji\cdots(c_{ij}=0),\quad\cdots
iji\cdots\to\cdots jij\cdots(c_{ij}=1),$ $\displaystyle\cdots
ijij\cdots\to\cdots jiji\cdots(c_{ij}=2),\quad\cdots ijijij\cdots\to\cdots
jijiji\cdots(c_{ij}=3),$

which are called by 2-move, 3-move, 4-move, 6-move respectively.

###  3.5. Cellular Crystal
$\mathbb{B}_{\mathbf{i}}=\mathbb{B}_{i_{1}i_{2}\cdots
i_{k}}=B_{i_{1}}\otimes\cdots\otimes B_{i_{k}}$

For a reduced word ${\mathbf{i}}=i_{1}i_{2}\cdots i_{k}$ of some Weyl group
element, we call the crystal
$\mathbb{B}_{\mathbf{i}}:=B_{i_{1}}\otimes\cdots\otimes B_{i_{k}}$ a cellular
crystal associated with a reduced word ${\mathbf{i}}$. Indeed, it is obtained
by applying the tropicalization functor to the geometric crystal on the
Langlands-dual Schubert cell ${}^{L}X_{w}$, where $w=s_{i_{1}}\cdots
s_{i_{k}}$ is an element of the Well group $W$ ([25]). It is immediate from
the braid-type isomorphisms that for any $w\in W$ and its reduced words
$i_{1}\cdots i_{l}$ and $j_{1}\cdots j_{l}$, we get the following isomorphism
of crystals:

(3.23) $B_{i_{1}}\otimes\cdots\otimes B_{i_{l}}\cong
B_{j_{1}}\otimes\cdots\otimes B_{j_{l}}.$

###  3.6. Half potential and the crystal $B(\infty)$

For a Laurent polynomial $\phi(x_{1},\cdots,x_{n})$ with positive
coefficients, the tropicalization of $\phi$ is denoted by
$\widetilde{\phi}:={\rm Trop}(\phi)$, which is given by the rule: ${\rm
Trop}(ax+by)={\rm min}(x,y)$ with $a,b>0$, ${\rm Trop}(xy)=x+y$ and ${\rm
Trop}(x/y)=x-y$ and ${\rm Trop}(c)=0$ for $c>0$. In [17], the crystal
$B(\infty)$ has been realized as a certain subset of $\mathbb{B}_{\mathbf{i}}$
defined as follows:

######  Theorem 3.6 ([17, Theorem 5.11]).

Define the subset of $\mathbb{B}_{\mathbf{i}}$:

$(\widetilde{\mathbb{B}}^{-}_{w_{0}})_{\Phi^{(+)},\Theta_{\mathbf{i}}}=\\{x=(x_{1},\cdots,x_{N})\in\mathbb{B}_{\mathbf{i}}\mid\widetilde{\Phi}^{(+)}(x)\geq
0\\},$

where $\mathbb{B}^{-}_{w_{0}}$ is a certain geometric crystal,
$\widetilde{\Phi}^{(+)}$ is a tropicalization of the half potential
$\Phi^{(+)}$ which is a Laurent polynomial with positive coefficients in $N$
variables and $\Theta_{\mathbf{i}}$ is a certain positive structure on the
geometric crystal $\mathbb{B}^{-}_{w_{0}}$. Then,
$(\widetilde{\mathbb{B}}^{-}_{w_{0}})_{\Phi^{(+)},\Theta_{\mathbf{i}}}\cong
B(\infty)$.

######  Remark 3.7.

To define the crystal structure on
$(\widetilde{\mathbb{B}}^{-}_{w_{0}})_{\Phi^{(+)},\Theta_{\mathbf{i}}}$, it is
supposed that if
$\tilde{e}_{i}x\not\in(\widetilde{\mathbb{B}}^{-}_{w_{0}})_{\Phi^{(+)},\Theta_{\mathbf{i}}}$,
then $\tilde{e}_{i}x=0$. Thus, in this sense, the embedding
$B(\infty)\cong(\widetilde{\mathbb{B}}^{-}_{w_{0}})_{\Phi^{(+)},\Theta_{\mathbf{i}}}\hookrightarrow\mathbb{B}_{\mathbf{i}}$
is not a strict embedding. In [26, 25], it has been given the strict embedding
of $B(\infty)\hookrightarrow\mathbb{B}_{\mathbf{i}}$, which is called
”Kashiwara embedding” and the method to describe the image of this embedding,
called ”polyhedral realization”.

###  3.7. Subspace ${\mathcal{H}}_{\mathbf{i}}$

Fix a reduced longest word ${\mathbf{i}}=i_{1}\cdots i_{N}$ and take the
function $\beta^{({\mathbf{i}})}_{k}(x)=x_{k}+\sum_{k<j<k^{+}}\langle
h_{i_{k}}\alpha_{i_{j}}\rangle x_{j}+x_{k^{+}}\quad(1\leq k\leq N)$ as in
(3.14). In what follows, let us identify the $\mathbb{Z}$-lattice
$\mathbb{Z}^{N}$ with $B_{\mathbf{i}}$ and then we define the summation of
elements $x=(x_{1},\cdots,x_{N})$ and $y=(y_{1},\cdots,y_{N})$ by
$x+y=(x_{1}+y_{1},\cdots,x_{N}+y_{N})$ as a standard one in $\mathbb{Z}^{N}$.
Here, we define the subspace ${\mathcal{H}}_{\mathbf{i}}\subset\mathbb{Z}^{N}$
by

(3.24)
${\mathcal{H}}_{\mathbf{i}}:=\\{x\in\mathbb{Z}^{N}(=\mathbb{B}_{\mathbf{i}})\,|\,\beta^{({\mathbf{i}})}_{k}(x)=0\hbox{
for any }k{\hbox{ such that }}k^{+}\leq N\\}\subset\mathbb{B}_{\mathbf{i}}.$

The following result was presented in [17]:

######  Proposition 3.8 ([17]).

For ${\mathbf{i}}=i_{1}i_{2}\cdots i_{N}$, $k=1,2,\cdots,N$ and a fundamental
weight $\Lambda_{i}$, set

(3.25) $h^{(k)}_{i}:=\langle h_{i_{k}},s_{i_{k+1}}\cdots
s_{i_{N}}\Lambda_{i}\rangle\quad\hbox{ and
}\quad{\mathbf{h}}_{i}:=(h^{(1)}_{i},h^{(2)}_{i},\ldots,h^{(N)}_{i})\in\mathbb{B}_{\mathbf{i}}$

Then, we obtain that $\\{{\mathbf{h}}_{1},\cdots,{\mathbf{h}}_{N}\\}$ is a
$\mathbb{Z}$-basis of ${\mathcal{H}}_{\mathbf{i}}$, namely,

(3.26)
${\mathcal{H}}_{\mathbf{i}}=\mathbb{Z}{\mathbf{h}}_{1}\oplus\mathbb{Z}{\mathbf{h}}_{2}\oplus\cdots\oplus\mathbb{Z}{\mathbf{h}}_{n}.$

######  Example 3.9.

In $\mathfrak{g}=G_{2}$-case. Set $a_{12}=-1$ and $a_{21}=-3$. Taking a
reduced longest word ${\mathbf{i}}=121212$, one has

$\beta^{({\mathbf{i}})}_{1}(x)=x_{1}-x_{2}+x_{3},\quad\beta^{({\mathbf{i}})}_{2}(x)=x_{2}-3x_{3}+x_{4},\quad\beta^{({\mathbf{i}})}_{3}(x)=x_{3}-x_{4}+x_{5},\quad\beta^{({\mathbf{i}})}_{4}(x)=x_{4}-3x_{5}+x_{6}.$

By the formula (3.25), one gets

${\mathbf{h}}_{1}=(1,3,2,3,1,0),\qquad{\mathbf{h}}_{2}=(0,1,1,2,1,1).$

Then the solution space ${\mathcal{H}}_{\mathbf{i}}$ of
$\beta^{({\mathbf{i}})}_{1}(x)=\beta^{({\mathbf{i}})}_{2}(x)=\beta^{({\mathbf{i}})}_{3}(x)=\beta^{({\mathbf{i}})}_{4}(x)=0$
is given by

${\mathcal{H}}_{\mathbf{i}}=\\{{c_{1}}{\mathbf{h}}_{1}+{c_{2}}{\mathbf{h}}_{2}=({c_{1}},{c_{2}}+{3c_{1}},{c_{2}}+{2c_{1}},{2c_{2}}+{3c_{1}},{c_{2}}+{c_{1}},{c_{2}})\mid
c_{1},c_{2}\in\mathbb{Z}\\}.$

######  Lemma 3.10.

The braid-type isomorphisms are well-defined on ${\mathcal{H}}_{\mathbf{i}}$,
that is,
$\phi_{ij}^{(k)}({\mathcal{H}}_{\mathbf{i}})={\mathcal{H}}_{{\mathbf{i}}^{\prime}}$,
where ${\mathbf{i}}^{\prime}$ is the reduced word obtained by applying the
corresponding braid-moves. We also obtain the following formula:

  1. (1)

For any
$h=(\cdots,x,y,\cdots)=\cdots\otimes(-x)_{i}\otimes(-y)_{j}\otimes\cdots\in{\mathcal{H}}_{\mathbf{i}}$,
assume that $a_{ij}=a_{ji}=0$. Applying the braid-type isomorphism
$\phi_{ij}^{(0)}$ on $(x,y)$ in $h$, we have

(3.27)
$\phi^{(0)}_{ij}(h)=(\cdots,y,x,\cdots)=\cdots\otimes(-y)_{j}\otimes(-x)_{i}\otimes\cdots\in{\mathcal{H}}_{{\mathbf{i}}^{\prime}}$

  2. (2)

For any
$h=(\cdots,x,y,z,\cdots)=\cdots\otimes(-x)_{i}\otimes(-y)_{j}\otimes(-z)_{i}\otimes\cdots\in{\mathcal{H}}_{\mathbf{i}}$,
assume that $a_{ij}=a_{ji}=-1$. Applying the braid-type isomorphism
$\phi_{ij}^{(1)}$ on $(x,y,z)$ in $h$, we have

(3.28)
$\phi^{(1)}_{ij}(h)=(\cdots,z,y,x,\cdots)=\cdots\otimes(-z)_{j}\otimes(-y)_{i}\otimes(-x)_{j}\otimes\cdots\in{\mathcal{H}}_{{\mathbf{i}}^{\prime}}$

  3. (3)

For
$h=(\cdots,x,y,z,w,\cdots)=\cdots\otimes(-x)_{i}\otimes(-y)_{j}\otimes(-z)_{i}\otimes(-w)_{j}\cdots\in{\mathcal{H}}_{\mathbf{i}}$,
assume that $a_{ij}\cdot a_{ji}=2$. Applying the braid-type isomorphism
$\phi_{ij}^{(2)}$ on $(x,y,z,w)$ in $h$, we have

(3.29)
$\phi^{(2)}_{ij}(h)=(\cdots,w,z,y,x,\cdots)=\cdots\otimes(-w)_{j}\otimes(-z)_{i}\otimes(-y)_{j}\otimes(-x)_{i}\otimes\cdots\in{\mathcal{H}}_{{\mathbf{i}}^{\prime}}$

  4. (4)

For
$h=(\cdots,x,y,z,u,v,w,\cdots)=\cdots\otimes(-x)_{i}\otimes(-y)_{j}\otimes(-z)_{i}\otimes(-u)_{j}\otimes(-v)_{i}\otimes(-w)_{j}\cdots\in{\mathcal{H}}_{\mathbf{i}}$,
assume that $a_{ij}\cdot a_{ji}=3$. Applying the braid-type isomorphism
$\phi_{ij}^{(3)}$ on $(x,y,z,u,v,w)$ in $h$, we have

(3.30)
$\phi^{(3)}_{ij}(h)=(\cdots,w,v,u,z,y,x,\cdots)=\cdots\otimes(-w)_{j}\otimes(-z)_{i}\otimes(-y)_{j}\otimes(-x)_{i}\otimes\cdots\in{\mathcal{H}}_{{\mathbf{i}}^{\prime}}$

Proof. The formula (3.27)–(3.30) are obtained easily by applying the braid-
type isomorphisms (3.15), (3.16),(3.17),(3.22) on
${\mathcal{H}}_{\mathbf{i}}$. For example, take
$h=(\cdots,x,y,z,w,\cdots)\in{\mathcal{H}}_{\mathbf{i}}$ as in (2). Then, it
satisfies $-x+c_{1}y+z=0=-y+c_{2}z-w$ and then one has $X$ in (3.18) as

$\displaystyle X$ $\displaystyle=$
$\displaystyle-w+(c_{2}x-y+w+c_{2}(-x+c_{1}y-z)_{+})_{+}=-w+(c_{2}x-y+w)_{+}=-w+(y-c_{2}z+w)_{+}=-w$

and similarly, one has $Y=-z$, $Z=-y$ and $W=-x$. Then (3.29) has been shown.

Next, let us show that
$\phi_{ij}^{(k)}(h)\in{\mathcal{H}}_{{\mathbf{i}}^{\prime}}$ for any
$h\in{\mathcal{H}}_{\mathbf{i}}$. We shall see the case of $\phi_{ij}^{(2)}$.
The other cases will be shown similarly to the case of $\phi_{ij}^{(2)}$. Let
$x$ be $m$-th entry in $h$, namely,
$(i_{m},i_{m+1},i_{m+2},i_{m+3})=(i,j,i,j)$. For
${\mathbf{i}}=i_{1}\cdots\stackrel{{\scriptstyle
m}}{{i}}\stackrel{{\scriptstyle\,m+1}}{{j}}\stackrel{{\scriptstyle
m+2}}{{i}}\stackrel{{\scriptstyle\,m+3}}{{j}}\cdots i_{N}$, let
${\mathbf{i}}^{\prime}:=i_{1}\cdots\stackrel{{\scriptstyle
m}}{{j}}\stackrel{{\scriptstyle\,m+1}}{{i}}\stackrel{{\scriptstyle\,m+2}}{{j}}\stackrel{{\scriptstyle\,m+3}}{{i}}\cdots
i_{N}$. For any $h=(\cdots\stackrel{{\scriptstyle
m}}{{x}},y,z,w,\cdots)\in{\mathcal{H}}_{\mathbf{i}}$ and $k\in[1,N]$ with
$k^{+}\leq N$ let us show that

(3.31) $\beta_{k}^{({\mathbf{i}}^{\prime})}(\phi_{ij}^{(2)}(h))=0.$

To see all $\beta_{k}^{({\mathbf{i}}^{\prime})}(x)$’s with the variables
$x_{m},x_{m+1},x_{m+2},x_{m+3}$, it suffices to investigate the following
cases:

  1. (1)

$k=m$, that is, $\beta_{m}^{({\mathbf{i}}^{\prime})}(\phi_{ij}^{(2)}(h))=0$.

  2. (2)

$k=m+1$, that is,
$\beta_{m+1}^{({\mathbf{i}}^{\prime})}(\phi_{ij}^{(2)}(h))=0$.

  3. (3)

$k=m+2$, that is,
$\beta_{m+2}^{({\mathbf{i}}^{\prime})}(\phi_{ij}^{(2)}(h))=0$.

  4. (4)

$k=m+3$, that is,
$\beta_{m+3}^{({\mathbf{i}}^{\prime})}(\phi_{ij}^{(2)}(h))=0$.

  5. (5)

$k=m^{-}$, that is,
$\beta_{m^{-}}^{({\mathbf{i}}^{\prime})}(\phi_{ij}^{(2)}(h))=0$.

  6. (6)

$k=(m+1)^{-}$, that is,
$\beta_{(m+1)^{-}}^{({\mathbf{i}}^{\prime})}(\phi_{ij}^{(2)}(h))=0$.

  7. (7)

$\beta_{k}^{({\mathbf{i}}^{\prime})}(\phi_{ij}^{(2)}(h))=0$ for $k$ such that
$i_{k}\neq i,j$ and $[m,m+3]\subset[k,k^{+}]$.

Since all cases are almost trivial, we shall only show the cases (6) and (7).
The case (6): Note that one has
$i_{m}=i_{m^{-}}=i^{\prime}_{m+1}=i^{\prime}_{(m+1)^{-}}=i$ for
${\mathbf{i}}^{\prime}=(i^{\prime}_{1},\cdots,i^{\prime}_{N})$. For
$h=(h_{1},\cdots,h_{m^{-}},\cdots,h_{m-1},x,y,z,w,\cdots,h_{N})$, set
$h^{\prime}=\phi_{ij}^{(2)}(h)=(h^{\prime}_{1},\cdots,h^{\prime}_{N})$. One
has

$(h^{\prime}_{(m+1)^{-}},h^{\prime}_{m},h^{\prime}_{m+1},h^{\prime}_{m+2},h^{\prime}_{m+3})=(h_{m^{-}},w,z,y,x),\qquad
h^{\prime}_{j}=h_{j}\hbox{ for }j\not\in[m,m+3],$

by the formula (3.29) that is,
$h^{\prime}=\phi_{ij}^{(2)}(h)=(\cdots,h_{(m+1)^{-}},\cdots,h_{m-1},w,z,y,x,\cdots)$.
Since $\beta^{({\mathbf{i}})}_{(m+1)^{-}}(h)=0$, one has

$\displaystyle\beta^{({\mathbf{i}})}_{m^{-}}(h)=h_{m^{-}}+\cdots+a_{i\,i_{m-1}}h_{m-1}+x=0,$
$\displaystyle\beta^{({\mathbf{i}}^{\prime})}_{(m+1)^{-}}(\phi_{ij}^{(2)}(h))=h^{\prime}_{(m+1)^{-}}+\cdots+a_{i\,i_{m-1}}h^{\prime}_{m-1}+(-c_{1}w+z)=h_{m^{-}}+\cdots+a_{i\,i_{m-1}}h_{m-1}+(-c_{1}w+z).$

Thus, to show
$\beta^{({\mathbf{i}}^{\prime})}_{(m+1)^{-}}(\phi_{ij}^{(2)}(h))=0$, one may
get $x=-c_{1}w+z$, which is immediate from the fact $x-c_{1}y+z=y-c_{2}z+w=0$
and $c_{1}c_{2}=2$.
The case (7): Set $i_{k}=l$. Then one has

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\beta^{({\mathbf{i}})}_{k}(h)=h_{k}+a_{li_{k+1}}h_{k+1}+\cdots+a_{li}x+a_{lj}y+a_{li}z+a_{lj}w+\cdots+h_{k^{+}}$
$\displaystyle=$ $\displaystyle
h_{k}+a_{li_{k+1}}h_{k+1}+\cdots+a_{lj}w+a_{li}z+a_{lj}y+a_{li}x+\cdots+h_{k^{+}}=\beta^{({\mathbf{i}}^{\prime})}_{k}(\phi_{ij}^{(2)}(h)).$

Therefore, we obtain that
$\phi_{ij}^{(p)}({\mathcal{H}}_{\mathbf{i}})\subset{\mathcal{H}}_{{\mathbf{i}}^{\prime}}$
for $p=0,1,2,3$. Since $\phi_{ij}^{(p)}\circ\phi_{ji}^{(p)}={\rm id}$, it
yields
$\phi_{ij}^{(p)}({\mathcal{H}}_{\mathbf{i}})={\mathcal{H}}_{{\mathbf{i}}^{\prime}}$
for $p=0,1,2,3$.

In [17, Sect.8], we have shown the following statements under the condition
”${\bf H}_{\mathbf{i}}$”, where we omit the explicit form of ${\bf
H}_{\mathbf{i}}$ since we do not need it here. But, we succeed in showing the
following proposition without the condition ${\bf H}_{\mathbf{i}}$ since in
[17] we have shown that there exists a specific reduced longest word
${\mathbf{i}}_{0}$ satisfying the condition ${\bf H}_{{\mathbf{i}}_{0}}$ for
each simple Lie algebra $\mathfrak{g}$ and we got Lemma 3.10.

######  Proposition 3.11.

Let ${\mathbf{i}}=i_{1}i_{2}\cdots i_{N}$ be an arbitrary reduced longest
word. Here if the crystal $B(\infty)$ is realized in $\mathbb{B}_{\mathbf{i}}$
as in 3.6, we shall denote it by $B(\infty)_{\mathbf{i}}$ to emphasize the
word ${\mathbf{i}}$. For $h\in{\mathcal{H}}_{\mathbf{i}}$, define

$B^{h}(\infty)_{\mathbf{i}}:=\\{x+h\in\mathbb{Z}^{N}(=\mathbb{B}_{\mathbf{i}})\,|\,x\in
B(\infty)_{\mathbf{i}}\\}\subset\mathbb{B}_{\mathbf{i}}.$

  1. (1)

For any $x+h\in B^{h}(\infty)_{\mathbf{i}}$ and $i\in I$, we obtain

(3.32)
$\tilde{e}_{i}(x+h)=\tilde{e}_{i}(x)+h,\qquad\tilde{f}_{i}(x+h)=\tilde{f}_{i}(x)+h.$

  2. (2)

For any $h\in{\mathcal{H}}_{\mathbf{i}}$, we have $B(\infty)_{\mathbf{i}}\cap
B^{h}(\infty)_{\mathbf{i}}\neq\emptyset$.

  3. (3)

$\mathbb{B}_{\mathbf{i}}=\bigcup_{h\in{\mathcal{H}}_{\mathbf{i}}}B^{h}(\infty)_{\mathbf{i}}$

######  Remark 3.12.

In the setting of the half-potential method in [17], as mentioned in Remark
3.7, the crystal $B(\infty)$ is realized as a subset of
$\mathbb{B}_{\mathbf{i}}$ and it is supposed that $\tilde{e}_{i}x=0$ if
$\tilde{e}_{i}x\not\in(\widetilde{\mathbb{B}}^{-}_{w_{0}})_{\Phi^{(+)},\Theta_{\mathbf{i}}}\cong
B(\infty)$. At the statement (2), since $x\in B(\infty)_{\mathbf{i}}$ is
considered as an element of $\mathbb{B}_{\mathbf{i}}$, $\tilde{e}_{i}x$ is
also considered as an element in $\mathbb{B}_{\mathbf{i}}$. That is, even if
$\tilde{e}_{i}x\not\in B(\infty)$, we consider that
$\tilde{e}_{i}x\in\mathbb{B}_{\mathbf{i}}$ and then it never vanishes.

Proof. First, note that in [17], it has been shown the statement (1)-(3) for
some specific reduced longest word ${\mathbf{i}}_{0}$ satisfying the condition
${\bf H}_{{\mathbf{i}}_{0}}$.

(1) One can show (3.32) easily since $\beta^{({\mathbf{i}})}_{k}(h)=0$ for any
$k\in[1,N]$ and $h\in{\mathcal{H}}_{\mathbf{i}}$ and then one has
$\widetilde{m}^{(i)}_{e}(x+h)=\widetilde{m}^{(i)}_{e}(x)$ and
$\widetilde{m}^{(i)}_{f}(x+h)=\widetilde{m}^{(i)}_{f}(x)$.

(2) For any reduced longest word ${\mathbf{i}}$, there exists a braid-move
$\xi$ on ${\mathbf{i}}_{0}$ such that $\xi({\mathbf{i}}_{0})={\mathbf{i}}$.
Let $\Xi$ be the composition of braid-type isomorphisms associated with $\xi$
such that $\Xi(B_{{\mathbf{i}}_{0}})=B_{\mathbf{i}}$. Since $\Xi$ is an
isomorphism of crystals and then commutes with the actions of $\tilde{e}_{i}$
and $\tilde{f}_{i}$ and $B(\infty)_{{\mathbf{i}}_{0}}\cap
B^{h}(\infty)_{{\mathbf{i}}_{0}}\neq\emptyset$ for any
$h\in{\mathcal{H}}_{{\mathbf{i}}_{0}}$, one gets that
$\Xi(B^{h}(\infty)_{{\mathbf{i}}_{0}})=B^{\xi(h)}(\infty)_{\mathbf{i}}$ for
any $h\in{\mathcal{H}}_{{\mathbf{i}}_{0}}$, which implies that
$B(\infty)_{\mathbf{i}}\cap B^{H}(\infty)_{\mathbf{i}}\neq\emptyset$ for any
$H\in{\mathcal{H}}_{\mathbf{i}}$ since
$\Xi({\mathcal{H}}_{{\mathbf{i}}_{0}})={\mathcal{H}}_{\mathbf{i}}$ by Lemma
3.10.

(3) Let $\xi$ and $\Xi$ be as in (2). Then, since
$\Xi({\mathcal{H}}_{{\mathbf{i}}_{0}})={\mathcal{H}}_{\mathbf{i}}$ and
$\Xi(B^{h}(\infty)_{{\mathbf{i}}_{0}})=B^{\xi(h)}(\infty)_{\mathbf{i}}$ for
any $h\in{\mathcal{H}}_{{\mathbf{i}}_{0}}$, we obtain

$\qquad\qquad\qquad\mathbb{B}_{\mathbf{i}}=\Xi(\mathbb{B}_{{\mathbf{i}}_{0}})=\bigcup_{h\in{\mathcal{H}}_{{\mathbf{i}}_{0}}}\Xi(B^{h}(\infty)_{{\mathbf{i}}_{0}})=\bigcup_{h\in{\mathcal{H}}_{{\mathbf{i}}_{0}}}B^{\xi(h)}(\infty)_{{\mathbf{i}}}=\bigcup_{H\in{\mathcal{H}}_{{\mathbf{i}}}}B^{H}(\infty)_{{\mathbf{i}}}.\qquad\qquad\qquad\qquad\hfill\framebox[5.69054pt]{}$

It is immediate from this proposition that one has the following theorem:

######  Theorem 3.13 ([17]).

For any simple Lie algebra $\mathfrak{g}$ and any reduced word
$i_{1}i_{2}\cdots i_{k}$, the cellular crystal $\mathbb{B}_{i_{1}i_{2}\cdots
i_{k}}=B_{i_{1}}\otimes B_{i_{2}}\otimes\cdots\otimes B_{i_{k}}$ is connected
as a crystal graph.

##  4\. Quiver Hecke Algebra and its modules

###  4.1. Definition of Quiver Hecke Algebra

We review basics of the quiver Hecke algebras (see e.g.,[10, 12, 14, 28]). For
a finite index set $I$ and a field $\bf k$, let
$({\mathscr{Q}}_{i,j}(u,v))_{i,j\in I}\in{\bf k}[u,v]$ be polynomials
satisfying:

  1. (1)

${\mathscr{Q}}_{i,j}(u,v)={\mathscr{Q}}_{j,i}(v,u)$ for any $i,j\in I$.

  2. (2)

${\mathscr{Q}}_{i,j}(u,v)$ is in the form:

${\mathscr{Q}}_{i,j}(u,v)=\begin{cases}\displaystyle\sum_{a(\alpha_{i},\alpha_{i})+b(\alpha_{j},\alpha_{j})=-2(\alpha_{i},\alpha_{j})}t_{i,j;a,b}u^{a}v^{b}&\hbox{
if }i\neq j,\\\ 0&\hbox{ if }i=j,\end{cases}$

where $t_{i,j;-a_{ij},0}\in{\bf k}^{\times}$.

For $\beta=\sum_{i}m_{i}\alpha_{i}\in Q_{+}$ with $|\beta|:=\sum_{i}m_{i}=m$,
set $I^{\beta}:=\\{\nu=(\nu_{1},\cdots,\nu_{m})\in
I^{m}\mid\sum_{k=1}^{m}\alpha_{\nu_{k}}=\beta\\}$.

######  Definition 4.1.

For $\beta\in Q_{+}$, the quiver Hecke algebra $R(\beta)$ associated with a
Cartan matrix $A$ and polynomials ${\mathscr{Q}}_{i,j}(u,v)$ is the ${\bf
k}$-algebra generated by

$\\{e(\nu)|\nu\in I^{\beta}\\},\quad\\{x_{k}|1\leq k\leq
n\\},\quad\\{\tau_{i}|1\leq i\leq n-1\\}$

with the following relations:

$\displaystyle
e(\nu)e(\nu^{\prime})=\delta_{\nu,\nu^{\prime}}e(\nu),\quad\sum_{\nu\in
I^{\beta}}e(\nu)=1,\quad e(\nu)x_{k}=x_{k}e(\nu),\quad x_{k}x_{l}=x_{l}x_{k},$
$\displaystyle\tau_{l}e(\nu)=e(s_{l}(\nu))\tau_{l},\quad\tau_{k}\tau_{l}=\tau_{l}\tau_{k}\,\,\hbox{
if }|k-l|>1,$
$\displaystyle\tau_{k}^{2}e(\nu)={\mathscr{Q}}_{\nu_{k},\nu_{k+1}}(x_{k},x_{k+1})e(\nu),$
$\displaystyle(\tau_{k}x_{l}-x_{s_{k}(l)}\tau_{k})e(\nu)=\begin{cases}-e(\nu)&\hbox{
if }l=k,\,\,\nu_{k}=\nu_{k+1},\\\ e(\nu)&\hbox{ if
}l=k+1,\,\,\nu_{k}=\nu_{k+1},\\\ 0&\hbox{otherwise},\end{cases}$
$\displaystyle(\tau_{k+1}\tau_{k}\tau_{k+1}-\tau_{k}\tau_{k+1}\tau_{k})e(\nu)=\begin{cases}\overline{\mathscr{Q}}_{\nu_{k},\nu_{k+1}}(x_{k},x_{k+1},x_{k+2})e(\nu)&\hbox{
if }\nu_{k}=\nu_{k+2},\\\ 0&\hbox{ otherwise},\end{cases}$

where
$\overline{\mathscr{Q}}_{i,j}(u,v,w)=\frac{{\mathscr{Q}}_{i,j}(u,v)-{\mathscr{Q}}_{i,j}(w,v)}{u-w}\in{\bf
k}[u,v,w]$.

  1. (1)

The relations above are homogeneous if we define

${\rm deg}(e(\nu))=0,\quad{\rm
deg}(x_{k}e(\nu))=(\alpha_{\nu_{k}},\alpha_{\nu_{k}}),\quad{\rm
deg}(\tau_{l}e(\nu))=-(\alpha_{\nu_{l}},\alpha_{\nu_{l+1}}).$

Thus, $R(\beta)$ becomes a $\mathbb{Z}$-graded algebra. Here we define the
weight of $R(\beta)$-module $M$ as ${\rm wt}(M)=-\beta$.

  2. (2)

Let $M=\bigoplus_{k\in\mathbb{Z}}M_{k}$ be a $\mathbb{Z}$-graded
$R(\beta)$-module. Define a grading shift functor $q$ on the category of
graded $R(\beta)$-modules $R(\beta)$-Mod by

$qM:=\bigoplus_{k\in\mathbb{Z}}(qM)_{k},\quad\hbox{where }(qM)_{k}=M_{k-1}.$

  3. (3)

For $M,\,N\in R(\beta)$-Mod, let ${\rm Hom}_{R(\beta)}(M,N)$ be the space of
degree preserving morphisms and define
$\textsc{Hom}_{R(\beta)}(M,N):=\bigoplus_{k\in\mathbb{Z}}{\rm
Hom}_{R(\beta)}(q^{k}M,N)$, which is a space of morphisms up to grading shift.
We define ${\rm deg}(f)=k$ for $f\in{\rm Hom}_{R(\beta)}(q^{k}M,N)$.

  4. (4)

Let $\psi$ be the anti-automorphism of $R(\beta)$ preserving all generators.
For $M\in R(\beta)$-Mod, define $M^{*}:=\textsc{Hom}_{\bf k}(M,{\bf k})$ with
the $R(\beta)$\- module structure by $(r\cdot f)(u):=f(\psi(r)u)$ for $r\in
R(\beta)$, $u\in M$ and $f\in M^{*}$, which is called a dual module of $M$. In
particular, if $M\cong M^{*}$ we call $M$ is self-dual.

  5. (5)

For $\beta,\gamma\in Q_{+}$, set $e(\beta,\gamma)=\sum_{\nu\in
I^{\beta},\nu^{\prime}\in I^{\gamma}}e(\nu,\nu^{\prime})$. We define an
injective homomorphism $\xi_{\beta,\gamma}:R(\beta)\otimes R(\gamma)\to
e(\beta,\gamma)R(\beta+\gamma)e(\beta,\gamma)$ by
$\xi(\beta,\gamma)(e(\nu)\otimes e(\nu^{\prime}))=e(\nu,\nu^{\prime})$,
$\xi(\beta,\gamma)(x_{k}e(\beta)\otimes 1)=x_{k}e(\beta,\gamma)$,
$\xi(\beta,\gamma)(1\otimes x_{k}e(\gamma))=x_{k+|\beta|}e(\beta,\gamma)$,
$\xi(\beta,\gamma)(\tau_{k}e(\beta)\otimes 1)=\tau_{k}e(\beta,\gamma)$,
$\xi(\beta,\gamma)(1\otimes\tau_{k}e(\gamma))=\tau_{k+|\beta|}e(\beta,\gamma)$.

  6. (6)

For $M\in R(\beta)$-Mod and $N\in R(\gamma)$-Mod, define the convolution
product $\circ$ by

$M\circ N:=R(\beta+\gamma)e(\beta,\gamma)\otimes_{R(\beta)\otimes
R(\gamma)}(M\otimes N)$

For simple $M\in R(\beta)$-Mod and simple $N\in R(\gamma)$-Mod, we say $M$ and
$N$ strongly commutes if $M\circ N$ is simple and $M$ is real if $M\circ M$ is
simple.

  7. (7)

For $M\in R(\beta)$-Mod and $N\in R(\gamma)$-Mod, denote by $M\nabla N:={\rm
hd}(M\circ N)$ the head of $M\circ N$ and $M\Delta N:={\rm soc}(M\circ N)$ the
socle of $M\circ N$, where the head of module $M$ is the quotient by its
radical and the socle of module $M$ is the summation of all simple submodules.

###  4.2. Categorification of quantum coordinate ring
${\mathcal{A}}_{q}(\mathfrak{n})$

Let $R(\beta)$-gmod be the full subcategory of $R(\beta)$-Mod whose objects
are finite-dimensional graded $R(\beta)$-modules and set
$R$-gmod$=\bigoplus_{\beta\in Q_{+}}R(\beta)$-gmod. Define the functors

$E_{i}:R(\beta)\hbox{-gmod}\to R(\beta-\alpha_{i})\hbox{-gmod},\qquad
F_{i}:R(\beta)\hbox{-gmod}\to R(\beta+\alpha_{i})\hbox{-gmod },$

by $E_{i}(M):=e(\alpha_{i},\beta-\alpha_{i})M,\quad F_{i}(M)=L(i)\circ M$,
where $e(\alpha_{i},\beta-\alpha_{i}):=\sum_{\nu\in
I^{\beta},\nu_{1}=i}e(\nu)$ and $L(i):=R(\alpha_{i})/R(\alpha_{i})x_{1}$ is a
1-dimensional simple $R(\alpha_{i})$-module. Let ${\mathcal{K}}(R\hbox{\rm-
gmod})$ be the Grothendieck ring of $R$-gmod and then
${\mathcal{K}}(R\hbox{\rm-gmod})$ becomes a $\mathbb{Z}[q,q^{-1}]$-algebra
with the multiplication induced by the convolution product and
$\mathbb{Z}[q,q^{-1}]$-action induced by the grading shift functor $q$. Here,
one obtain the following:

######  Theorem 4.2 ([10, 28]).

As a $\mathbb{Z}[q,q^{-1}]$-algebra there exists an isomorphism

${\mathcal{K}}(R\hbox{\rm-
gmod})\cong{\mathcal{A}}_{q}(\mathfrak{n})_{\mathbb{Z}[q,q^{-1}]}.$

###  4.3. Categorification of the crystal $B(\infty)$ by Lauda and Vazirani
[19]

The following lemma is given in [10]:

######  Lemma 4.3 ([10]).

For any simple $R(\beta)$-module $M$, ${\rm soc}(E_{i}M)$, ${\rm hd}(E_{i}M)$
and ${\rm hd}(F_{i}M)$ are all simple modules. Here we also have that ${\rm
soc}(E_{i}M)\cong{\rm hd}(E_{i}M)$ up to grading shift.

For $M\in R(\beta)$-gmod, define

(4.1) $\displaystyle\mathrm{wt}(M)=-\beta,\quad\varepsilon_{i}(M)={\rm
max}\\{n\in\mathbb{Z}\,|\,E_{i}^{n}M\neq
0\\},\quad\varphi_{i}(M)=\varepsilon_{i}(M)+\langle
h_{i},\mathrm{wt}(M)\rangle,$ (4.2)
$\displaystyle\widetilde{E}_{i}M:=q_{i}^{1-\varepsilon_{i}(M)}{\rm
soc}(E_{i}M)\cong q_{i}^{\varepsilon_{i}(M)-1}{\rm
hd}(E_{i}M),\qquad\widetilde{F}_{i}M:=q_{i}^{\varepsilon_{i}(M)}{\rm
hd}(F_{i}M).$

Set $\mathbb{B}(\hbox{$R$-\hbox{gmod}}):=\\{S\,|\,S\hbox{ is a self-dual
simple module in }R\hbox{-gmod}\\}$. Then, it follows from Lemma 4.3 that
$\widetilde{E}_{i}$ and $\widetilde{F}_{i}$ are well-defined on
$\mathbb{B}(\hbox{$R$-\hbox{gmod}})$.

######  Theorem 4.4 ([19]).

The 6-tuple
$(\mathbb{B}(\hbox{$R$-\hbox{gmod}}),\\{\widetilde{E}_{i}\\},\\{\widetilde{F}_{i}\\},\mathrm{wt},\\{\varepsilon_{i}\\},\\{\varphi_{i}\\})_{i\in
I}$ holds a crystal structure and there exists the following isomorphism of
crystals:

$\Psi:\mathbb{B}(\hbox{$R$-\hbox{gmod}})\quad\smash{\mathop{\longrightarrow}\limits^{\sim}}\quad
B(\infty).$

######  Remark 4.5.

Note that Lauda and Vasirani showed this theorem under more general setting
that $\mathfrak{g}$ is arbitrary symmetrizable Kac-Moody Lie algebra. Here we
assume that $\mathfrak{g}$ is a simple Lie algebra. The definition of
$\widetilde{E}_{i}$ and $\widetilde{F}_{i}$ in (4.2) differs from the one in
[19], which follows the one in [14].

##  5\. Localization of monoidal category

In this section, we shall review briefly the basics on localization of
monoidal category following [12].

###  5.1. Braiders and Real Commuting Family

Let $\Lambda$ be $\mathbb{Z}$-lattice and
${\mathcal{T}}=\oplus_{\lambda\in\Lambda}{\mathcal{T}}_{\lambda}$ be a ${\bf
k}$-linear $\Lambda$-graded monoidal category with a data consisting of a
bifunctor
$\otimes:{\mathcal{T}}_{\lambda}\times{\mathcal{T}}_{\mu}\to{\mathcal{T}}_{\lambda+\mu}$,
an isomorphism $a(X,Y,Z):(X\otimes Y)\otimes
Z\smash{\mathop{\longrightarrow}\limits^{\sim}}X\otimes(Y\otimes Z)$
satisfying $a(X,Y,Z\otimes W)\circ a(X\otimes Y,Z,W)={\rm id}_{X}\otimes
a(Y,Z,W)\circ a(X,Y\otimes Z,W)\circ a(X,Y,Z)\otimes{\rm id}_{W}$ and an
object ${\bf 1}\in{\mathcal{T}}_{0}$ endowed with an isomorphism
$\epsilon:{\bf 1}\otimes{\bf
1}\smash{\mathop{\longrightarrow}\limits^{\sim}}{\bf 1}$ such that the functor
$X\mapsto X\otimes{\bf 1}$ and $X\mapsto{\bf 1}\otimes X$ are fully-faithful.

######  Definition 5.1 ([12]).

Let $q$ be the grading shift functor on ${\mathcal{T}}$. A graded braider is a
triple $(C,R_{C},\phi)$, where $C\in{\mathcal{T}}$, $\mathbb{Z}$-linear map
$\phi:\Lambda\to\mathbb{Z}$ and a morphism:

$R_{C}:C\otimes X\to q^{\phi(\lambda)}X\otimes
C\quad(X\in{\mathcal{T}}_{\lambda}),$

satisfying the following commutative diagram:

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Y}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 54.45581pt\raise
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95.04869pt\raise-19.69443pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
95.04869pt\raise-28.88887pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
51.70198pt\raise-39.38887pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{q^{\phi(\lambda+\mu)}(X\otimes Y)\otimes
C}$}}}}}}}\ignorespaces}}}}\ignorespaces\qquad(X\in{\mathcal{T}}_{\lambda},\,\,Y\in{\mathcal{T}}_{\mu})$

and being functorial, that is, for any $X,Y\in{\mathcal{T}}$ and $f\in{\rm
Hom}_{\mathcal{T}}(X,Y)$ it satisfies the following commutative diagram:

$\textstyle{C\otimes
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
id}\otimes f}$$\scriptstyle{R_{C}(X)}$$\textstyle{C\otimes
Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R_{C}(Y)}$$\textstyle{X\otimes
C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f\otimes{\rm
id}}$$\textstyle{Y\otimes C}$

######  Definition 5.2 ([12]).

Let $I$ be an index set and $(C_{i},R_{C_{i}},\phi_{i})_{i\in I}$ a family of
graded braiders in ${\mathcal{T}}$. We say that
$(C_{i},R_{C_{i}},\phi_{i})_{i\in I}$ is a real commuting family of graded
braiders in ${\mathcal{T}}$ if

  1. (1)

$C_{i}\in{\mathcal{T}}_{\lambda_{i}}$ for some $\lambda_{i}\in\Lambda$, and
$\phi_{i}(\lambda_{i})=0$, $\phi_{i}(\lambda_{j})+\phi_{j}(\lambda_{i})=0$ for
any $i,j\in I$.

  2. (2)

$R_{C_{i}}(C_{i})\in{\bf k}^{\times}{\rm id}_{C_{i}\otimes C_{i}}$ for any
$i\in I$.

  3. (3)

$R_{C_{i}}(C_{j})\otimes R_{C_{j}}(C_{i})\in{\bf k}^{\times}{\rm
id}_{C_{i}\otimes C_{j}}$ for any $i,j\in I$.

Note that $R_{C_{i}}$’s satisfy so-called ”Yang-Baxter equation”, such as,

$R_{C_{i}}(C_{j})\circ R_{C_{i}}(C_{k})\circ
R_{C_{j}}(C_{k})=R_{C_{j}}(C_{k})\circ R_{C_{i}}(C_{k})\circ
R_{C_{i}}(C_{j})\,\,\hbox{ on }\,\,C_{i}\circ C_{j}\circ C_{k}.$

For a finite index set $I$, set $\Gamma:=\oplus_{i\in I}\mathbb{Z}e_{i}$ and
$\Gamma_{+}:=\oplus_{i\in I}\mathbb{Z}_{\geq 0}e_{i}$.

######  Lemma 5.3 ([12]).

Suppose that we have a real commuting family of graded braiders
$(C_{i},R_{C_{i}},\phi_{i})_{i\in I}$. We can choose a bilinear map
$H:\Gamma\times\Gamma\to\mathbb{Z}$ such that
$\phi_{i}(\lambda_{j})=H(e_{i},e_{j})-H(e_{j},e_{i})$ and there exist

  1. (1)

an object $C^{\alpha}$ for any $\alpha\in\Gamma_{+}$.

  2. (2)

an isomorphism $\xi_{\alpha,\beta}:C^{\alpha}\otimes
C^{\beta}\smash{\mathop{\longrightarrow}\limits^{\sim}}q^{H(\alpha,\beta)}C^{\alpha+\beta}$
for any $\alpha,\beta\in\Gamma_{+}$

such that $C^{0}=1$ and $C^{e_{i}}=C_{i}$.

###  5.2. Localization

Let ${\mathcal{T}}$ and $(C_{i},R_{C_{i}},\phi_{i})_{i\in I}$ be as above and
$\\{C^{\alpha}\\}_{\alpha\in\Gamma_{+}}$ objects as in the previous lemma. We
define a partial order $\preceq$ on $\Gamma$ by

$\alpha\preceq\beta\Longleftrightarrow\beta-\alpha\in\Gamma_{+}$

For $\alpha_{1},\alpha_{2},\cdots\in\Gamma$, define

${\mathcal{D}}_{\alpha_{1},\alpha_{2},\cdots}:=\\{\delta\in\Gamma\,|\,\alpha_{j}+\delta\in\Gamma_{+}\,\,\hbox{
for any }j=1,2,\cdots\\}.$

For $X\in{\mathcal{T}}_{\lambda}$, $Y\in{\mathcal{T}}_{\mu}$ and
$\delta\in{\mathcal{D}}_{\alpha,\beta}$, set

$H_{\delta}((X,\alpha),(Y,\beta)):={\rm
Hom}_{{\mathcal{T}}}(C^{\delta+\alpha}\otimes
X,q^{P(\alpha,\beta,\delta,\mu)}Y\otimes C^{\beta+\delta}),$

where a $\mathbb{Z}$-valued function
$P(\alpha,\beta,\delta,\mu):=H(\delta,\beta-\alpha)+\phi(\delta+\beta,\mu)$
and the map $\phi:\Gamma\times\Lambda\to\mathbb{Z}$ is defined by
$\phi(\alpha,L(\beta))=H(\alpha,\beta)-H(\beta,\alpha)$ and
$L:\Gamma\to\Lambda$ is defined by $L(e_{i})=\lambda_{i}$ ([12]).

######  Lemma 5.4 ([12]).

For $\delta\preceq\delta^{\prime}$ there exists the map

$\zeta_{\delta,\delta^{\prime}}:H_{\delta}((X,\alpha),(Y,\beta))\to
H_{\delta^{\prime}}((X,\alpha),(Y,\beta))$

satisfying

$\zeta_{\delta,\delta^{\prime}}\circ\zeta_{\delta^{\prime},\delta^{\prime\prime}}=\zeta_{\delta,\delta^{\prime\prime}}\hbox{for
}\delta\preceq\delta^{\prime}\preceq\delta^{\prime\prime}.$

Therefore, we find that
$\\{H_{\delta}((X,\alpha),(Y,\beta))\\}_{\delta\in{\mathcal{D}}_{\alpha,\beta}}$
becomes an inductive system.

######  Definition 5.5 (Localization [12]).

We define the category $\widetilde{\mathcal{T}}$ by

$\displaystyle{\rm Ob}(\widetilde{\mathcal{T}}):={\rm
Ob}({\mathcal{T}})\times\Gamma,$
$\displaystyle\mathrm{Hom}_{\widetilde{\mathcal{T}}}((X,\alpha),(Y,\beta)):=\varinjlim_{\begin{array}[]{c}\scriptstyle\tiny\delta\in{\mathcal{D}}(\alpha,\beta),\\\
\scriptstyle\lambda+L(\alpha)=\mu+L(\beta)\end{array}}H_{\delta}((X,\alpha),(Y,\beta)),$

where $X\in{\mathcal{T}}_{\lambda}$, $Y\in{\mathcal{T}}_{\mu}$ and the
function $L:\Gamma\to\Lambda$ ($e_{i}\mapsto\lambda_{i}$) is as above. We call
this $\widetilde{\mathcal{T}}$ a localization of ${\mathcal{T}}$ by
$(C_{i},R_{C_{i}},\phi_{i})_{i\in I}$ and denote it by
${\mathcal{T}}[C_{i}^{\otimes-1}\,|\,i\in I]$ when we emphasize $\\{C_{i}\mid
i\in I\\}$.

######  Theorem 5.6 ([12]).

$\widetilde{\mathcal{T}}$ becomes a monoidal category. Moreover, there exists
a monoidal functor $\Upsilon:{\mathcal{T}}\to\widetilde{\mathcal{T}}$ such
that

  1. (1)

$\Upsilon(C_{i})$ is invertible in $\widetilde{\mathcal{T}}$ for any $i\in I$,
namely, the functors $X\mapsto X\otimes\Upsilon(C_{i})$ and
$X\mapsto\Upsilon(C_{i})\otimes X$ are equivalence of categories.

  2. (2)

For any $i\in I$ and $X\in{\mathcal{T}}$,
$\Upsilon(R_{C_{i}}(X)):\Upsilon(C_{i}\otimes X)\to\Upsilon(X\otimes C_{i})$
is an isomorphism.

  3. (3)

The functor $\Upsilon$ holds the following universality: If there exists
another monoidal category ${\mathcal{T}}^{\prime}$ and a monoidal fucntor
$\Upsilon^{\prime}:{\mathcal{T}}\to{\mathcal{T}}^{\prime}$ satisfying the
above statements (1) and (2), then there exists a monoidal functor
$F:\widetilde{\mathcal{T}}\to{\mathcal{T}}^{\prime}$ (unique up to iso.) such
that $\Upsilon^{\prime}=F\circ\Upsilon$.

######  Proposition 5.7 ([12]).

Under the setting above, we obtain

  1. (1)

$(X,\alpha+\beta)\cong q^{-H(\beta,\alpha)}(C^{\alpha}\otimes X,\beta)$,
$(1,\beta)\otimes(1,-\beta)\cong q^{-H(\beta,\beta)}(1,0)$ for
$\alpha\in\Gamma_{+}$, $\beta\in\Gamma$ and $X\in\widetilde{\mathcal{T}}$.

  2. (2)

If ${\mathcal{T}}$ is an abelian category, then so is
$\widetilde{\mathcal{T}}$.

  3. (3)

The functors $\Upsilon:{\mathcal{T}}\to\widetilde{\mathcal{T}}$ is exact.

  4. (4)

If the functor $-\otimes Y$ and $Y\otimes-$ are exact for any $Y$ in
${\mathcal{T}}$, then the functors
$\widetilde{\mathcal{T}}\to\widetilde{\mathcal{T}}$ ($X\mapsto X\otimes Y$
(resp. $X\to Y\otimes X$)) are exact for any $Y$ in $\widetilde{\mathcal{T}}$.

##  6\. Localization of the category $R$-gmod

###  6.1. Determinantial Modules

Here we just go back to the setting as in Sect.4. Let
$L(i^{n}):=q_{i}^{\frac{n(n-1)}{2}}L(i)^{\circ n}$ be a simple
$R(n\alpha_{i})$-module satisfying ${\rm
qdim}(L(i^{n}))=[n]_{i}!:=\prod_{k=1}^{n}\frac{q_{i}^{k}-q_{i}^{-k}}{q_{i}-q_{i}^{-1}}$
($q_{i}:=q^{\frac{(\alpha_{i},\alpha_{i})}{2}}$).

######  Definition 6.1 ([12, 14]).

For $M\in R$-gmod, define

$\widetilde{F}_{i}^{n}(M):=L(i^{n})\nabla M.$

For a Weyl group element $w$, let $s_{i_{1}}\cdots s_{i_{l}}$ be its reduced
expression. For a dominant weight $\Lambda\in P_{+}$, set

$m_{k}:=\langle h_{i_{k}},s_{i_{k+1}}\cdots
s_{i_{l}}\Lambda\rangle\qquad(k=1,\cdots,l).$

We define the determinantial module associated with $w$ and $\Lambda$ by

${\bf
M}(w\Lambda,\Lambda):=\widetilde{F}_{i_{1}}^{m_{1}}\cdots\widetilde{F}_{i_{l}}^{m_{l}}{\bf
1},$

where ${\bf 1}$ is a trivial $R(0)$-module.

Note that in general, one can define determinantial modules ${\bf
M}(w\Lambda,u\Lambda)$ ($w,u\in W$) which corresponds to the generalized minor
$\Delta_{w\Lambda,u\Lambda}$.

Now, let us see some similarity between the family of determinantial modules
$\\{{\bf M}(w_{0}\Lambda,\Lambda)\\}_{\Lambda\in P_{+}}$ and the subspace
${\mathcal{H}}_{\mathbf{i}}$. As has seen above that for a reduced longest
word ${\mathbf{i}}=i_{1}\cdots i_{N}$, the subspace
${\mathcal{H}}_{\mathbf{i}}\subset\mathbb{B}_{\mathbf{i}}$ is presented by

${\mathcal{H}}_{\mathbf{i}}=\bigoplus_{i\in
I}\mathbb{Z}{\mathbf{h}}_{i},\quad{\mathbf{h}}_{i}=((h^{(k)}_{i}:=\langle
h_{i_{k}},s_{i_{k+1}}\cdots s_{i_{N}}\Lambda_{i}\rangle)_{k=1,\cdots,N}.$

Furthermore, we also get

######  Proposition 6.2.

For any reduced longest word ${\mathbf{i}}=i_{1}i_{2}\cdots i_{N}$ and
$\Lambda\in P_{+}$, set

$m_{k}:=\langle h_{i_{k}},s_{i_{k+1}}s_{i_{k+2}}\cdots
s_{i_{l}}\Lambda\rangle\quad(k=1,2,\cdots,N)\quad\hbox{ and }\quad{\bf
h}_{\Lambda}:=(m_{1},\cdots,m_{N}).$

Then we obtain

$\displaystyle{\bf
h}_{\Lambda}=\tilde{f}_{i_{1}}^{m_{1}}\tilde{f}_{i_{2}}^{m_{2}}\cdots\tilde{f}_{i_{N}}^{m_{N}}((0)_{i_{1}}\otimes(0)_{i_{2}}\otimes\cdots\otimes(0)_{i_{N}})$
$\displaystyle=$
$\displaystyle\tilde{f}_{i_{1}}^{m_{1}}(0)_{i_{1}}\otimes\tilde{f}_{i_{2}}^{m_{2}}(0)_{i_{2}}\otimes\cdots\otimes\tilde{f}_{i_{N}}^{m_{N}}(0)_{i_{N}}\in{\mathcal{H}}_{\mathbf{i}},$

where note that for $\Lambda=\sum_{i}a_{i}\Lambda_{i}$, one has ${\bf
h}_{\Lambda}=\sum_{i}a_{i}{\bf h}_{\Lambda_{i}}$.

Proof. We shall show the following by descending induction on $k\in[1,N]$:

(6.1)
$\tilde{f}_{i_{k}}^{m_{k}}\cdots\tilde{f}_{i_{N}}^{m_{N}}((0)_{i_{1}}\otimes(0)_{i_{2}}\otimes\cdots\otimes(0)_{i_{N}})=(0)_{i_{1}}\otimes\cdots\otimes(0)_{i_{k-1}}\otimes\tilde{f}_{i_{k}}^{m_{k}}(0)_{i_{k}}\otimes\cdots\otimes\tilde{f}_{i_{N}}^{m_{N}}(0)_{i_{N}}=:M_{k}$

By Proposition 3.8, we find that
$(m_{1},\cdots,m_{N})\in{\mathcal{H}}_{\mathbf{i}}$ and for
$\\{k=k_{0}<k_{1}<\cdots<k_{L}\\}:=\\{l\mid i_{l}=i_{k},l\geq k\\}$

$\sigma_{k}(M_{k+1})=\sigma_{k_{1}}(M_{k+1})=\cdots=\sigma_{k_{L}}(M_{k+1}).$

Then, we get

$0=\sigma_{k}(M_{k+1})>-m_{k}=\sigma_{k_{1}}(M_{k+1})=\cdots=\sigma_{k_{L}}(M_{k+1}),$

which implies

$\tilde{f}_{i_{k}}^{m_{k}}M_{k+1}=\tilde{f}_{i_{k}}^{m_{k}}\cdots\tilde{f}_{i_{N}}^{m_{N}}((0)_{i_{1}}\otimes(0)_{i_{2}}\otimes\cdots\otimes(0)_{i_{N}})=(0)_{i_{1}}\otimes\cdots\otimes(0)_{i_{k-1}}\otimes\tilde{f}_{i_{k}}^{m_{k}}(0)_{i_{k}}\otimes\cdots\otimes\tilde{f}_{i_{N}}^{m_{N}}(0)_{i_{N}}=M_{k}.$

Thus, we finish the proof.

By this proposition, one observes that there would exist a certain
correspondence

(6.2) ${\bf
M}(w_{0}\Lambda,\Lambda)=\widetilde{F}_{i_{1}}^{m_{1}}\cdots\widetilde{F}_{i_{l}}^{m_{l}}{\bf
1}\quad\longleftrightarrow\quad{\bf
h}_{\Lambda}=\tilde{f}_{i_{1}}^{m_{1}}\cdots\tilde{f}_{i_{N}}^{m_{N}}((0)_{i_{1}}\otimes(0)_{i_{2}}\otimes\cdots\otimes(0)_{i_{N}}).$

######  Definition 6.3 ([12]).

For $\beta\in Q_{+}$, define a central element in $R(\beta)$ by
${\mathfrak{p}}_{i}:=\sum_{\nu\in
I^{\beta}}\left(\prod_{a\in\\{1,2,\cdots,{\rm
ht}(\beta)\\},\nu_{a}=i}x_{a}\right)e(\nu)\in R(\beta)$. For a simple $M\in
R(\beta)$-gmod, define an affinization $\widehat{M}$ of $M$ with degree $d$:

  1. (1)

There is an endomorphism $z:\widehat{M}\to\widehat{M}$ of degree $d>0$ such
that $\widehat{M}$ is finitely generated free module of ${\bf k}[z]$ and
$\widehat{M}/z\widehat{M}\cong M$.

  2. (2)

${\mathfrak{p}}_{i}\widehat{M}\neq 0$ for any $i\in I$.

######  Theorem 6.4 ([12, Theorem 3.26]).

For any $\Lambda\in P_{+}$ and $w\in W$, the determinantial module ${\bf
M}(w\Lambda,\Lambda)$ is a real simple module and admits an affinization
$\widehat{\bf M}(w\Lambda,\Lambda)$.

Note that indeed, if $\mathfrak{g}$ is simply-laced, then the affinization
$\widehat{M}$ always exists for any simple $M\in R(\beta)$-gmod as ([9]),

$\widehat{M}={\bf k}[z]\otimes_{\bf k}M.$

###  6.2. Localization

######  Definition 6.5 ([12]).

Let $M$ be a simple $R$-module. A graded braider $(M,R_{M},\phi)$ is non-
degenerate if $R_{M}(L(i)):M\circ L(i)\to L(i)\circ M$ is a non-zero
homomorphism.

For $R$-gmod, there exists a non-degenerate real commuting family of graded
braiders $(C_{i},R_{C_{i}},\phi_{i})_{i\in I}$([12]). Set $C_{\Lambda}:={\bf
M}(w_{0}\Lambda,\Lambda)$ and denote $C_{\Lambda_{i}}$ by $C_{i}$.

######  Proposition 6.6 ([15]).

For $\Lambda=\sum_{i}m_{i}\Lambda_{i}\in P_{+}$, we obtain the following
isomorphism up to grading shift:

(6.3) $C_{\Lambda}:={\bf M}(w_{0}\Lambda,\Lambda)\cong C_{1}^{\circ
m_{1}}\circ\cdots\circ C_{n}^{\circ m_{n}}.$

######  Theorem 6.7 ([12, Proposition 5.1]).

Define the function $\phi_{i}:Q\to\mathbb{Z}$ by

$\phi_{i}(\beta):=-(\beta,w_{0}\Lambda_{i}+\Lambda_{i}).$

Then there exists $\\{(C_{i},R_{C_{i}},\phi_{i})\\}_{i\in I}$ a non-degenerate
real commuting family of graded braiders of the monoidal category $R$-gmod.

Now, we take $\Gamma=P=\bigoplus_{i}\mathbb{Z}\Lambda_{i}$ and
$\Gamma_{+}=P_{+}=\bigoplus_{i}\mathbb{Z}_{\geq 0}\Lambda_{i}$. Here, we
obtain the localization $R\hbox{-gmod}[C_{i}^{\circ-1}\,|\,i\in I]$ by
$\\{(C_{i},R_{C_{i}},\phi_{i})\\}_{i\in I}$, which will be denoted by
$\widetilde{R}\hbox{-gmod}$.

By the above Proposition, it holds the following properties:

######  Proposition 6.8 ([12]).

Let $\Phi:\hbox{$R$-\hbox{gmod}}\to\widetilde{R}\hbox{-gmod}$ be the canonical
functor. Then,

  1. (1)

$\widetilde{R}\hbox{-gmod}$ is an abelian category and the functor $\Phi$ is
exact.

  2. (2)

For any simple object $S\in\hbox{$R$-\hbox{gmod}}$, $\Phi(S)$ is simple in
$\widetilde{R}\hbox{-gmod}$.

  3. (3)

$\widetilde{C}_{i}:=\Phi(C_{i})$ ($i\in I$) is invertible central graded
braider in $\widetilde{R}\hbox{-gmod}$.

For $\mu\in P$, define $\widetilde{C}_{\mu}$ such that
$\widetilde{C}_{\mu}:=\Phi(C_{\mu})$ for $\mu\in P_{+}$,
$\widetilde{C}_{-\Lambda_{i}}=C_{i}^{\circ-1}$ and
$\widetilde{C}_{\lambda+\mu}=\widetilde{C}_{\lambda}\circ\widetilde{C}_{\mu}$
for $\lambda,\mu\in P$ up to grading shift.

  4. (4)

Any simple object in $\widetilde{R}\hbox{-gmod}$ is isomorphic to
$\widetilde{C}_{\Lambda}\circ\Phi(S)$ for some simple module
$S\in\hbox{$R$-\hbox{gmod}}$ and $\Lambda\in P$.

Note that in (4) $\Lambda\in P$ and $S\in\hbox{$R$-\hbox{gmod}}$ are not
necessarily unique.

######  Remark 6.9.

In [12], the localization is applied to more general category
${\mathscr{C}}_{w}$, which is the full subcategory of $R$-gmod associated with
a Weyl group element $w$. The category $R$-gmod here coincides with
${\mathscr{C}}_{w_{0}}$ associated with the longest element $w_{0}$ in $W$.

######  Definition 6.10.

The category $\widetilde{R}\hbox{-gmod}$ is abelian and monoidal. Therefore,
its Grothendieck ring ${\mathcal{K}}(\widetilde{R}\hbox{-gmod})$ holds a
natural $\mathbb{Z}[q,q^{-1}]$-algebra structure, which defines a localized
quantum coordinate ring
$\widetilde{\mathcal{A}}_{q}(\mathfrak{n}):=\mathbb{Q}(q)\otimes_{\mathbb{Z}[q,q^{-1}]}{\mathcal{K}}(\widetilde{R}\hbox{-gmod})$.

Indeed, the Grothendieck ring ${\mathcal{K}}(\widetilde{R}\hbox{-gmod})$ is
described as follows:

######  Proposition 6.11 ([12, Corollary 5.4]).

The Grothendieck ring ${\mathcal{K}}(\widetilde{R}\hbox{-gmod})$ is isomorphic
to the left ring of quotients of the ring
${\mathcal{K}}(\hbox{$R$-\hbox{gmod}})$ with respect to the multiplicative set

${\mathcal{S}}:=\\{q^{k}\prod_{i\in I}[C_{i}]^{a_{i}}\mid
k\in\mathbb{Z},\,\,(a_{i})_{i\in I}\in\mathbb{Z}^{I}_{\geq 0}\\},$

that is,
${\mathcal{K}}(\widetilde{R}\hbox{-gmod})\cong{\mathcal{S}}^{-1}{\mathcal{K}}(\hbox{$R$-\hbox{gmod}})$.

##  7\. Crystal Structure on localized quantum coordinate rings

In this section, one of the main theorems will be introduced, which claims
that we can define a certain crystal structure on the localized quantum
coordinate ring $\widetilde{\mathcal{A}}_{q}(\mathfrak{n})$, more precisely, a
crystal structure will be defined on the family of all self-dual simple
modules in $\widetilde{R}\hbox{-gmod}$.

The following lemma is essential for this section.

######  Lemma 7.1 ([10, Proposition 2.18]).

For any $i\in I$, $\beta,\gamma\in Q_{+}$, any modules $M\in R(\beta)$-gmod
and $N\in R(\gamma)$-gmod, one has the following exact sequence in
$R(\beta+\gamma-\alpha_{i})$-gmod:

(7.1) $0\longrightarrow E_{i}M\circ N\longrightarrow E_{i}(M\circ
N)\longrightarrow q^{-(\alpha,\,\beta)}M\circ E_{i}N\longrightarrow 0.$

For $i\in I$, let $i^{*}\in I$ be a unique index satisfying
$\Lambda_{i^{*}}=-w_{0}\Lambda_{i}$.

######  Lemma 7.2.

  1. (1)

For $S\in\hbox{$R$-\hbox{gmod}}$ and $i\in I$, if $E_{i}S=0$, then the module
$E_{i}C_{\Lambda_{i^{*}}}\circ S$ is a simple module.

  2. (2)

If $E_{i}S=0$ for $S\in\hbox{$R$-\hbox{gmod}}$, then we get for $\Lambda\in
P_{+}$ with $\langle h_{i^{*}},\Lambda\rangle>0$,

(7.2) ${\rm soc}(E_{i}(C_{\Lambda}\circ S))\cong
C_{\Lambda-\Lambda_{i^{*}}}\circ(E_{i}C_{i^{*}}\circ S),$

up to grading shift.

Proof. (1) By applying (7.1) in Lemma 7.1 to $C_{\Lambda_{i^{*}}}\circ S\cong
S\circ C_{\Lambda_{i^{*}}}$, we know that

$E_{i}C_{\Lambda_{i^{*}}}\circ S\cong E_{i}(C_{\Lambda_{i^{*}}}\circ S)\cong
E_{i}(S\circ C_{\Lambda_{i^{*}}})\cong S\circ E_{i}C_{\Lambda_{i^{*}}}$

up to grading shift. Now it follows from Proposition 3.13 in [12] that
$E_{i}C_{\Lambda_{i^{*}}}\circ S$ holds a simple socle $S\Delta
E_{i}C_{\Lambda_{i^{*}}}$ and a simple head $E_{i}C_{\Lambda_{i^{*}}}\nabla S$
and

$S\Delta E_{i}C_{\Lambda_{i^{*}}}\cong E_{i}C_{\Lambda_{i^{*}}}\nabla
S\cong{\rm Im}\,\,{\bf r}_{E_{i}C_{\Lambda_{i^{*}}},\,S}\neq 0,$

where ${\bf r}_{M,N}:M\circ N\to N\circ M$ is a R-matrix for
$M,N\in\hbox{$R$-\hbox{gmod}}$ introduced in [12, 3.2]. By Proposition 3.13 in
[12] we also have that

${\rm H\textsc{om}}_{R}(E_{i}C_{\Lambda_{i^{*}}}\circ
S,E_{i}C_{\Lambda_{i^{*}}}\circ S)={\bf k}\,{\rm
id}_{E_{i}C_{\Lambda_{i^{*}}}\circ S},\qquad{\rm
H\textsc{om}}_{R}(E_{i}C_{\Lambda_{i^{*}}}\circ S,S\circ
E_{i}C_{\Lambda_{i^{*}}})={\bf k}\,{\bf r}_{E_{i}C_{\Lambda_{i^{*}}},\,S}.$

Here let us denote the isomorphism $\xi:S\circ
E_{i}C_{\Lambda_{i^{*}}}\smash{\mathop{\longrightarrow}\limits^{\sim}}E_{i}C_{\Lambda_{i^{*}}}\circ
S$. Since $\xi\circ{\bf r}_{E_{i}C_{\Lambda_{i^{*}}},\,S}\in{\rm
H\textsc{om}}(E_{i}C_{\Lambda_{i^{*}}}\circ S,E_{i}C_{\Lambda_{i^{*}}}\circ
S)$ and ${\bf r}_{E_{i}C_{\Lambda_{i^{*}}},\,S}$ never vanishes, we obtain
$\xi\circ{\bf r}_{E_{i}C_{\Lambda_{i^{*}}},\,S}=c\,{\rm
id}_{E_{i}C_{\Lambda_{i^{*}}}\circ S}$ for some $c\in{\bf k}^{\times}$, which
implies that

${\rm Im}\,\,{\bf r}_{E_{i}C_{\Lambda_{i^{*}}},S}\cong S\circ
E_{i}C_{\Lambda_{i^{*}}}\cong E_{i}C_{\Lambda_{i^{*}}}\circ S,$

and then $E_{i}C_{\Lambda_{i^{*}}}\circ S$ is a simple module in $R$-gmod.

(2) Since $E_{i}S=0$, by (7.1) one has $E_{i}C_{\Lambda}\circ S\cong
E_{i}(C_{\Lambda}\circ S)$ and then

(7.3) ${\rm soc}(E_{i}C_{\Lambda}\circ S)\cong{\rm soc}(E_{i}(C_{\Lambda}\circ
S)).$

Let us show

(7.4) ${\rm soc}(E_{i}C_{\Lambda})\cong C_{\Lambda-\Lambda_{i^{*}}}\circ
E_{i}C_{i^{*}},$

up to grading shift. By the result of (1) and (7.1), one knows that
$C_{\Lambda-\Lambda_{i^{*}}}\circ E_{i}C_{i^{*}}\cong E_{i}C_{i^{*}}\circ
C_{\Lambda-\Lambda_{i^{*}}}$ is a simple submodule of $E_{i}(C_{\Lambda})\cong
E_{i}(C_{i^{*}}\circ C_{\Lambda-\Lambda_{i^{*}}})$. It is also known that
$E_{i}(C_{\Lambda})$ holds a simple socle [16] and then we obtain (7.4). Due
to (1) and (7.4) one finds that $C_{\Lambda-\Lambda_{i^{*}}}\circ
E_{i}C_{\Lambda_{i^{*}}}\circ S$ is a simple submodule of
$E_{i}C_{\Lambda}\circ S$ and since $E_{i}C_{\Lambda}\circ S$ possesses a
simple socle, we get (7.2).

We set

$\mathbb{B}(\widetilde{R}\hbox{-gmod}):=\\{L\,|\,L\hbox{ is a self-dual simple
module in }\widetilde{R}\hbox{-gmod}\\}.$

######  Lemma 7.3 ([12]).

For any simple $L\in\widetilde{R}\hbox{-gmod}$, there exists a unique
$n\in\mathbb{Z}$ such that $q^{n}L$ is self-dual simple. For a simple module
$L\in\widetilde{R}\hbox{-gmod}$ we define $\delta(L)$ to be this integer $n$.

Then by this lemma, we find that $\mathbb{B}(\widetilde{R}\hbox{-gmod})$
includes all simple modules in $\widetilde{R}\hbox{-gmod}$ up to grading
shift. For a simple object
$\widetilde{C}_{\Lambda}\circ\Phi(S)\in\widetilde{R}\hbox{-gmod}$ we write
simply $C_{\Lambda}\circ S$ if there is no confusion.

Now let us define the Kashiwara operators $\widetilde{F}_{i}$ and
$\widetilde{E}_{i}$ ($i\in I$) on $\mathbb{B}(\widetilde{R}\hbox{-gmod})$ by

(7.5) $\displaystyle\widetilde{F}_{i}(C_{\Lambda}\circ
S)=q^{\delta(C_{\Lambda}\circ\widetilde{F}_{i}S)}C_{\Lambda}\circ\widetilde{F}_{i}S,$
(7.6) $\displaystyle\widetilde{E}_{i}(C_{\Lambda}\circ
S)=\begin{cases}q^{\delta(C_{\Lambda}\circ\widetilde{E}_{i}S)}C_{\Lambda}\circ\widetilde{E}_{i}S&\hbox{
if }E_{i}S\neq 0,\\\
q^{\delta(C_{\Lambda-\Lambda_{i^{*}}}\circ(\widetilde{E}_{i}C_{\Lambda_{i^{*}}}\circ
S))}C_{\Lambda-\Lambda_{i^{*}}}\circ(\widetilde{E}_{i}C_{\Lambda_{i^{*}}}\circ
S)&\hbox{ if }E_{i}S=0,\end{cases}$

where $C_{\Lambda}\circ S$ is a self-dual simple module in
$\widetilde{R}\hbox{-gmod}$, the actions $\widetilde{E}_{i}S$ and
$\widetilde{F}_{i}S$ are given in (4.2), which is defined on the family of all
self-dual simple modules in $R$-gmod and in (7.6) the module
$\widetilde{E}_{i}C_{\Lambda_{i^{*}}}\circ S$ is simple by Lemma 7.2. Note
that for any $m>0$, $\widetilde{E}_{i}^{m}(C_{\Lambda}\circ S)\neq
0,\,\,\widetilde{F}_{i}^{m}(C_{\Lambda}\circ S)\neq 0$.

Let
$\Psi:\mathbb{B}(\hbox{$R$-\hbox{gmod}})\smash{\mathop{\longrightarrow}\limits^{\sim}}B(\infty)$
be as in Theorem 4.4. For $C_{\Lambda}\circ
S\in\mathbb{B}(\widetilde{R}\hbox{-gmod})$, we also define

(7.7) $\begin{array}[]{ll}&\varepsilon_{i}(C_{\Lambda}\circ
S)=\varepsilon_{i}(\Psi(S))-\langle
h_{i},w_{0}\Lambda\rangle,\qquad\mathrm{wt}(C_{\Lambda}\circ
S)=\mathrm{wt}(\Psi(S))+w_{0}\Lambda-\Lambda,\\\ &\varphi_{i}(C_{\Lambda}\circ
S)=\varepsilon_{i}(\Psi(C_{\Lambda}\circ S))+\langle
h_{i},\mathrm{wt}(C_{\Lambda}\circ S)\rangle.\end{array}$

######  Theorem 7.4.

The 6-tuple
$(\mathbb{B}(\widetilde{R}\hbox{-gmod}),\mathrm{wt},\\{\varepsilon_{i}\\},\\{\varphi_{i}\\},\\{\widetilde{E}_{i}\\},\\{\widetilde{F}_{i}\\})_{i\in
I}$ is a crystal.

Proof. First, we should show the well-definedness of all data
$(\mathrm{wt},\\{\varepsilon_{i}\\},\\{\varphi_{i}\\},\\{\widetilde{E}_{i}\\},\\{\widetilde{F}_{i}\\})_{i\in
I}$, i.e., these data do not depend on the presentation of a simple module
$L=C_{\Lambda}\circ S\cong C_{\Lambda^{\prime}}\circ S^{\prime}$. Let us
assume that for a simple module $L\in\mathbb{B}(\widetilde{R}\hbox{-gmod})$
there exist $\Lambda,\Lambda^{\prime}\in P$ and
$S,\,S^{\prime}\in\mathbb{B}(\hbox{$R$-\hbox{gmod}})$ such that
$L=C_{\Lambda}\circ S\cong C_{\Lambda^{\prime}}\circ S^{\prime}$. By the
definition of localization, one finds that there is a weight $\mu\in P$ such
that $\Lambda+\mu,\,\Lambda^{\prime}+\mu\in P_{+}$ and $C_{\Lambda+\mu}\circ
S\cong C_{\Lambda^{\prime}+\mu}\circ S^{\prime}$ in $R$-gmod up to grading
shift. Let us show the following lemma:

######  Lemma 7.5.

For a dominant weight $\mu\gg 0$, $i\in I$ and a simple module
$S\in\hbox{$R$-\hbox{gmod}}$, one has the following up to grading shift:

(7.8) $C_{\mu}\circ(L(i)\nabla S)\cong L(i)\nabla(C_{\mu}\circ S).$

Proof. Since $C_{\mu}$ is a central object by Theorem 5.2 in [12], one has

(7.9) $C_{\mu}\circ L(i)\circ S\cong L(i)\circ C_{\mu}\circ S,$

up to grading shift. Both $C_{\mu}\circ(L(i)\nabla S)$ and
$L(i)\nabla(C_{\mu}\circ S)$ are simple quotients in $C_{\mu}\circ L(i)\circ
S\cong L(i)\circ C_{\mu}\circ S$. Since $C_{\mu}$ is a real simple module,
from [14] and [16] we have that $C_{\mu}\circ S$ is simple and $L(i)\circ
C_{\mu}\circ S$ holds a simple head and then it is isomorphic to
$C_{\mu}\circ(L(i)\nabla S)$ and $L(i)\nabla(C_{\mu}\circ S)$, which implies
that $C_{\mu}\circ(L(i)\nabla S)\cong L(i)\nabla(C_{\mu}\circ S)$ up to
grading shift.

By this lemma, we obtain

$C_{\Lambda+\mu}\circ\widetilde{F}_{i}S\cong C_{\Lambda+\mu}\circ(L(i)\nabla
S)\cong L(i)\nabla(C_{\Lambda+\mu}\circ S)\cong
L(i)\nabla(C_{\Lambda^{\prime}+\mu}\circ S^{\prime})\cong
C_{\Lambda^{\prime}+\mu}\circ(L(i)\nabla S^{\prime})\cong
C_{\Lambda^{\prime}+\mu}\circ(\widetilde{F}_{i}S^{\prime})$

up to grading shift. It shows the well-definedness of $\widetilde{F}_{i}$.

Next, let us see the one for $\widetilde{E}_{i}$. By Lemma 7.1, it yields the
following exact sequence:

(7.10) $0\longrightarrow E_{i}C_{\Lambda+\mu}\circ S\longrightarrow
E_{i}(C_{\Lambda+\mu}\circ S)\longrightarrow
q^{(\alpha_{i},\xi)}C_{\Lambda+\mu}\circ E_{i}S\longrightarrow 0,$

where we set $\xi=w_{0}(\Lambda+\mu)-(\Lambda+\mu)$. Since
$C_{\Lambda+\mu}\circ S$ is simple, by Lemma 4.3 one knows that
$E_{i}(C_{\Lambda+\mu}\circ S)$ possesses a simple head and then a unique
maximal submodule $M$, which includes the module $E_{i}C_{\Lambda+\mu}\circ
S$. Set $M^{\prime}=M/E_{i}C_{\Lambda+\mu}\circ S$. Now, we get the following
commutative diagram:

(7.11) $\begin{array}[]{ccccccccc}&&0&&0&&0&&\\\
&&\downarrow&&\downarrow&&\downarrow&&\\\ 0&\to&E_{i}C_{\Lambda+\mu}\circ
S&\smash{\mathop{\longrightarrow}\limits^{=}}&E_{i}C_{\Lambda+\mu}\circ
S&\to&0&\to&0\\\ &&\downarrow&&\downarrow&&\downarrow&&\\\
0&\to&M&\smash{\mathop{\longrightarrow}\limits^{\,h\,}}&E_{i}(C_{\Lambda+\mu}\circ
S)&\to&{\rm hd}(E_{i}(C_{\Lambda+\mu}\circ S))&\to&0\\\
&&\downarrow&&\quad\downarrow\psi&&\downarrow&&\\\
0&\to&M^{\prime}&\smash{\mathop{\longrightarrow}\limits^{\,g\,}}&q^{(\alpha_{i},\xi)}C_{\Lambda+\mu}\circ
E_{i}S&\to&q^{(\alpha_{i},\xi)}(C_{\Lambda+\mu}\circ
E_{i}S)/M^{\prime}&\to&0\\\ &&\downarrow&&\downarrow&&\downarrow&&\\\
&&0&&0&&0&&\end{array}.$

In this diagram, all of three rows are exact and the first two columns are
also exact. Therefore, applying the nine lemma to this diagram, we obtain

${\rm hd}(E_{i}(C_{\Lambda+\mu}\circ S))\cong
q^{(\alpha_{i},\xi)}(C_{\Lambda+\mu}\circ E_{i}S)/M^{\prime}.$

######  Lemma 7.6.

The module $M^{\prime}$ is a unique maximal submodule of
$q^{(\alpha_{i},\xi)}C_{\Lambda+\mu}\circ E_{i}S$.

Proof. First let us show the maximality of $M^{\prime}$. Assume that there
exists a submodule $\overline{M}$ such that $M^{\prime}\cong
g(M^{\prime})\subsetneqq\overline{M}\subsetneqq
q^{(\alpha_{i},\xi)}C_{\Lambda+\mu}\circ E_{i}S$. Since
$\psi^{-1}(\overline{M})$ is a submodule of $E_{i}(C_{\Lambda+\mu}\circ S)$
and $\psi\circ h(M)\cong g(M^{\prime})$, we have
$h(M)\subsetneqq\psi^{-1}(\overline{M})$. But, since $h(M)$ is a maximal
submodule of $E_{i}(C_{\Lambda+\mu}\circ S)$, one has
$\psi^{-1}(\overline{M})=E_{i}(C_{\Lambda+\mu}\circ S)$ and then
$M^{\prime}=q^{(\alpha_{i},\xi)}C_{\Lambda+\mu}\circ E_{i}S$, which derives a
contradiction.

Next, let us see the uniqueness of $M^{\prime}$. Suppose that there exists a
maximal submoudle $\widetilde{M}\subsetneqq
q^{(\alpha_{i},\xi)}C_{\Lambda+\mu}\circ E_{i}S$. It is evident that
$\psi^{-1}(\widetilde{M})$ is a maximal submodule of
$E_{i}(C_{\Lambda+\mu}\circ S)$. Since $h(M)$ is a unique maximal submodule of
$E_{i}(C_{\Lambda+\mu}\circ S)$, we get $\psi^{-1}(\widetilde{M})=h(M)$ and
then $\widetilde{M}=\psi\circ h(M)=g(M^{\prime})\cong M^{\prime}$

By this lemma one has that $M^{\prime}$ is a unique maximal submodule of
$q^{(\alpha_{i},\xi)}C_{\Lambda+\mu}\circ E_{i}S$ and then ${\rm
hd}(q^{(\alpha_{i},\xi)}C_{\Lambda+\mu}\circ E_{i}S)\cong
q^{(\alpha_{i},\xi)}(C_{\Lambda+\mu}\circ E_{i}S)/M^{\prime}$. Therefore, one
obtains

${\rm hd}(E_{i}(C_{\Lambda+\mu}\circ S))\cong{\rm
hd}(q^{(\alpha_{i},\xi)}C_{\Lambda+\mu}\circ E_{i}S).$

Hence, for $C_{\Lambda+\mu}\circ S\cong C_{\Lambda^{\prime}+\mu}\circ
S^{\prime}$ (up to grading shift) it follows that

${\rm hd}(C_{\Lambda+\mu}\circ E_{i}S)\cong{\rm hd}(E_{i}(C_{\Lambda+\mu}\circ
S))\cong{\rm hd}(E_{i}(C_{\Lambda^{\prime}+\mu}\circ S^{\prime}))\cong{\rm
hd}(C_{\Lambda^{\prime}+\mu}\circ E_{i}S^{\prime}),$

up to grading shift. We also have

(7.12) $C_{\Lambda+\mu}\circ\widetilde{E}_{i}S\cong{\rm
hd}(C_{\Lambda+\mu}\circ E_{i}S),\qquad
C_{\Lambda^{\prime}+\mu}\circ\widetilde{E}_{i}S^{\prime}\cong{\rm
hd}(C_{\Lambda^{\prime}+\mu}\circ E_{i}S^{\prime}),$

up to grading shift. Thus, if $E_{i}S,E_{i}S^{\prime}\neq 0$, we obtain
$C_{\Lambda+\mu}\circ\widetilde{E}_{i}S\cong
C_{\Lambda^{\prime}+\mu}\circ\widetilde{E}_{i}S^{\prime}$ up to grading shift.
Let us assume that $E_{i}S=0$. Owing to (7.1) one has
$E_{i}C_{\Lambda+\mu}\circ S\cong E_{i}(C_{\Lambda+\mu}\circ S)$. By [12,
Proposition 3.23] one has

(7.13) $\varepsilon_{i}(C_{j})=\begin{cases}1&\hbox{ if }j=i^{*},\\\ 0&\hbox{
otherwise.}\end{cases}$

Since $E_{i}C_{\Lambda_{i^{*}}}\cong\widetilde{E}_{i}C_{\Lambda_{i^{*}}}$ by
[14, Corollary 10.1.7], one has

(7.14) ${\rm soc}(E_{i}C_{\Lambda+\mu})\cong
C_{\Lambda+\mu-\Lambda_{i^{*}}}\circ E_{i}C_{\Lambda_{i^{*}}}\cong
C_{\Lambda+\mu-\Lambda_{i^{*}}}\circ\widetilde{E}_{i}C_{\Lambda_{i^{*}}},$

up to grading shift. Hence, if $E_{i}S^{\prime}=0$, one gets

$\displaystyle
C_{\Lambda+\mu-\Lambda_{i^{*}}}\circ\widetilde{E}_{i}C_{\Lambda_{i^{*}}}\circ
S$ $\displaystyle\cong$ $\displaystyle C_{\Lambda+\mu-\Lambda_{i^{*}}}\circ
E_{i}C_{\Lambda_{i^{*}}}\circ S\cong{\rm soc}(E_{i}C_{\Lambda+\mu})\circ
S\cong{\rm soc}(E_{i}(C_{\Lambda+\mu}\circ S))$ $\displaystyle\cong$
$\displaystyle{\rm soc}(E_{i}(C_{\Lambda^{\prime}+\mu}\circ
S^{\prime}))\cong{\rm soc}(E_{i}C_{\Lambda^{\prime}+\mu})\circ S^{\prime}\cong
C_{\Lambda^{\prime}+\mu-\Lambda_{i^{*}}}\circ E_{i}C_{\Lambda_{i^{*}}}\circ
S^{\prime}$ $\displaystyle\cong$ $\displaystyle
C_{\Lambda^{\prime}+\mu-\Lambda_{i^{*}}}\circ\widetilde{E}_{i}C_{\Lambda_{i^{*}}}\circ
S^{\prime},$

up to grading shift. Even if $E_{i}S^{\prime}\neq 0$, it follows from (7.12)
and (7.14) that the simple module
$C_{\Lambda^{\prime}+\mu}\circ\widetilde{E}_{i}S^{\prime}$ is isomorphic to

${\rm soc}E_{i}(C_{\Lambda^{\prime}+\mu}\circ S^{\prime})\cong{\rm
soc}E_{i}(C_{\Lambda+\mu}\circ S)\cong
C_{\Lambda+\mu-\Lambda_{i^{*}}}\circ(\widetilde{E}_{i}C_{\Lambda_{i^{*}}}\circ
S)$

up to grading shift, which implies the well-definedness of $\widetilde{E}_{i}$
on $\mathbb{B}(\widetilde{R}\hbox{-gmod})$.

Let us investigate the well-definedness of the function $\varepsilon_{i}$. For
$n>0$, $\mu\in P_{+}$ and a simple module $S\in R$-gmod, the following formula
is an immediate consequence of (7.10):

(7.15) $[E_{i}^{n}(C_{\mu}\circ
S)]=\bigoplus_{k=0}^{n}a_{k}(q)[E_{i}^{k}C_{\mu}\circ E_{i}^{n-k}S],$

where $[M]$ means an equivalence class of a module $M$ in Grothendieck ring
${{\mathcal{K}}}(\hbox{$R$-\hbox{gmod}})$ and $a_{k}(q)$ is a non-zero Laurent
polynomial in $q$ with non-negative coefficients. It implies that
$\varepsilon_{i}(C_{\mu}\circ S)=\varepsilon_{i}(C_{\mu})+\varepsilon_{i}(S)$.
Therefore, since $\varepsilon_{i}(C_{\Lambda^{*}})=1$ and then
$\varepsilon_{i}(C_{\mu})$ coincides with the number of
$C_{\Lambda_{i^{*}}}=C_{-w_{0}\Lambda_{i}}$ in $C_{\mu}$, one gets
$\varepsilon_{i}(C_{\mu})=\langle h_{i},-w_{0}\mu\rangle$. Thus,

$\varepsilon_{i}(C_{\mu}\circ S)=\varepsilon_{i}(S)-\langle
h_{i},w_{0}\mu\rangle.$

Hence, if $C_{\Lambda+\mu}\circ S\cong C_{\Lambda^{\prime}+\mu}\circ
S^{\prime}$ for $\Lambda+\mu,\Lambda^{\prime}+\mu\in P_{+}$, then
$\varepsilon_{i}(C_{\Lambda+\mu}\circ
S)=\varepsilon(C_{\Lambda+^{\prime}\mu}\circ S^{\prime})$ and then

$\varepsilon_{i}(S)-\langle
h_{i},w_{0}(\Lambda+\mu)\rangle=\varepsilon_{i}(S^{\prime})-\langle
h_{i},w_{0}(\Lambda^{\prime}+\mu)\rangle,$

holds, which shows $\varepsilon_{i}(S)-\langle
h_{i},w_{0}\Lambda\rangle=\varepsilon_{i}(S^{\prime})-\langle
h_{i},w_{0}\Lambda^{\prime}\rangle$. Now, we get the well-definedness of
$\varepsilon_{i}$. The well-definedness of $\varphi_{i}$ and $\mathrm{wt}$ is
trivial, which completes the proof for the well-definedness of
$\widetilde{E}_{i},\widetilde{F}_{i},\varepsilon_{i},\varphi_{i}$ and
$\mathrm{wt}$ on $\mathbb{B}(\widetilde{R}\hbox{-gmod})$.

Next, let us see that the 6-tuple satisfies the conditions of crystal as in
Definition 3.1 (1)–(5). The condition (1) is automatically satisfied by the
definition of $\varphi_{i}$. As for the condition (5),
$\varphi_{i}(b)=-\infty$ never happen. So we shall see the condition (2). For
$C_{\Lambda}\circ S\in\mathbb{B}(\widetilde{R}\hbox{-gmod})$, suppose that
$\widetilde{E}_{i}S\neq 0$.

$\varepsilon(\widetilde{E}_{i}(C_{\Lambda}\circ
S))=\varepsilon_{i}(C_{\Lambda}\circ\widetilde{E}_{i}S)=\varepsilon_{i}(\widetilde{E}_{i}S)-\langle
h_{i},w_{0}\Lambda\rangle=\varepsilon_{i}(S)-1-\langle
h_{i},w_{0}\Lambda\rangle=\varepsilon_{i}(C_{\Lambda}\circ S)-1.$

It is trivial that
$\mathrm{wt}(C_{\Lambda}\circ\widetilde{E}_{i}S)=\mathrm{wt}(\widetilde{E}_{i})+w_{0}\Lambda-\Lambda=\mathrm{wt}(S)+\alpha_{i}w_{0}\Lambda-\Lambda=\mathrm{wt}(C_{\Lambda}\circ
S)+\alpha_{i}$. Suppose that $E_{i}S=0$, which implies that
$0=\varepsilon_{i}(S)=\varepsilon_{i}(\widetilde{E}_{i}C_{\Lambda_{i^{*}}}\circ
S)$ and $\varepsilon_{i}(C_{\Lambda}\circ
S)=\varepsilon_{i}(C_{\Lambda})+\varepsilon_{i}(S)=-\langle
h_{i},w_{0}\Lambda\rangle$. Then, we have

$\varepsilon(\widetilde{E}_{i}(C_{\Lambda}\circ
S))=\varepsilon_{i}(C_{\Lambda-\Lambda_{i^{*}}}\circ(\widetilde{E}_{i}C_{\Lambda_{i^{*}}}\circ
S))=\varepsilon_{i}(\widetilde{E}_{i}C_{\Lambda_{i^{*}}}\circ S)-\langle
h_{i},w_{0}({\Lambda-\Lambda_{i^{*}}})\rangle=-\langle
h_{i},w_{0}({\Lambda})\rangle-1=\varepsilon_{i}(C_{\Lambda}\circ S)-1.$

We also get

$\displaystyle\mathrm{wt}(C_{\Lambda-\Lambda_{i^{*}}}\circ(\widetilde{E}_{i}C_{\Lambda_{i^{*}}}\circ
S))$ $\displaystyle=$
$\displaystyle\mathrm{wt}(\widetilde{E}_{i}C_{\Lambda_{i^{*}}}\circ
S)+w_{0}(\Lambda-\Lambda_{i^{*}})-\Lambda+\Lambda_{i^{*}}$ $\displaystyle=$
$\displaystyle\mathrm{wt}(S)+\mathrm{wt}(\widetilde{E}_{i}C_{\Lambda_{i^{*}}})+w_{0}(\Lambda-\Lambda_{i^{*}})-\Lambda+\Lambda_{i^{*}}$
$\displaystyle=$
$\displaystyle\mathrm{wt}(S)+w_{0}(\Lambda_{i^{*}})-\Lambda_{i^{*}}+\alpha_{i}+w_{0}(\Lambda-\Lambda_{i^{*}})-\Lambda+\Lambda_{i^{*}}$
$\displaystyle=$
$\displaystyle\mathrm{wt}(S)+w_{0}\Lambda-\Lambda+\alpha_{i}=\mathrm{wt}(C_{\Lambda}\circ
S)+\alpha_{i}$

The condition (3) will be shown by similar way to the condition (2). Let us
check the condition (4). To do that, it suffices to show that
$\widetilde{E}_{i}\widetilde{F}_{i}={\rm id}_{\mathbb{B}(\widetilde{R}{\rm-
gmod})}=\widetilde{F}_{i}\widetilde{E}_{i}$. At first, since
$\widetilde{E}_{i}(\widetilde{F}_{i}S)=S\neq 0$ for any
$S\in\mathbb{B}(\hbox{$R$-\hbox{gmod}})$, it is trivial that
$\widetilde{E}_{i}\widetilde{F}_{i}(C_{\Lambda}\circ
S)=\widetilde{E}_{i}(C_{\Lambda}\circ\widetilde{F}_{i}S)=C_{\Lambda}\circ S$.
In the case $E_{i}S\neq 0$, it is also evident that
$\widetilde{F}_{i}\widetilde{E}_{i}(C_{\Lambda}\circ
S)=\widetilde{F}_{i}(C_{\Lambda}\circ\widetilde{E}_{i}S)\cong
C_{\Lambda}\circ\widetilde{F}_{i}\widetilde{E}_{i}S\cong C_{\Lambda}\circ S$.
So, assuming $E_{i}S=0$, let us show the following lemma:

######  Lemma 7.7.

For any simple $S\in\hbox{$R$-\hbox{gmod}}$ and $i\in I$, if $E_{i}S=0$, then
we obtain

(7.16) $\widetilde{F}_{i}(\widetilde{E}_{i}C_{\Lambda_{i^{*}}}\circ S)\cong
C_{\Lambda_{i^{*}}}\circ S,$

up to grading shift.

Proof. It follows from (7.1) that there exists the following exact sequence

$0\longrightarrow(E_{i}C_{\Lambda_{i^{*}}})\circ S\longrightarrow
E_{i}(C_{\Lambda_{i^{*}}}\circ S)\longrightarrow
q^{-(\alpha_{i},\Lambda_{i^{*}}-w_{0}\Lambda_{i^{*}})}C_{\Lambda_{i^{*}}}\circ
E_{i}S\longrightarrow 0.$

Since $E_{i}S=0$ and
$E_{i}C_{\Lambda_{i^{*}}}\cong\widetilde{E}_{i}C_{\Lambda_{i^{*}}}$, one has
$\widetilde{E}_{i}(C_{\Lambda_{i^{*}}})\circ S\cong
E_{i}(C_{\Lambda_{i^{*}}}\circ S)$. Since
$\widetilde{E}_{i}(C_{\Lambda_{i^{*}}})\circ S$ is simple by Lemma 7.2, one
obtains $\widetilde{E}_{i}(C_{\Lambda_{i^{*}}})\circ
S\cong\widetilde{E}_{i}(C_{\Lambda_{i^{*}}}\circ S)$ in $R$-gmod up to grading
shift, which shows (7.16).

Therefore, by this lemma under the assumption $E_{i}S=0$ we get

$\widetilde{F}_{i}\widetilde{E}_{i}(C_{\Lambda}\circ
S)\cong\widetilde{F}_{i}(C_{\Lambda-\Lambda_{i^{*}}}\circ(\widetilde{E}_{i}C_{\Lambda_{i^{*}}}\circ
S))\cong
C_{\Lambda-\Lambda_{i^{*}}}\circ\widetilde{F}_{i}(\widetilde{E}_{i}C_{\Lambda_{i^{*}}}\circ
S)\cong C_{\Lambda-\Lambda_{i^{*}}}\circ C_{\Lambda_{i^{*}}}\circ S\cong
C_{\Lambda}\circ S.$

Thus, we get that the condition (4) holds and then we have completed the proof
of the theorem.

##  8\. Cellular Crystal $\mathbb{B}_{\mathbf{i}}$ and
$\mathbb{B}(\widetilde{R}$-gmod$)$

In this section, we shall describe the explicit crystal structure on
$\mathbb{B}(\widetilde{R}\hbox{-gmod})$ defined by constructing the
isomorphism to the cellular crystal $\mathbb{B}_{\mathbf{i}}$.

Here, by Proposition 6.2 we observe that there seems to exist a certain
correspondence:

$\displaystyle\\{C_{\Lambda}\,|\,\Lambda\in
P_{+}\\}\subset\hbox{$R$-\hbox{gmod}}$ $\displaystyle\longleftrightarrow$
$\displaystyle{\mathcal{H}}_{\mathbf{i}}$ $\displaystyle
C_{\Lambda}=\widetilde{F}_{i_{1}}^{m_{1}}\cdots\widetilde{F}_{i_{N}}^{m_{N}}{\bf
1}$ $\displaystyle\longleftrightarrow$ $\displaystyle{\bf
h}_{\Lambda}=\tilde{f}_{i_{1}}^{m_{1}}\tilde{f}_{i_{2}}^{m_{2}}\cdots\tilde{f}_{i_{N}}^{m_{N}}((0)_{i_{1}}\otimes(0)_{i_{2}}\otimes\cdots\otimes(0)_{i_{N}})$

Together with the result of Proposition 3.11, we obtain the following:

######  Theorem 8.1.

For any reduced longest word ${\mathbf{i}}=i_{1}i_{2}\cdots i_{N}$, there
exists an isomorphism of crystals:

$\displaystyle\widetilde{\Psi}:\mathbb{B}(\widetilde{R}\hbox{-gmod})$
$\displaystyle\smash{\mathop{\longrightarrow}\limits^{\sim}}$
$\displaystyle\mathbb{B}_{\mathbf{i}}=\bigcup_{h\in{\mathcal{H}}_{\mathbf{i}}}B^{h}(\infty)$
$\displaystyle C_{\Lambda}\circ S$ $\displaystyle\longmapsto$
$\displaystyle{\mathbf{h}}_{\Lambda}+\Psi(S)\in
B^{{\mathbf{h}}_{\Lambda}}(\infty),$

where
$\Psi:\mathbb{B}(\hbox{$R$-\hbox{gmod}})\smash{\mathop{\longrightarrow}\limits^{\sim}}B(\infty)$
is the isomorphism of crystals given in Theorem 4.4, $S$ is simple in
$\mathbb{B}(\hbox{$R$-\hbox{gmod}})$ and for
$\Lambda=\sum_{i}a_{i}\Lambda_{i}$ set
${\mathbf{h}}_{\Lambda}=\sum_{i}a_{i}{\mathbf{h}}_{i}$.

Proof. We shall show:

  1. (1)

Well-definedness of $\widetilde{\Psi}$, that is, if $C_{\Lambda}\circ S\cong
C_{\Lambda^{\prime}}\circ S^{\prime}$, then
${\mathbf{h}}_{\Lambda}+\Psi(S)={\mathbf{h}}_{\Lambda^{\prime}}+\Psi(S^{\prime})$.

  2. (2)

Bijectivity of $\widetilde{\Psi}$.

  3. (3)

$\mathrm{wt}(C_{\Lambda}\circ S)=\mathrm{wt}({\mathbf{h}}_{\Lambda}+\Psi(S))$.

  4. (4)

$\varepsilon_{i}(C_{\Lambda}\circ
S)=\varepsilon_{i}({\mathbf{h}}_{\Lambda}+\Psi(S))$.

  5. (5)

$\widetilde{\Psi}\circ\widetilde{F}_{i}=\tilde{f}_{i}\,\circ\widetilde{\Psi}$
and
$\widetilde{\Psi}\circ\widetilde{E}_{i}=\tilde{e}_{i}\,\circ\widetilde{\Psi}$.

First, let us see (1). We may show that ${\bf h}_{\Lambda+\mu}+\Psi(S)={\bf
h}_{\Lambda^{\prime}+\mu}+\Psi(S^{\prime})$ in case of $C_{\Lambda+\mu}\circ
S\cong C_{\Lambda^{\prime}+\mu}\circ S^{\prime}$ for $\mu\gg 0$ satisfying
$\Lambda+\mu,\Lambda^{\prime}+\mu\in P_{+}$. Thus, we shall show if
$S^{\prime}\cong C_{\Lambda}\circ S$ for simple modules
$S,S^{\prime}\in\hbox{$R$-\hbox{gmod}}$ and $\Lambda\in P_{+}$, then
$\Psi(S^{\prime})=\Psi(S)+{\bf h}_{\Lambda}$. Here we denote
$(0)_{i_{1}}\otimes\cdots\otimes(0)_{i_{N}}$ by $\bf 0_{\mathbf{i}}$ and
identify $u_{\infty}$ with $\bf 0_{\mathbf{i}}$. By the definition of the
determinantial module $C_{\Lambda}={\bf M}(w_{0}\Lambda,\Lambda)$ and the
isomorphism of crystals
$\Psi:\hbox{$R$-\hbox{gmod}}\smash{\mathop{\longrightarrow}\limits^{\sim}}B(\infty)$,
we know that there exists a sequence of indices
$j_{1},\cdots,j_{l},j_{l+1},\cdots,j_{l+m}$ such that
$\widetilde{E}_{j_{l+m}}\cdots\widetilde{E}_{j_{l+1}}C_{\Lambda}={\bf 1}$ and
$\widetilde{E}_{j_{l}}\cdots\widetilde{E}_{j_{1}}S={\bf 1}$. Then, we have

$\displaystyle\widetilde{E}_{j_{l+m}}\cdots\widetilde{E}_{j_{1}}(C_{\Lambda}\circ
S)$ $\displaystyle\cong$
$\displaystyle\widetilde{E}_{j_{l+m}}\cdots\widetilde{E}_{j_{l+1}}(C_{\Lambda}\circ\widetilde{E}_{j_{l}}\cdots\widetilde{E}_{j_{1}}S)$
$\displaystyle\cong$
$\displaystyle\widetilde{E}_{j_{l+m}}\cdots\widetilde{E}_{j_{l+1}}(C_{\Lambda}\circ{\bf
1})\cong\widetilde{E}_{j_{l+m}}\cdots\widetilde{E}_{j_{l+1}}(C_{\Lambda})\cong{\bf
1},$

which also means
$\widetilde{E}_{j_{l+m}}\cdots\widetilde{E}_{j_{1}}S^{\prime}={\bf 1}$. Then,
since $\Psi:\hbox{$R$-\hbox{gmod}}\to B(\infty)$ is an isomorphism of
crystals, we find that
$\Psi(S^{\prime})=\tilde{f}_{j_{1}}\cdots\tilde{f}_{j_{l+m}}{\bf
0}_{\mathbf{i}}$. It follows from Proposition 3.11 (1) and Proposition 6.2
that

$\tilde{f}_{j_{1}}\cdots\tilde{f}_{j_{l+m}}{\bf
0}_{\mathbf{i}}=\tilde{f}_{j_{1}}\cdots\tilde{f}_{j_{l}}(\tilde{f}_{j_{l+1}}\cdots\tilde{f}_{j_{l+m}}{\bf
0}_{\mathbf{i}})=\tilde{f}_{j_{1}}\cdots\tilde{f}_{j_{l}}({\bf
h}_{\Lambda})=\tilde{f}_{j_{1}}\cdots\tilde{f}_{j_{l}}({\bf
0}_{\mathbf{i}}+{\bf h}_{\Lambda})=\Psi(S)+{\bf h}_{\Lambda}.$

Hence, we obtain $\Psi(S^{\prime})=\Psi(S)+{\bf h}_{\Lambda}$ and then (1).

Next, let us show (2) the bijectivity of $\widetilde{\Psi}$. For any
$x\in\mathbb{B}_{\mathbf{i}}$ by Proposition 3.11, there exists $\Lambda\in P$
and $b\in B(\infty)$ such that $x={\bf h}_{\Lambda}+b\in B^{{\bf
h}_{\Lambda}}(\infty)$. Then, the fact
$\widetilde{\Psi}(C_{\Lambda}\circ\Psi^{-1}(b))={\bf h}_{\Lambda}+b$ shows the
surjectivity of $\widetilde{\Psi}$. The following lemma is almost evident by
the property of the localization method:

######  Lemma 8.2.

For any simple modules $M,M^{\prime}\in\widetilde{R}\hbox{-gmod}$, there exist
$\xi\in P$ and simple modules $L,L^{\prime}\in\hbox{$R$-\hbox{gmod}}$ such
that $M=C_{\xi}\circ L$ and $M^{\prime}=C_{\xi}\circ L^{\prime}$.

Proof. For $M,\,M^{\prime}$ there exist $\Lambda,\,\Lambda^{\prime}\in P$ and
$S,\,S^{\prime}\in\hbox{$R$-\hbox{gmod}}$ such that $M\cong C_{\Lambda}\circ
S$ and $M^{\prime}\cong C_{\Lambda^{\prime}}\circ S^{\prime}$. Then, there
exists a dominant weight $\mu\gg 0$ such that
$\Lambda+\mu,\,\Lambda^{\prime}+\mu\in P_{+}$ and $L:=C_{\mu}\circ M\cong
C_{\Lambda+\mu}\circ S,\,L^{\prime}:=C_{\Lambda^{\prime}+\mu}\circ
S^{\prime}\cong C_{\mu}\circ M^{\prime}\in\hbox{$R$-\hbox{gmod}}$ up to
grading shift. Note that both $L,\,L^{\prime}$ are simple. Thus, setting
$\xi=-\mu$, since $C_{\mu}$ is invertible, we get the desired result.

For any $M,M^{\prime}\in\mathbb{B}(\widetilde{R}\hbox{-gmod})$, by the above
lemma we can set $M=C_{\mu}\circ L$ and $M^{\prime}=C_{\mu}\circ L^{\prime}$
for some $\mu\in P$, $L,L^{\prime}\in\mathbb{B}(\hbox{$R$-\hbox{gmod}})$. Here
we assume $\widetilde{\Psi}(C_{\mu}\circ L)=\widetilde{\Psi}(C_{\mu}\circ
L^{\prime})$. Then we obtain ${\bf h}_{\mu}+\Psi(L)={\bf
h}_{\mu}+\Psi(L^{\prime})$, which implies $\Psi(L)=\Psi(L^{\prime})$ and then
$L\cong L^{\prime}$ since $\Psi$ is bijective. Then, one has $M\cong
M^{\prime}$, which means the injectivity of $\widetilde{\Psi}$.

Let us see (3). By the definition of $\mathrm{wt}$ in (7.7) and Proposition
6.2, it is immediate that $\mathrm{wt}(C_{\Lambda}\circ
S)=w_{0}\Lambda-\Lambda+\mathrm{wt}(\Psi(S))=\mathrm{wt}({\bf
h}_{\Lambda}+\Psi(S))$.

As for (4), first by the definition of $\varepsilon_{i}$ on
$\mathbb{B}(\widetilde{R}\hbox{-gmod})$ in (7.7) one has
$\varepsilon_{i}(C_{\Lambda}\circ S)=\varepsilon_{i}(\Psi(S))-\langle
h_{i},w_{0}\Lambda\rangle$. Now, set ${\bf
h}_{\Lambda}=(m_{1},\cdots,m_{N})\in{\mathcal{H}}_{\mathbf{i}}$ and
$\Psi(S)=(x_{1},\cdots,x_{N})\in B(\infty)$, where $m_{k}:=\langle
h_{i_{k}},s_{i_{k+1}}\cdots s_{i_{N}}\Lambda\rangle$. Let
$\\{k_{1},k_{2},\cdots\\}=\\{k\mid i=i_{k}\\}$ where $k_{1}<k_{2}<\cdots$. By
the definition of ${\mathcal{H}}_{\mathbf{i}}$, one has $\sigma_{k_{1}}({\bf
h}_{\Lambda})=\sigma_{k_{2}}({\bf h}_{\Lambda})=\cdots$ and then

$\varepsilon_{i}({\bf h}_{\Lambda})=\sigma_{k_{1}}({\bf
h}_{\Lambda})=m_{k_{1}}+\sum_{j<k_{1}}a_{i,i_{j}}m_{j}.$

Using this formula, one gets

$\displaystyle\varepsilon_{i}({\bf h}_{\Lambda}+\Psi(S))$ $\displaystyle=$
$\displaystyle{\rm max}\\{\sigma_{k}({\bf h}_{\Lambda}+\Psi(S))\mid
i=i_{k}\\}={\rm max}\\{x_{k}+m_{k}+\sum_{j<k}a_{i,i_{j}}(x_{j}+m_{j})\mid
i=i_{k}\\}$ $\displaystyle=$ $\displaystyle{\rm
max}\\{x_{k}+\sum_{j<k}a_{i,i_{j}}x_{j}\mid
i=i_{k}\\}+m_{k_{1}}+\sum_{j<k_{1}}a_{i,i_{j}}m_{j}=\varepsilon_{i}(\Psi(S))+\varepsilon_{i}({\bf
h}_{\Lambda}).$

It yields that

$\displaystyle\langle h_{i},w_{0}\Lambda\rangle$ $\displaystyle=$
$\displaystyle\langle h_{i},s_{i_{1}}\cdots s_{i_{N}}\Lambda\rangle=\langle
h_{i},s_{i_{2}}\cdots s_{i_{N}}\Lambda-\langle h_{i_{1}},s_{i_{2}}\cdots
s_{i_{N}}\Lambda\rangle\alpha_{i_{1}}\rangle$ $\displaystyle=$
$\displaystyle\langle h_{i},s_{i_{2}}\cdots s_{i_{N}}\Lambda\rangle-\langle
h_{i_{1}},\alpha_{i_{2}}\rangle m_{1}=\langle h_{i},s_{i_{3}}\cdots
s_{i_{N}}\Lambda-\langle h_{i_{2}},s_{i_{2}}\cdots
s_{i_{N}}\Lambda\rangle\alpha_{i_{2}}\rangle-\langle
h_{i},\alpha_{i_{1}}\rangle m_{1}$ $\displaystyle=$ $\displaystyle\langle
h_{i},s_{i_{3}}\cdots s_{i_{N}}\Lambda\rangle-\langle
h_{i},\alpha_{i_{2}}\rangle m_{2}-\langle h_{i},\alpha_{i_{1}}\rangle
m_{1}=\cdots=\langle h_{i},s_{i_{k_{1}+1}}\cdots
s_{i_{N}}\Lambda\rangle-\sum_{j\leq k_{1}}\langle h_{i},\alpha_{i_{j}}\rangle
m_{j}$ $\displaystyle=$ $\displaystyle
m_{k_{1}}-2m_{k_{1}}-\sum_{j<k_{1}}\langle h_{i},\alpha_{i_{j}}\rangle
m_{j}=-m_{k_{1}}-\sum_{j<k_{1}}\langle h_{i},\alpha_{i_{j}}\rangle
m_{j}=-\varepsilon_{i}({\bf h}_{\Lambda}),$

which shows (4), that is,

$\varepsilon_{i}(C_{\Lambda}\circ S)=\varepsilon_{i}(\Psi(S))-\langle
h_{i},w_{0}\Lambda\rangle=\varepsilon_{i}(\Psi(S))+\varepsilon_{i}({\bf
h}_{\Lambda})=\varepsilon_{i}({\bf h}_{\Lambda}+\Psi(S)).$

Let us investigate (5). For a simple $C_{\Lambda}\circ
S\in\mathbb{B}(\widetilde{R}\hbox{-gmod})$, it yields

$\widetilde{\Psi}\widetilde{F}_{i}(C_{\Lambda}\circ
S)=\widetilde{\Psi}(C_{\Lambda}\circ\widetilde{F}_{i}S)={\bf
h}_{\Lambda}+\Psi(\widetilde{F}_{i}S)={\bf h}_{\Lambda}+\tilde{f}_{i}\Psi(S),$

and by (3.32) in Proposition 3.11, one has

$\tilde{f}_{i}\widetilde{\Psi}(C_{\Lambda}\circ S)=\tilde{f}_{i}({\bf
h}_{\Lambda}+\Psi(S))={\bf h}_{\Lambda}+\tilde{f}_{i}\Psi(S).$

It shows
$\widetilde{\Psi}\circ\widetilde{F}_{i}=\tilde{f}_{i}\,\circ\widetilde{\Psi}$.
By use of the formula
$\widetilde{\Psi}\circ\widetilde{F}_{i}=\tilde{f}_{i}\,\circ\widetilde{\Psi}$
and
$\widetilde{F}_{i}\widetilde{E}_{i}=\widetilde{E}_{i}\widetilde{F}_{i}={\rm
id}_{\mathbb{B}(\widetilde{R}{\rm-gmod})}$, it is clear that

${\bf
h}_{\Lambda}+\Psi(S)=\widetilde{\Psi}(\widetilde{F}_{i}\widetilde{E}_{i}(C_{\Lambda}\circ
S))=\tilde{f}_{i}\,\widetilde{\Psi}(\widetilde{E}_{i}(C_{\Lambda}\circ S))$

and then applying $\tilde{e}_{i}$ on the both sides, it follows from
$\tilde{f}_{i}\tilde{e}_{i}=\tilde{e}_{i}\tilde{f}_{i}={\rm
id}_{\mathbb{B}_{\mathbf{i}}}$ that

$\tilde{e}_{i}({\bf
h}_{\Lambda}+\Psi(S))=\widetilde{\Psi}(\widetilde{E}_{i}(C_{\Lambda}\circ
S)),$

which implies
$\tilde{e}_{i}\widetilde{\Psi}=\widetilde{\Psi}\widetilde{E}_{i}$. Now, we
have completed the proof of the theorem.

##  9\. Application and further problems

In this section, we shall see several miscellaneous results as applications of
the crystal structure on $\mathbb{B}(\widetilde{R}\hbox{-gmod})$.

###  9.1. Operator $\tilde{\mathfrak{a}}$

Define the $\mathbb{Q}(q)$-linear anti-automorphism $\star$ of
$U_{q}(\mathfrak{g})$ by

$(q^{h})^{\star}=q^{-h},\quad e_{i}^{\star}=e_{i},\quad f_{i}^{\star}=f_{i}.$

######  Theorem 9.1 ([8]).

Set $L^{\star}(\infty):=\\{u^{\star}\,|\,u\in L(\infty)\\}$,
$B^{\star}(\infty):=\\{b^{\star}\,|\,b\in B(\infty)\\}$. Then we have

$\displaystyle L^{\star}(\infty)=L(\infty),$ $\displaystyle
B^{\star}(\infty)=B(\infty).$

From the proof of Theorem 5.13 in [12] we get

######  Proposition 9.2 ([12]).

For $\nu=(\nu_{1},\nu_{2},\cdots,\nu_{m-1},\nu_{m})\in I^{\beta}$
($m:=|\beta|$) set
$\overline{\nu}=(\nu_{m},\nu_{m-1},\cdots,\nu_{2},\nu_{1})$. Define the
automorphism ${\mathfrak{a}}$ on $R(\beta)$ by

${\mathfrak{a}}(e(\nu))=e(\overline{\nu}),\quad{\mathfrak{a}}(x_{i}e(\nu))=x_{m-i+1}e(\overline{\nu}),\quad{\mathfrak{a}}(\tau_{j}e(\nu))=-\tau_{m-j}e(\overline{\nu}).$

Then, there exists the functor
$\mathfrak{a}:\hbox{$R$-\hbox{gmod}}\to\hbox{$R$-\hbox{gmod}}$ such that
$\mathfrak{a}(C_{i})=C_{i^{*}}$ ($\forall i\in I$), $\mathfrak{a}^{2}\cong{\rm
id}$ and $\mathfrak{a}(X\circ Y)\cong\mathfrak{a}(Y)\circ\mathfrak{a}(X)$ for
$X,Y\in\hbox{$R$-\hbox{gmod}}$. Furthermore, it is extended to the functor
$\tilde{\mathfrak{a}}:\widetilde{R}\hbox{-gmod}\to\widetilde{R}\hbox{-gmod}$
which satisfies

(9.1) $\tilde{\mathfrak{a}}^{2}\cong{\rm
id},\quad\hbox{and}\quad\tilde{\mathfrak{a}}(X\circ
Y)\cong\tilde{\mathfrak{a}}(Y)\circ\tilde{\mathfrak{a}}(X)\quad\hbox{ for
}X,Y\in\widetilde{R}\hbox{-gmod}.$

Note that $\mathfrak{a}$(resp. $\tilde{\mathfrak{a}}$) induces the operation
$\star$ on ${\mathcal{A}}_{q}(\mathfrak{n})$ (resp.
$\widetilde{{\mathcal{A}}_{q}}(\mathfrak{n})$) since $\mathfrak{a}(L(i))=L(i)$
and then one has $\mathfrak{a}(f_{i})=f_{i}$ (resp.
$\tilde{\mathfrak{a}}(f_{i})=f_{i}$) on ${\mathcal{A}}_{q}(\mathfrak{n})$
(resp. $\widetilde{{\mathcal{A}}_{q}}(\mathfrak{n})$). Now, we obtain the
following:

######  Proposition 9.3.

Let
$\tilde{\mathfrak{a}}:\widetilde{R}\hbox{-gmod}\to\widetilde{R}\hbox{-gmod}$
be the functor as above. It yields

(9.2)
$\tilde{\mathfrak{a}}(\mathbb{B}(\widetilde{R}\hbox{-gmod}))=\mathbb{B}(\widetilde{R}\hbox{-gmod}).$

Proof. It is evident that for any simple module
$L\in\widetilde{R}\hbox{-gmod}$, $\tilde{\mathfrak{a}}(L)$ is also simple.
Therefore, it suffices to show that if a simple module $L$ satisfies $L\cong
L^{*}$, then $\tilde{\mathfrak{a}}(L)\cong\tilde{\mathfrak{a}}(L)^{*}$. To do
that, we show the following lemma:

######  Lemma 9.4.

Let $\psi$ be the anti-automorphism on $R(\beta)$ as in 4.1. We obtain

(9.3) $\psi\circ\mathfrak{a}=\mathfrak{a}\circ\psi.$

Proof. For $\beta$ with $|\beta|=m$, it is trivial that for the generators
$x_{i},e(\nu)$ of $R(\beta)$,

$\psi\circ\mathfrak{a}(e(\nu))=e(\overline{\nu})=\mathfrak{a}\circ\psi(e(\nu)),\qquad\psi\circ\mathfrak{a}(x_{i}e(\nu))=e(\overline{\nu})x_{m+1-i}=x_{m+1-i}e(\overline{\nu})=\mathfrak{a}\circ\psi(x_{i}(e(\nu))$

One also gets

$\displaystyle\psi\circ\mathfrak{a}(\tau_{j}e(\nu))$ $\displaystyle=$
$\displaystyle-\psi(\tau_{m-j}e(\overline{\nu}))=-e(\overline{\nu})\tau_{m-j}=-\tau_{m-j}e(s_{m-j}(\overline{\nu}))=-\tau_{m-j}e(\overline{s_{j}\nu})$
$\displaystyle=$
$\displaystyle\mathfrak{a}(\tau_{j}e(s_{j}\nu))=\mathfrak{a}(e(\nu)\tau_{j})=\mathfrak{a}\circ\psi(\tau_{j}e(\nu)),$

which completes the proof.

By this lemma, we find that

(9.4) $\tilde{\mathfrak{a}}(L)^{*}\cong\tilde{\mathfrak{a}}(L^{*}).$

Since $\tilde{\mathfrak{a}}(L)$ is simple, by Lemma 7.3 there exists a unique
$n\in\mathbb{Z}$ such that $(q^{n}\tilde{\mathfrak{a}}(L))^{*}\cong
q^{n}\tilde{\mathfrak{a}}(L)\cong\tilde{\mathfrak{a}}(q^{n}L)$ and then
$q^{-n}(\tilde{\mathfrak{a}}(L))^{*}\cong(q^{n}\tilde{\mathfrak{a}}(L))^{*}\cong\tilde{\mathfrak{a}}(q^{n}L)$.
Therefore, owing to (9.4) we get

$\tilde{\mathfrak{a}}(q^{n}L)\cong
q^{-n}\tilde{\mathfrak{a}}(L^{*})\cong\tilde{\mathfrak{a}}(q^{-n}L^{*})\quad\hbox{
and then }\quad q^{2n}L\cong L^{*},$

which means $n=0$ and then we obtain
$\tilde{\mathfrak{a}}(L)\cong\tilde{\mathfrak{a}}(L)^{*}$.

Here note that Proposition 9.3 can be seen as a generalization of Theorem 9.1.

Since as crystals
$\mathbb{B}(\widetilde{R}\hbox{-gmod})\cong\mathbb{B}_{\mathbf{i}}$ for any
reduced longest word ${\mathbf{i}}$, the proposition above gives rise to the
following problem.
Problem 1. Can we describe $\tilde{\mathfrak{a}}$-operation on
$\mathbb{B}_{\mathbf{i}}=B_{i_{1}}\otimes\cdots\otimes B_{i_{N}}$ explicitly?

Of course, this problem is non-trivial since even for the case $B(\infty)$ the
explicit description has not yet been done before in
$\mathbb{B}_{\mathbf{i}}$.

###  9.2. Summation $\oplus_{\mathbf{i}}$

By use of the map $\widetilde{\Psi}$ we can define a certain additive
structure on $\mathbb{B}(\widetilde{R}\hbox{-gmod})$ as follows: For
$L,L^{\prime}\in\mathbb{B}(\widetilde{R}\hbox{-gmod})$ and a reduced longest
word ${\mathbf{i}}$, we define

(9.5)
$L\oplus_{\mathbf{i}}L^{\prime}:=\widetilde{\Psi}^{-1}(\widetilde{\Psi}(L)+\widetilde{\Psi}(L^{\prime})),$

where the summation in the right hand side of (9.5) is the natural summation
of $\mathbb{B}_{\mathbf{i}}$ by identifying with the free-$\mathbb{Z}$ lattice
$\mathbb{Z}^{N}$. The following is trivial from the defnition of
$\oplus_{\mathbf{i}}$

######  Proposition 9.5.

By the summation $\oplus_{\mathbf{i}}$,
$\mathbb{B}(\widetilde{R}\hbox{-gmod})$ becomes an additive group. Here let us
denote the inverse of $L$ with respect to $\oplus_{\mathbf{i}}$ by
$L^{\ominus_{\mathbf{i}}}$, where we define
$L^{\ominus_{\mathbf{i}}}:=\widetilde{\Psi}^{-1}(-\widetilde{\Psi}(L))$.

Note that this summation DOES depned on a choice of reduced word
${\mathbf{i}}$. In this sense, it is not canonical. The following example
implies that.

######  Example 9.6.

For $\mathfrak{g}=\mathfrak{s}l_{3}$ and ${\mathbf{i}}=121$, let us see the
inverse of $L(1)\in R(\alpha_{1})$-gmod, which corresponds to
$(0,0,1)\in\mathbb{B}_{121}$. As has seen in Proposition 3.8, ${\bf
h}_{1}=(0,1,1)$ and ${\bf h}_{2}=(1,1,0)$, and then we obtain

$(0,0,-1)=\tilde{f}_{2}(0,-1,-1)=\tilde{f}_{2}(-{\bf h}_{1})=-{\bf
h}_{1}+\tilde{f}_{2}(0,0,0).$

This implies that

$L(1)^{\ominus_{\mathbf{i}}}\cong C_{-\Lambda_{1}}\circ L(2).$

For ${\mathbf{i}}^{\prime}=212$, $L(1)$ corresponds to
$(0,1,0)\in\mathbb{B}_{212}$. One has ${\bf h}_{1}=(1,1,0)$, ${\bf
h}_{2}=(0,1,1)$. Then, one gets
$(0,-1,0)=\tilde{f}_{2}(0,-1,-1)=\tilde{f}_{2}(-{\bf h}_{2})=-{\bf
h}_{2}+\tilde{f}_{2}(0,0,0)$ and then

$L(1)^{\ominus_{{\mathbf{i}}^{\prime}}}\cong C_{-\Lambda_{2}}\circ L(2).$

Problem 2. Describe $L\oplus_{\mathbf{i}}L$ and $L^{\ominus_{\mathbf{i}}}$ for
$L,\,L^{\prime}\in\widetilde{R}\hbox{-gmod}$ explicitly. Indeed, for ${\bf
h}_{\Lambda}\in{\mathcal{H}}_{\mathbf{i}}$ and $b\in B(\infty)$

${\bf h}_{\Lambda}+b\in B^{{\bf
h}_{\Lambda}}(\infty)\subset\mathbb{B}_{\mathbf{i}}\longleftrightarrow
C_{\Lambda}\circ\Psi^{-1}(b)\in\mathbb{B}(\widetilde{R}\hbox{-gmod})$

But, note that if $b$ corresponds to ”imaginary”(=non-real) simple module
$S_{b}$ in $R$-gmod, then $S_{b}\circ S_{b}$ is not simple and then it does
not appear in $\mathbb{B}_{\mathbf{i}}$.

###  9.3. Category $\widetilde{{\mathscr{C}}}_{w}$

In [12], it has been shown that for an arbitrary symmetrizable Kac-Moody Lie
algebra and any Weyl group element $w\in W$, there exists a subcategory
${{\mathscr{C}}}_{w}\subset$$R$-gmod and it admits a localization

$\widetilde{{\mathscr{C}}}_{w}={\mathscr{C}}_{w}[C_{i}^{\circ-1}\,|\,i\in
I],\quad(C_{i}=M(w\Lambda_{i},\Lambda_{i}))$

Indeed, note that for finite type Lie algebra setting,
${{\mathscr{C}}}_{w_{0}}=R$-gmod.

Problem 3. We conjecture that the localization $\widetilde{{\mathscr{C}}}_{w}$
possess a crystal $\mathbb{B}(\widetilde{{\mathscr{C}}}_{w})$. If so, we also
conjecture that there is an isomorphism of crystals

$\mathbb{B}(\widetilde{{\mathscr{C}}}_{w})\,\smash{\mathop{\longrightarrow}\limits^{\sim}}\,B_{i_{1}}\otimes\cdots\otimes
B_{i_{m}},$

where $i_{1}\cdots i_{m}$ is a reduced word of $w$.

###  9.4. Rigidity

######  Definition 9.7.

Let $X,Y$ be objects in a monoidal category ${\mathcal{T}}$, and
$\varepsilon:X\otimes Y\to 1$ and $\eta:1\to Y\otimes X$ morphisms in
${\mathcal{T}}$. We say that a pair $(X,Y)$ is dual pair or $X$ is a left dual
to $Y$ or $Y$ is a right dual to $X$ if the following compositions are
identities:

$X\simeq X\otimes 1\,\,\smash{\mathop{\longrightarrow}\limits^{{\rm
id}\otimes\eta}}\,\,X\otimes Y\otimes
X\,\,\smash{\mathop{\longrightarrow}\limits^{\varepsilon\otimes{\rm
id}}}\,\,1\otimes X\simeq X,\,\,Y\simeq 1\otimes
Y\,\,\smash{\mathop{\longrightarrow}\limits^{\eta\otimes{\rm id}}}\,\,Y\otimes
X\otimes Y\,\,\smash{\mathop{\longrightarrow}\limits^{{\rm
id}\otimes\varepsilon}}\,\,Y\otimes 1\simeq Y$

We denote a right dual to $X$ by ${\mathcal{D}}(X)$ and a left dual to $X$ by
${\mathcal{D}}^{-1}(X)$.

######  Theorem 9.8 ([12]).

For any finite type $R$, $\widetilde{R}\hbox{-gmod}$ is rigid, i.e., every
object in $\widetilde{R}\hbox{-gmod}$ has left and right duals.

Note that in [13], it is shown that for any symmetrizable Kac-Moody setting
the localized category $\widetilde{\mathscr{C}}_{w}$ is rigid.

Problem 4. For a simple object $C_{\Lambda}\circ
S\in\mathbb{B}(\widetilde{R}\hbox{-gmod})$, describe the right and left duals
explicitly:

$\widetilde{\Psi}({{\mathcal{D}}}(C_{\Lambda}\circ
S)),\quad\widetilde{\Psi}({{\mathcal{D}}}^{-1}(C_{\Lambda}\circ
S))\in\mathbb{B}_{\mathbf{i}}.$

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