2.1 Quantum affine algebras

The quintuple $( \mathsf {A}, \mathsf {P}, \Pi, \mathsf {P}^\vee, \Pi ^\vee )$ is called an affine Cartan datum if it consists of the following components:

1. (i) an affine Cartan matrix $\mathsf {A}=(a_{ij})_{i,j\in I}$ with a finite index set $I$;

2. (ii) a free abelian group $\mathsf {P}$ of rank $|I|+1$, called the weight lattice;

3. (iii) a set $\Pi =\{\alpha _i \in \mathsf {P} \mid i \in I\}$, whose elements are called simple roots;

4. (iv) the group $\mathsf {P}^\vee \mathbin {:=} \operatorname {Hom}_\mathbb {\mspace {1mu}Z\mspace {1mu}}(\mathsf {P},\mathbb {\mspace {1mu}Z\mspace {1mu}})$, called the coweight lattice;

5. (v) a set $\Pi ^\vee =\{h_i \mid i \in I\} \subset \mathsf {P}^\vee$, whose elements are simple coroots;

and if it satisfies the following properties:

1. (a) $\langle h_i, \alpha _j \rangle = a_{i,j}$ for any $i,j\in I$;

2. (b) for any $i\in I$, there exists $\Lambda _i\in \mathsf {P}$ such that $\langle {h_j,\Lambda _i}\rangle =\delta (i=j)$ for any $j\in I$;

3. (c) $\Pi$ is linearly independent.

Let ${\mathfrak {g}}$ be the affine Kac–Moody algebra associated with $(\mathsf {A},\mathsf {P},\Pi,\mathsf {P}^\vee,\Pi ^\vee )$. We set $\mathsf {Q} \mathbin {:=} \bigoplus _{i \in I}\mathbb {\mspace {1mu}Z\mspace {1mu}}\alpha _i \subset \mathsf {P}$, which is called the root lattice, and $\mathsf {Q}^+ \mathbin {:=} \sum _{i \in I}\mathbb {\mspace {1mu}Z\mspace {1mu}}_{\geqslant 0}\alpha _i\subset \mathsf {Q}$. For $\beta = \sum _{i\in I} b_i \alpha _i \in \mathsf {Q}^+$, we write $|\beta | = \sum _{i\in I} b_i$. We denote by $\delta \in \mathsf {Q}$ the imaginary root and by $c \in \mathsf {Q}^\vee$ the central element. Note that the positive imaginary root $\Delta _+^{\rm im}$ is equal to $\mathbb {\mspace {1mu}Z\mspace {1mu}}_{>0}\, \delta$ and the center of ${\mathfrak {g}}$ is generated by $c$. We write $\mathsf {P}_{\mathrm {cl}\mspace {1mu}} \mathbin {:=} \mathsf {P} / (\mathsf {P}\cap \mathbb {Q} \delta )$, which is called the classical weight lattice, and take $\rho \in \mathsf {P}$ (respectively $\rho ^\vee \in \mathsf {P}^\vee$) such that $\langle h_i,\rho \rangle =1$ (respectively $\langle \rho ^\vee,\alpha _i\rangle =1$) for any $i \in I$. There exists a $\mathbb {Q}$-valued non-degenerate symmetric bilinear form $( \ , \ )$ on $\mathsf {P}$ satisfying

\[ \langle h_i,\lambda \rangle= \frac{2(\alpha_i,\lambda)}{(\alpha_i,\alpha_i)} \quad \text{and} \quad \langle c,\lambda \rangle = (\delta,\lambda) \]

for any $i \in I$ and $\lambda \in \mathsf {P}$. We write $\mathsf {W} \mathbin {:=} \langle s_i \mid i \in I \rangle \subset \textrm {Aut}(\mathsf {P})$ for the Weyl group of $\mathsf {A}$, where $s_i (\lambda ) \mathbin {:=} \lambda - \langle h_i,\lambda \rangle \alpha _i$ for $\lambda \in \mathsf {P}$. We will use the standard convention in [Reference KacKac90] of choosing $0\in I$ except for type $A^{(2)}_{2n}$, in which case we take the longest simple root to be $\alpha _0$, and for types $B_2^{(1)}$, $A_3^{(2)}$ and $E_k^{(1)}$ ($k=6,7,8$), in which cases we take the following Dynkin diagrams.

(2.1)

Note that $B^{(1)}_2$ and $A^{(2)}_3$ in (

) are denoted by $C_2^{(1)}$ and $D_3^{(2)}$, respectively, in [Reference KacKac90].

Let ${\mathfrak {g}}_0$ be the subalgebra of ${\mathfrak {g}}$ generated by the Chevalley generators $e_i$, $f_i$ and $h_i$ for $i \in I_0 \mathbin {:=} I \setminus \{ 0 \}$, and let $\mathsf {W}_0$ be the subgroup of $\mathsf {W}$ generated by $s_i$ for $i \in I_0$. Note that ${\mathfrak {g}}_0$ is a finite-dimensional simple Lie algebra and $\mathsf {W}_0$ contains the longest element $w_0$.

Let $q$ be an indeterminate and ${{\mathbf {k}}}$ the algebraic closure of the subfield $\mathbb {C}(q)$ in the algebraically closed field $\hat {\mathbf {k}}\mathbin {:=}\bigcup _{m >0}\mathbb {C}((q^{1/m}))$. For $m,n \in \mathbb {\mspace {1mu}Z\mspace {1mu}}_{\geqslant 0}$ and $i\in I$, we define $q_i = q^{(\alpha _i,\alpha _i)/2}$ and

\[ [n]_i =\frac{ q^n_{i} - q^{{-}n}_{i} }{ q_{i} - q^{{-}1}_{i} }, \quad [n]_i! = \prod^{n}_{k=1} [k]_i , \quad \Big[\begin{matrix}m \\ n\\ \end{matrix} \Big]_i= \frac{ [m]_i! }{[m-n]_i! [n]_i! }. \]

Let $d$ be the smallest positive integer such that $d ({(\alpha _i, \alpha _i)}/{2})\in \mathbb {\mspace {1mu}Z\mspace {1mu}}$ for all $i\in I$.

Let us denote by $U_q'({\mathfrak {g}})$ the $\mathbf {k}$-subalgebra of $U_q({\mathfrak {g}})$ generated by $e_i,f_i$ and $K^{\pm 1}_i$ $(i \in I)$. The coproduct $\Delta$ of $U_q'({\mathfrak {g}})$ is given by

\[ \Delta(q^h)=q^h \mathop\otimes q^h, \quad \Delta(e_i)=e_i \mathop\otimes K_i^{{-}1}+1 \mathop\otimes e_i, \quad \Delta(f_i)=f_i \mathop\otimes 1 +K_i \mathop\otimes f_i, \]

and the bar involution $\bar{}$ of $U_q'({\mathfrak {g}})$ is defined as

\[ q^{1/m} \to q^{{-}1/m}, \quad e_i \mapsto e_i, \quad f_i \mapsto f_i, \quad K_i \mapsto K_i^{{-}1}. \]

Let $\mathscr {C}_{\mathfrak {g}}$ be the category of finite-dimensional integrable $U_q'({\mathfrak {g}})$-modules, i.e. finite-dimensional modules $M$ with a weight decomposition

\[ M=\bigoplus_{\lambda\in\mathsf{P}_{\mathrm{cl}\mspace{1mu}}}M_\lambda \quad \text{where } M_\lambda=\{{u\in M\mid K_iu=q_i^{\langle{h_i,\lambda}\rangle}u }\}. \]

Note that the trivial module $\mathbf {1}$ is contained in $\mathscr {C}_{\mathfrak {g}}$ and the tensor product $\otimes$ gives a monoidal category structure on $\mathscr {C}_{\mathfrak {g}}$. It is known that the Grothendieck ring $K(\mathscr {C}_{\mathfrak {g}})$ is a commutative ring [Reference Frenkel and ReshetikhinFR99]. A simple module $L$ in $\mathscr {C}_{\mathfrak {g}}$ contains a non-zero vector $u \in L$ of weight $\lambda \in \mathsf {P}_{\mathrm {cl}\mspace {1mu}}$ such that (i) $\langle h_i,\lambda \rangle \geqslant 0$ for all $i \in I_0$ and (ii) all the weights of $L$ are contained in $\lambda - \sum _{i \in I_0} \mathbb {\mspace {1mu}Z\mspace {1mu}}_{\geqslant 0}\, {\mathrm {cl}\mspace {1mu}}(\alpha _i)$, where ${\mathrm {cl}\mspace {1mu}}\colon \mathsf {P}\to \mathsf {P}_{\mathrm {cl}\mspace {1mu}}$ is the canonical projection. Such a $\lambda$ is unique, and $u$ is unique up to a constant multiple. We call $\lambda$ the dominant extremal weight of $L$ and $u$ a dominant extremal weight vector of $L$.

Let $\mathsf {P}_{\mathrm {cl}\mspace {1mu}}^0\mathbin {:=}\{\lambda \in \mathsf {P}_{\mathrm {cl}\mspace {1mu}}\mid \langle {c,\lambda }\rangle =0\}$. For each $i \in I_0$ we set

\[ \varpi_i \mathbin{:=} {\rm gcd}(\mathsf{c}_0,\mathsf{c}_i)^{{-}1}{\mathrm{cl}\mspace{1mu}}(\mathsf{c}_0\Lambda_i-\mathsf{c}_i \Lambda_0) \in \mathsf{P}_{\mathrm{cl}\mspace{1mu}}^0, \]

where the central element $c$ is equal to $\sum _{ i\in I} \mathsf {c}_i h_i$. Note that $\mathsf {P}_{\mathrm {cl}\mspace {1mu}}^0= \bigoplus _{i\in I_0}\mathbb {\mspace {1mu}Z\mspace {1mu}}\varpi _i$. For any $i\in I_0$, there exists a unique simple module $V(\varpi _i)$ in $\mathscr {C}_{\mathfrak {g}}$ satisfying certain good conditions (see [Reference KashiwaraKas02, § 5.2]), which is called the $i$th fundamental representation. Note that the dominant extremal weight of $V(\varpi _i)$ is $\varpi _i$.

For simple modules $M$ and $N$ in $\mathscr {C}_{\mathfrak {g}}$, we say that $M$ and $N$ commute or $M$ commutes with $N$ if $M\mathop \otimes N\simeq N\mathop \otimes M$. We say that $M$ and $N$ strongly commute or $M$ strongly commutes with $N$ if $M\mathop \otimes N$ is simple. Note that $M\mathop \otimes N$ is simple if and only if $N\mathop \otimes M$ is simple, since $K(\mathscr {C}_{\mathfrak {g}})$ is a commutative ring. It is clear that if simple modules $M$ and $N$ strongly commute, then they commute. We say that a simple module $M$ is real if $M$ strongly commutes with itself.

For an integrable $U_q'({\mathfrak {g}})$-module $M$, we denote by $M_z$ the affinization of $M$ and by $z_M \colon M_z \to M_z$ the $U_q'({\mathfrak {g}})$-module automorphism of weight $\delta$. Note that $M_z \simeq \mathbf {k}[z^{\pm 1}]\otimes _{\mathbf {k}} M$ for an indeterminate $z$ as a $\mathbf {k}$-vector space. For $x \in {{\mathbf {k}}}^\times$, we define

\[ M_x \mathbin{:=} M_z / (z_M -x)M_z. \]

We call $x$ a spectral parameter. The functor $T_x$ defined by $T_x(M)=M_x$ is an endofunctor of $\mathscr {C}_{\mathfrak {g}}$ that commutes with tensor products (see [Reference KashiwaraKas02, § 4.2] for details).

It is known that a fundamental representation is a good module, which is a simple $U_q'({\mathfrak {g}})$-module with good properties including a bar involution, a crystal basis with simple crystal graph, and a global basis (see [Reference KashiwaraKas02] for the precise definition). We say that a $U_q'({\mathfrak {g}})$-module $M$ is quasi-good if

\[ M \simeq V_c \]

for some good module $V$ and $c \in \mathbf {k}^\times$. Note that any quasi-good module is a simple $U_q'({\mathfrak {g}})$-module. Moreover the tensor product $M^{\otimes k} \mathbin {:=} \underbrace {M \mathop \otimes \cdots \mathop \otimes M}_{k\text { times}}$ for a quasi-good module $M$ and $k \in \mathbb {\mspace {1mu}Z\mspace {1mu}}_{\geqslant 1}$ is again quasi-good.

For a $U_q'({\mathfrak {g}})$-module $M$, we denote by $\bar {M}=\{ \bar {u} \mid u \in M \}$ the $U_q'({\mathfrak {g}})$-module defined by $x \bar {u} \mathbin {:=} \overline { \bar {x} u }$ for $x \in U_q'({\mathfrak {g}})$. Then we have

\[ \overline{M_a} \simeq (\bar{M})_{\,\bar{a}} \quad \text{and} \quad \overline{M \otimes N} \simeq \bar{N} \otimes \bar{M}\quad\text{for any $M,N\in\mathscr{C}_{\mathfrak{g}}$ and $a\in\mathbf{k}^\times$.}\quad \]

Note that $V(\varpi _i)$ is bar-invariant, i.e. $\overline {V(\varpi _i)}\simeq V(\varpi _i)$ (see [Reference Akasaka and KashiwaraAK97, Appendix A]).

Let $m_i$ be a positive integer such that

\[ \mathsf{W}\pi_i\cap\bigl(\pi_i+\mathbb{\mspace{1mu}Z\mspace{1mu}}\delta\bigr)=\pi_i+\mathbb{\mspace{1mu}Z\mspace{1mu}} m_i\delta, \]

where $\pi _i$ is an element of $\mathsf {P}$ such that ${\mathrm {cl}\mspace {1mu}}(\pi _i)=\varpi _i$. Note that $m_i=(\alpha _i,\alpha _i)/2$ in the case where ${\mathfrak {g}}$ is the dual of an untwisted affine algebra, and $m_i=1$ otherwise. Then for $x,y\in {{\mathbf {k}}}^\times$ we have (see [Reference Akasaka and KashiwaraAK97, § 1.3])

(2.2)\begin{equation} V(\varpi_i)_x \simeq V(\varpi_i)_y \quad \text{if and only if} \quad \text{$x^{m_i}=y^{m_i}$}. \end{equation}

We set

(2.3)\begin{equation} {\sigma({\mathfrak{g}})} \mathbin{:=} I_0 \times \mathbf{k}^\times{/} \sim,\end{equation}

where the equivalence relation is given by

(2.4)\begin{equation} (i,x) \sim (j,y) \iff V(\varpi_i)_x \simeq V(\varpi_j)_y \iff \text{$i=j$ and $x^{m_i}=y^{m_j}$.} \end{equation}

We denote by $[(i,a)]$ the equivalence class of $(i,a)$ in ${\sigma ({\mathfrak {g}})}$. When confusion is unlikely to arise, we simply write $(i,a)$ for the equivalence class $[(i,a)]$.

The monoidal category $\mathscr {C}_{\mathfrak {g}}$ is rigid. For $M\in \mathscr {C}_{\mathfrak {g}}$, we denote by ${}^*\mspace {-3mu} M$ and $M^*$ the right and left duals of $M$, respectively. We set

(2.5)\begin{equation} p^* \mathbin{:=} ({-}1)^{ \langle \rho^\vee, \delta \rangle} q^{\langle c, \rho \rangle} \quad \text{and}\quad \tilde{p} \mathbin{:=} (p^*)^2 = q^{2 \langle c, \rho \rangle}. \end{equation}

The integer $\langle \rho ^\vee, \delta \rangle$ is called the Coxeter number, and $\langle c, \rho \rangle$ is called the dual Coxeter number (see [Reference KacKac90, Ch. 6]). For the reader's convenience we list $p^*$ for all types in the following table.

(2.6) \begin{align} \small \begin{array}{|c||c|c|c|c|c|c|c|} \hline \text{Type of ${\mathfrak{g}}$} & A_n^{(1)^{\vphantom{1}}} & B_n^{(1)} & C_n^{(1)} & D_n^{(1)} & A_{2n}^{(2)} & A_{2n-1}^{(2)} & D_{n+1}^{(2)} \\ & (n\geqslant1) & (n\geqslant2) & (n\geqslant3) & (n\geqslant4) & (n\geqslant1) & (n\geqslant2) & (n\geqslant3)\\ \hline p^* & ({-}q)^{n+1^{\vphantom{1}}} & q^{2n-1} & q^{n+1} & q^{2n-2} & -q^{2n+1} & -q^{2n} & ({-}1)^{n+1} q^{2n} \\ \hline \hline \text{Type of ${\mathfrak{g}}$} & E_6^{(1)} & E_7^{(1)} & E_8^{(1)^{\vphantom{1}}} & F_4^{(1)} & G_{2}^{(1)} & E_{6}^{(2)} & D_{4}^{(3)} \\ \hline p^* & q^{12} & q^{18^{\vphantom{1}}} & q^{30} & q^9 & q^4 & -q^{12} & q^6 \\ \hline\end{array} \end{align}

Then for any $M \in \mathscr {C}_{\mathfrak {g}}$ we have

\[ M^{**} \simeq M_{ (\tilde{p})^{{-}1} }\quad \text{and}\quad {}^{**} M \simeq M_{\tilde{p} }, \]

and for $i\in I_0$ and $x\in \mathbf {k}^\times$ we have

(2.7)\begin{equation} \bigl( V(\varpi_i)_x\bigr)^* \simeq V(\varpi_{i^*})_{(p^*)^{{-}1}x} \quad \text{and}\quad {}^*\bigl(V(\varpi_i)_x \bigr) \simeq V(\varpi_{i^*})_{p^*x}, \end{equation}

where $i^*\in I_0$ is defined by $\alpha _{i^*}=-w_0\,\alpha _i$ (see [Reference Akasaka and KashiwaraAK97, Appendix A]). Note that the involution $i \mapsto i^*$ is the identity for all types except $A_n, D_n$ and $E_6$, which are given as follows:

1. (a) (type $A_n$) $i^* = n+1-i$;

2. (b) (type $D_n$) $i^* = \begin {cases} n-(1-\epsilon ) & \text {if } n \hbox { is odd and } i=n-\epsilon\ (\epsilon =0,1), \\ i & \text {otherwise;} \end {cases}$

3. (c) (type $E_6$) the map $i \mapsto i^*$ is determined by

   \[ i^* = \begin{cases} 6 & \text{if } i=1, \\ i & \text{if } i=2,4, \\ 5 & \text{if } i=3, \end{cases} \]
   where the Dynkin diagram of type $E_6$ is given in (A.3) in Appendix A.

2.2 R-matrices

We recall the notion of R-matrices on $U_q'({\mathfrak {g}})$-modules and their coefficients (see [Reference Drinfel'dDri86], as well as [Reference Akasaka and KashiwaraAK97, Appendices A and B] and [Reference KashiwaraKas02, § 8], for details). Choose a basis $\{P_\nu \}_\nu$ of $U_q^+({\mathfrak {g}})$ and a basis $\{Q_\nu \}_\nu$ of $U_q^-({\mathfrak {g}})$ that are dual to each other with respect to a suitable coupling between $U_q^+({\mathfrak {g}})$ and $U_q^-({\mathfrak {g}})$. For $U_q'({\mathfrak {g}})$-modules $M$ and $N$, we define

\[ R^{\mspace{1mu}\mathrm{univ}}_{M,N}(u\otimes v) \mathbin{:=} q^{(\operatorname{wt}(u),\operatorname{wt}(v))} \sum_\nu P_\nu v\otimes Q_\nu u \quad\text{for $u\in M$ and $v\in N$,}\quad \]

so that $R^{\mspace {1mu}\mathrm {univ}}_{M,N}$ gives a $U_q'({\mathfrak {g}})$-linear homomorphism $M\otimes N \rightarrow N\otimes M$, called the universal R-matrix, provided that the infinite sum has a meaning. As $R^{\mspace {1mu}\mathrm {univ}}_{M,N_z}$ converges in the $z$-adic topology for $M,N\in \mathscr {C}_{\mathfrak {g}}$, we have a morphism of ${{\mathbf {k}}}((z))\mathop \otimes U_q'({\mathfrak {g}})$-modules

\[ R^{\mspace{1mu}\mathrm{univ}}_{M,N_z} \colon {{\mathbf{k}}}((z))\mathop\otimes_{{{\mathbf{k}}}[z^{\pm1}]} (M \mathop\otimes N_z) \xrightarrow{\,{\hspace{2ex}}\,} {{\mathbf{k}}}((z))\mathop\otimes_{{{\mathbf{k}}}[z^{\pm1}]} (N_z\mathop\otimes M). \]

Note that $R^{\mspace {1mu}\mathrm {univ}}_{M,N_z}$ is an isomorphism.

Let $M$ and $N$ be non-zero modules in $\mathscr {C}_{\mathfrak {g}}$. The universal R-matrix $R^{\mspace {1mu}\mathrm {univ}}_{M,N_z}$ is rationally renormalizable if there exists $f(z) \in {{\mathbf {k}}}((z))^\times$ such that

\[ f(z) R^{\mspace{1mu}\mathrm{univ}}_{M,N_z}\bigl( M\mathop\otimes N_z\bigr)\subset N_z\mathop\otimes M. \]

In this case, we can choose $c_{M,N}(z) \in {{\mathbf {k}}}((z))^\times$ such that for any $x \in {{\mathbf {k}}}^\times$, the specialization of $R^{\mspace {1mu}\mathrm {ren}}_{M,N_z} \mathbin {:=} c_{M,N}(z)R^{\mspace {1mu}\mathrm {univ}}_{M,N_z} \colon M \otimes N_z \to N_z \otimes M$ at $z=x$,

\[ R^{\mspace{1mu}\mathrm{ren}}_{M,N_z}\big\vert_{z=x} \colon M \otimes N_x \to N_x \otimes M, \]

does not vanish. Note that $R^{\mspace {1mu}\mathrm {ren}}_{M,N_z}$ and $c_{M,N}(z)$ are unique up to a multiple of ${\mathbf {k}[z^{\pm 1}]}^\times = \bigsqcup _{n \in \mathbb {\mspace {1mu}Z\mspace {1mu}}}{{\mathbf {k}}}^\times z^n$. We call $c_{M,N}(z)$ the renormalizing coefficient. We denote by ${{\mathbf {r}}_{{\scriptstyle {M,N}}}}$ the specialization at $z=1$,

(2.8)\begin{equation} {{\mathbf{r}}_{\mspace{-2mu}{{\scriptstyle{M,N}}}}} \mathbin{:=} R^{\mspace{1mu}\mathrm{ren}}_{M,N_z}\vert_{z=1} \colon M \otimes N \to N \otimes M, \end{equation}

and call it the R-matrix. The R-matrix ${{\mathbf {r}}_ {{\scriptstyle {M,N}}}}$ is well-defined up to a constant multiple whenever $R^{\mspace {1mu}\mathrm {univ}}_{M,N_z}$ is rationally renormalizable. By the definition, ${{\mathbf {r}}_{{\scriptstyle {M,N}}}}$ never vanishes.

Suppose that $M$ and $N$ are simple $U_q'({\mathfrak {g}})$-modules in $\mathscr {C}_{\mathfrak {g}}$. Let $u$ and $v$ be dominant extremal weight vectors of $M$ and $N$, respectively. Then there exists $a_{M,N}(z)\in {{\mathbf {k}}}[[z]]^\times$ such that

\[ R^{\mspace{1mu}\mathrm{univ}}_{M,N_z}( u \mathop\otimes v_z)= a_{M,N}(z)( v_z\mathop\otimes u ). \]

Thus we have a unique ${{\mathbf {k}}}(z)\mathop \otimes U_q'({\mathfrak {g}})$-module isomorphism

\[ R^{\mathrm{norm}}_{M,N_z}\mathbin{:=} a_{M,N}(z)^{{-}1} R^{\mspace{1mu}\mathrm{univ}}_{M,N_z}\big \vert_{{{\mathbf{k}}}(z)\otimes_{{{\mathbf{k}}}[z^{\pm1}]} ( M \otimes N_z)} \]

from ${{\mathbf {k}}}(z)\otimes _{{{\mathbf {k}}}[z^{\pm 1}]} ( M \otimes N_z)$ to ${{\mathbf {k}}}(z)\otimes _{{{\mathbf {k}}}[z^{\pm 1}]} ( N_z \otimes M )$, which satisfies

\[ R^{\mathrm{norm}}_{M, N_z}( u \otimes v_z) = v_z\otimes u . \]

We call $a_{M,N}(z)$ the universal coefficient of $M$ and $N$, and call $R^{\mathrm {norm}}_{M,N_z}$ the normalized $R$-matrix.

Let $d_{M,N}(z) \in {{\mathbf {k}}}[z]$ be a monic polynomial of the smallest degree such that the image of $d_{M,N}(z) R^{\mathrm {norm}}_{M,N_z}(M\mathop \otimes N_z)$ is contained in $N_z \otimes M$; we call it the denominator of $R^{\mathrm {norm}}_{M,N_z}$. Then we have

\[ R^{\mspace{1mu}\mathrm{ren}}_{M,N_z} = d_{M,N}(z)R^{\mathrm{norm}}_{M,N_z} \colon M \otimes N_z \xrightarrow{\,{\hspace{2ex}}\,} N_z \otimes M \quad\text{up to a multiple of ${\mathbf{k}[z^{\pm1}]}^\times$.}\quad \]

Thus

\[ R^{\mspace{1mu}\mathrm{ren}}_{M,N_z} =a_{M,N}(z)^{{-}1}d_{M,N}(z)R^{\mspace{1mu}\mathrm{univ}}_{M,N_z} \quad \text{and} \quad c_{M,N}(z)= \frac{d_{M,N}(z)}{a_{M,N}(z)} \]

up to a multiple of ${{\mathbf {k}}}[z^{\pm 1}]^\times$. In particular, $R^{\mspace {1mu}\mathrm {univ}}_{M,N_z}$ is rationally renormalizable whenever $M$ and $N$ are simple.

The denominator formulas between fundamental representations are summarized for all types in Appendix A.

The next theorem follows from the results of [Reference Akasaka and KashiwaraAK97, Reference ChariCha10, Reference KashiwaraKas02, Reference Kang, Kashiwara, Kim and OhKKKO15]. In the theorem, (ii) follows essentially from [Reference Kang, Kashiwara, Kim and OhKKKO15, Corollary 3.16] together with properties of R-matrices (see also [Reference Kashiwara, Kim, Oh and ParkKKOP20, Proposition 3.16 and Corollary 3.17]), and (i), (iii) and (iv) were conjectured in [Reference Akasaka and KashiwaraAK97, § 2] and proved in [Reference Akasaka and KashiwaraAK97, § 4] for affine types $A$ and $C$, in [Reference KashiwaraKas02, § 9] for general cases in terms of good modules, and in [Reference ChariCha10, §§ 4 and 6] using the braid group actions.

2.3 Hernandez–Leclerc categories

Recall ${\sigma ({\mathfrak {g}})}$ in (2.3). For $(i,x)$ and $(j,y) \in {\sigma ({\mathfrak {g}})}$, we put $d$ arrows from $(i,x)$ to $(j,y)$, where $d$ is the order of the zeros of $d_{ V(\varpi _i), V(\varpi _j) } ( z_{V(\varpi _j)} / z_{V(\varpi _i)} )$ at $z_{V(\varpi _j)} / z_{V(\varpi _i)} = y/x$. Then ${\sigma ({\mathfrak {g}})}$ has a quiver structure. Note that $(i, x)$ and $(j,y)$ are linked in ${\sigma ({\mathfrak {g}})}$ if and only if the tensor product $V(\varpi _i)_x \otimes V(\varpi _j)_y$ is reducible [Reference Akasaka and KashiwaraAK97, Corollary 2.4]. The denominator formulas are explicitly given in Appendix A.

We choose a connected component ${\sigma _0({\mathfrak {g}})}$ of ${\sigma ({\mathfrak {g}})}$. Since a connected component of ${\sigma ({\mathfrak {g}})}$ is unique up to a spectral parameter shift, ${\sigma _0({\mathfrak {g}})}$ is uniquely determined up to a quiver isomorphism. We set

(2.9)\begin{equation} q_s=q^{1/2}\quad\text{{and}}\quad q_t=q^{1/3}. \end{equation}

The distance $\mathrm {d}(u,v)$ between two vertices $u$ and $v$ in a finite Dynkin diagram is the length of the path connecting them. For example, $\mathrm {d}(1,4)=2$ in a Dynkin diagram of type $D_4$, and $\mathrm {d}(1,3)=2$ in a Dynkin diagram of type $F_4$. We denote by $\mathrm {d}_\circ (i,j)$ the distance between $i$ and $j$ in the Dynkin diagram of ${\mathfrak {g}}_0$. For the rest of this paper, we make the following choices of $\sigma _0({\mathfrak {g}})$ (see table (2.6) for the range of $n$):

\begin{align*} &\sigma_0(X) \mathbin{:=} \{(i,({-}q)^{p}) \in I_0 \times {{\mathbf{k}}}^{{\times}} \mid p \equiv_2 \mathrm{d}_\circ(1,i) \} \quad ( X=A^{(1)}_{n}, D^{(1)}_{n}, E^{(1)}_{k} (k=6,7,8) ), \\ & \sigma_0({B_{n}^{(1)}}) \mathbin{:=} \{{ (i,({-}1)^{ n+i} q_s q^m), (n,q^m)}\mid{1\leqslant i \leqslant n-1,\: m \in \mathbb{\mspace{1mu}Z\mspace{1mu}}}\}, \\ &\sigma_0(C^{(1)}_{n}) \mathbin{:=} \{(i,({-}q_s)^{p}) \in I_0 \times {{\mathbf{k}}}^{{\times}} \mid p \equiv_2 \mathrm{d}_\circ(1,i) \}, \\ &\sigma_0(F^{(1)}_{4}) \mathbin{:=} \{(i,({-}1)^iq_s^{2p-\delta_{i,3}}) \in I_0 \times {{\mathbf{k}}}^{{\times}} \mid p \in \mathbb{\mspace{1mu}Z\mspace{1mu}} \}, \\ &\sigma_0(G^{(1)}_{2}) \mathbin{:=} \{(i,({-}q_t)^{p}) \in I_0 \times {{\mathbf{k}}}^{{\times}} \mid p \equiv_2 \mathrm{d}_\circ(2, i ) \}, \\ &\sigma_0(A^{(2)}_{2n}) \mathbin{:=} \{(i,({-}q)^{p}) \in I_0 \times {{\mathbf{k}}}^{{\times}} \mid p \in \mathbb{\mspace{1mu}Z\mspace{1mu}} \}, \\ &\sigma_0(A^{(2)}_{2n-1}) \mathbin{:=} \{( i,\pm({-}q)^{p}), ( n, ({-}q)^{r}) \mid 1 \leqslant i < n,\: p \equiv_2 i+1, \: r \equiv_2 n+1 \}, \\ & \sigma_0(D^{(2)}_{n+1}) \mathbin{:=} \{ (i, (\sqrt{-1}^{n+1-i}) ({-}q)^p), (n, \pm ({-}q)^{r}) \mid 1\leqslant i < n,\: p \equiv_2 i+1 ,\: r \equiv_2 n+1 \}, \\ & \sigma_0(E^{(2)}_{6}) \mathbin{:=} \{ (i,\pm q^r), (i', \sqrt{-1} ({-}q )^{r'}) \mid i \in \{ 1,2 \}, \: i' \in \{3,4\}, \: r \equiv_2 i+1, \: r'\equiv_2 i'+1 \}, \\ & \sigma_0(D^{(3)}_{4}) \mathbin{:=} \{ (1, q^r),(1, \omega q^r), (1, \omega^2 q^r), (2,-q^{r+1}) \mid r\equiv_2 0 \} \quad (\omega^2+\omega+1=0), \end{align*}

where $a \equiv _2 b$ means that $a\equiv b\mspace {3mu}\mathbin {\mathrm {mod}}\mspace {1mu} 2$ (see [Reference Hernandez and LeclercHL10, § 3.7], [Reference Kang, Kashiwara, Kim and OhKKKO16, § 4.1], [Reference Kashiwara and OhKO19, § 6] and [Reference Oh and ScrimshawOS19a, § 6]). Note that in [Reference Oh and ScrimshawOS19a, § 6] the category ${\mathscr {C}_{\mathfrak {g}}^{Q}}$ and ${\sigma _Q({\mathfrak {g}})}$ were dealt with only in exceptional cases, but it is easy to obtain ${\sigma _0({\mathfrak {g}})}$ using ${\sigma _Q({\mathfrak {g}})}$. We use the notation $B_2^{(1)}$ and $A^{(2)}_{3}$ instead of $C_2^{(1)}$ and $D_{3}^{(2)}$, respectively. Here we use the standard convention for Dynkin diagrams in [Reference KacKac90, Ch. 4] except for $A^{(2)}_{2n}$, $A_3^{(2)}$, $B_2^{(1)}$ and $E^{(1)}_{k}$ ($k=6,7,8$), which are given in (2.1).

We define $\mathscr {C}_{\mathfrak {g}}^0$ to be the smallest full subcategory of $\mathscr {C}_{\mathfrak {g}}$ for which the following hold:

1. (a) $\mathscr {C}_{\mathfrak {g}}^0$ contains $V(\varpi _i)_x$ for all $(i,x) \in {\sigma _0({\mathfrak {g}})}$;

2. (b) $\mathscr {C}_{\mathfrak {g}}^0$ is stable by taking subquotients, extensions and tensor products.

For symmetric affine types, this category was introduced in [Reference Hernandez and LeclercHL10]. Note that every simple module in $\mathscr {C}_{\mathfrak {g}}$ is isomorphic to a tensor product of certain spectral parameter shifts of some simple modules in $\mathscr {C}_{\mathfrak {g}}^0$ (see [Reference Hernandez and LeclercHL10, § 3.7]).

2.4 The categories ${\mathscr {C}_{\mathfrak {g}}^{Q}}$

In this subsection, we recall very briefly a certain subcategory ${\mathscr {C}_{\mathfrak {g}}^{Q}}$ of $\mathscr {C}_{\mathfrak {g}}$ categorifying the coordinate ring $\mathbb {C}[N]$ of the maximal unipotent group $N$ associated with a certain simple Lie algebra.

This subcategory ${\mathscr {C}_{\mathfrak {g}}^{Q}}$ was introduced in [Reference Hernandez and LeclercHL15] for simply laced affine type ADE, in [Reference Kang, Kashiwara, Kim and OhKKKO16] for twisted affine types $A^{(2)}$ and $D^{(2)}$, in [Reference Kashiwara and OhKO19, Reference Oh and SuhOS19b] for untwisted affine types $B^{(1)}$ and $C^{(1)}$, and in [Reference Oh and ScrimshawOS19a] for exceptional affine type. The quantum affine Schur–Weyl duality functor between the finite-dimensional module category of a quiver Hecke algebra and ${\mathscr {C}_{\mathfrak {g}}^{Q}}$ was also constructed in [Reference Kang, Kashiwara and KimKKK15] for untwisted affine types $A^{(1)}$ and $D^{(1)}$, in [Reference Kang, Kashiwara, Kim and OhKKKO16] for twisted affine types $A^{(2)}$ and $D^{(2)}$, in [Reference Kashiwara and OhKO19] for untwisted affine types $B^{(1)}$ and $C^{(1)}$, in [Reference Oh and ScrimshawOS19a] for exceptional affine type, and in [Reference FujitaFuj20] for simply laced affine type ADE in a geometric manner.

We shall describe ${\sigma _Q({\mathfrak {g}})}$ and ${\mathscr {C}_{\mathfrak {g}}^{Q}}$ by using $Q$-data [Reference Fujita and OhFO21]. A $Q$-datum generalizes a Dynkin quiver with a height function, which provides a uniform way of describing the Hernandez–Leclerc category ${\mathscr {C}_{\mathfrak {g}}^{Q}}$. Our brief explanation follows [Reference Fujita and OhFO21, § 3] (see also [Reference Fujita, Hernandez, Oh and OyaFHOO21, § 4] and [Reference Kashiwara, Kim, Oh and ParkKKOP21, § 6]). Let ${\mathfrak {g}}$ be an affine Kac–Moody algebra and let ${{\mathfrak {g}}_{\mathrm {fin}}}$ be the simply laced finite-type Lie algebra corresponding to the affine type of ${\mathfrak {g}}$ in table (4.5). Let ${\mathrm {I_{fin}}}$ be the index set of ${{\mathfrak {g}}_{\mathrm {fin}}}$ and let $\mathsf {D}_{\rm fin}$ be the Dynkin diagram for ${{\mathfrak {g}}_{\mathrm {fin}}}$.

We first assume that ${\mathfrak {g}}$ is of untwisted type. We define an Dynkin diagram automorphism $\varrho$ of $\mathsf {D}_{\rm fin}$ as follows. For ${\mathfrak {g}}=A_n^{(1)}, D_n^{(1)}$ or $E_k^{(1)}$ type ($k=6,7,8$) we set $\varrho :={\rm id}$, and for the remaining types $\varrho$ is defined as follows (see [Reference Fujita and OhFO21, § 3.1]).

Let $I_0 =\{ 1,2,\ldots, n \}$ be the index set of ${\mathfrak {g}}_0$. Note that ${\mathrm {I_{fin}}} = I_0$ when ${\mathfrak {g}}=A_n^{(1)}, D_n^{(1)}, E_k^{(1)}$ ($k=6,7,8$). Let $\mathrm {ord}(\varrho )$ be the order of $\varrho$. For $i\in {\mathrm {I_{fin}}}$, we denote by $\mathrm {orb}(i)$ the orbit of $i$ under the action $\varrho$ and set $\mathsf {d}_i := | \mathrm {orb}(i) |$. We identify the set of orbits of ${\mathrm {I_{fin}}}$ with $I_0$ by mapping $\mathrm {orb}(i) \mapsto \min \{ \mathrm {orb}(i) \}$ for ${\mathfrak {g}} \ne F_4^{(1)}$ and mapping $\mathrm {orb}(1) \mapsto 1$, $\mathrm {orb}(3) \mapsto 2$, $\mathrm {orb}(4) \mapsto 3$ and $\mathrm {orb}(2) \mapsto 4$ for ${\mathfrak {g}} = F_4^{(1)}$. We write $\pi \colon {\mathrm {I_{fin}}}\to I_0$ for the projection via this identification.

For a $Q$-datum $Q = (\mathsf {D}_{\rm fin}, \varrho, \xi )$ associated to ${\mathfrak {g}}$, let

\[ \hat{I}_Q \mathbin{:=} \{ ( i ,p) \in {\mathrm{I_{fin}}} \times \mathbb{\mspace{1mu}Z\mspace{1mu}} \mid p -\xi_i \in 2\mathsf{d}_i\mathbb{\mspace{1mu}Z\mspace{1mu}}\}. \]

The generalized $\varrho$-Coxeter element $\tau _Q \in \mathsf {W}_{\rm fin} \rtimes \textrm {Aut} (\mathsf {D}_{\rm fin})$ associated with $Q$ is defined in [Reference Fujita and OhFO21, Definition 3.33] and can be understood as a generalization of a Coxeter element. Here $\mathsf {W}_{\rm fin}$ is the Weyl group of ${{\mathfrak {g}}_{\mathrm {fin}}}$.

For $i\in I_0$, we denote by $o(i)$ the corresponding orbit of ${\mathrm {I_{fin}}}$. For each $i\in I_0$, we denote by $i^\circ$ the unique vertex in the orbit $o( i)$ satisfying $\xi _{i^\circ } = \max \{ \xi _j \mid j\in o(i) \}$. In this paper, we assume further that the height function $\xi$ satisfies

(2.10)\begin{equation} \xi_{\varrho^k(i^\circ)} = \xi_{i^\circ} - 2k \quad \text{for each $i\in I_0$ and $0 \leqslant k < \mathsf{d}_i $.} \end{equation}

Let $\{ i_1, i_2, \ldots, i_n\}$ be a total order of $I_0$ satisfying $\xi _{i_1^\circ } \geqslant \xi _{i_2^\circ } \geqslant \cdots \geqslant \xi _{i_n^\circ }$. Then we have

\[ \tau_Q = s_{i_1^\circ} s_{i_2^\circ} \cdots s_{i_n^\circ} \varrho \in \mathsf{W}_{\rm fin} \rtimes \textrm{Aut} (\mathsf{D}_{\rm fin}) \]

(see [Reference Fujita and OhFO21, § 3.6] and also [Reference Fujita, Hernandez, Oh and OyaFHOO21, Proposition 4.4] for more details).

Let ${\Delta ^+_{Q}}$ be the set of positive roots of ${{\mathfrak {g}}_{\mathrm {fin}}}$, and let $\hat {\Phi } \mathbin {:=} {\Delta ^+_{Q}} \times \mathbb {\mspace {1mu}Z\mspace {1mu}}$. For each $i \in {\mathrm {I_{fin}}}$ we define

\[ \gamma^Q_i \mathbin{:=} (1-\tau_Q^{\mathsf{d}_i})\Lambda_i \in {\Delta^+_{Q}}, \]

where $\Lambda _i$ is the $i$th fundamental weight of ${{\mathfrak {g}}_{\mathrm {fin}}}$. It is shown in [Reference Hernandez and LeclercHL15, § 2.2] and [Reference Fujita and OhFO21, Theorem 3.35] that there exists a unique bijection $\psi _Q \colon \hat {I}_Q \to \hat {\Phi }$ defined inductively as follows:

1. (i) $\psi _Q(i,\xi _i)=(\gamma ^Q_i,0)$;

2. (ii) if $\psi _Q(i,p)=(\beta,m)$, then define:

   1. (a) $\psi _Q(i,p\pm 2\mathsf {d}_i) = (\tau _Q^{\mp \mathsf {d}_i} (\beta ),m)$ if $\tau _Q^{ \mp \mathsf {d}_i}(\beta ) \in {\Delta ^+_{Q}}$;

   2. (b) $\psi _Q(i,p\pm 2\mathsf {d}_i) = (-\tau _Q^{\mp \mathsf {d}_i}(\beta ),m\pm 1)$ if $\tau _Q^{\mp \mathsf {d}_i}(\beta ) \in - {\Delta ^+_{Q}}$.

Let $I_Q\mathbin {:=} \psi _Q^{-1}({\Delta ^+_{Q}}\times \{0\})\subset {\mathrm {I_{fin}}}\times \mathbb {\mspace {1mu}Z\mspace {1mu}}.$ Then one can describe

\[ I_Q=\{{(i,p)\in\hat{I}_Q\mid \xi_{ i^*} - \mathrm{ord}(\varrho)\mathsf{h}^\vee{<} p \leqslant \xi_{i}}\}, \]

where $\mathsf {h}^\vee$ is the dual Coxeter number of ${\mathfrak {g}}_0$ (see [Reference Fujita and OhFO21, Theorem 3.35] and also [Reference Fujita, Hernandez, Oh and OyaFHOO21, Proposition 4.15]). We define

\[ {\sigma_Q({\mathfrak{g}})} := \{ \zeta(i,p ) \mid (i,p) \in I_Q \}, \]

where we set ${{q}_{\mspace {1mu}{\scriptstyle {\mathrm {sh}}}}} \mathbin {:=} q^{1/\mathrm {ord}(\varrho )}$ and

\begin{align*} \zeta(i,p) := \begin{cases} (\pi(i), (-{{q}_{\mspace{1mu}{\scriptstyle{\mathrm{sh}}}}})^p) & \text{if } {\mathfrak{g}} = A^{(1)}_n, \; C_n^{(1)}, \; D^{(1)}_n, \; E_{6,7,8}^{(1)}, \; G_2^{(1)}, \\ (\pi(i), ({-}1)^{i+n}({{q}_{\mspace{1mu}{\scriptstyle{\mathrm{sh}}}}})^p) & \text{if } {\mathfrak{g}} = B_n^{(1)}, \\ (\pi(i), ({-}1)^{ \pi(i) }({{q}_{\mspace{1mu}{\scriptstyle{\mathrm{sh}}}}})^p) & \text{if } {\mathfrak{g}} = F_4^{(1)} \end{cases} \end{align*}

(see [Reference Fujita and OhFO21, § 5.4]). We define

(2.11)\begin{equation} \phi_Q\colon {\Delta^+_{Q}} \mathop{\xrightarrow{\sim}}{\sigma_Q({\mathfrak{g}})} \end{equation}

by $\phi _Q( \beta ) := \zeta \circ \psi _Q^{-1} (\beta, 0)$ for $\beta \in {\Delta ^+_{Q}}$. The map $\phi _Q$ is bijective.

For the rest of this paper, we make the following choices of $Q$-data:

• • for simply laced ADE type, $\mathrm {ord}(\varrho )=1$ and the height function $\xi$ is defined in Appendix A.1;

• • for ${\mathfrak {g}}=B_n^{(1)}$, $\mathrm {ord}(\varrho )=2$ and ;

• • for ${\mathfrak {g}}=C_n^{(1)}$, $\mathrm {ord}(\varrho )=2$ and ;

• • for ${\mathfrak {g}}=F_4^{(1)}$, $\mathrm {ord}(\varrho )=2$ and ;

• • for ${\mathfrak {g}}=G_2^{(1)}$, $\mathrm {ord}(\varrho )=3$ and .

Here an underlined integer stands for the value of $\xi _i$ at each vertex $i \in \mathsf {D}_{\rm fin}$ and an arrow $i \to j$ means that $\xi _i>\xi _j$ and $\mathrm {d}(i,j)=1$ in the Dynkin diagram $\mathsf {D}_{\rm fin}$. Note that our choice of $Q$ satisfies (2.10). Then $\tau _Q$ is given as follows:

• • for simply laced ADE type, $\tau _Q$ is the same as $\tau$ in Appendix A.1;

• • for ${\mathfrak {g}}=B_n^{(1)}, C_n^{(1)}$, $\tau _Q = s_1 s_2 \cdots s_n \varrho$;

• • for ${\mathfrak {g}}=F_4^{(1)}$, $\tau _Q = s_1 s_2 s_3 s_4 \varrho$;

• • for ${\mathfrak {g}}=G_2^{(1)}$, $\tau _Q = s_2 s_1 \varrho$.

In this case the set ${\sigma _Q({\mathfrak {g}})}$ is contained in ${\sigma _0({\mathfrak {g}})}$ in § 2.3 and can be written explicitly as follows (where $a\leqslant _2 b$ means that $a\leqslant b$ and $a\equiv b\mspace {3mu}\mathbin {\mathrm {mod}}\mspace {1mu} 2$):

\begin{align*} &\sigma_Q(A^{(1)}_{n }) \mathbin{:=} \{(i,({-}q)^{k}) \in \sigma_0(A^{(1)}_{n }) \mid i-2n +1 \leqslant_2 k \leqslant_2 -i +1 \}, \\ &\sigma_Q(B^{(1)}_{n}) \mathbin{:=} \{ (i,({-}1)^{n+i}q_s^{k}), (n,q^{k'}) \in \sigma_0(B^{(1)}_{n}) \mid i < n, -2n-2i+3 \leqslant_2 k \leqslant_2 2n-2i-1, \\ &\hspace{48ex} -\!2n+2 \leqslant k' \leqslant 0 \}, \\ & \sigma_Q({C_{n}^{(1)}}) \mathbin{:=} \{ (i, ({-}q_s)^{k}) \in \sigma_0(C^{(1)}_{n}) \mid{-}\mathrm{d}_\circ(1,i)-2n \leqslant_2 k \leqslant_2 -\mathrm{d}_\circ(1,i) \} , \\ & \sigma_Q({D_{n}^{(1)}}) \mathbin{:=} \{ (i, ({-}q)^{k}) \in \sigma_0(D^{(1)}_{n}) \mid{-}\mathrm{d}_\circ(1,i)-2n+4 \leqslant_2 k \leqslant_2 -\mathrm{d}_\circ(1,i) \} , \\ & \sigma_Q({E_{6}^{(1)}}) \mathbin{:=} \{ (i,({-}q)^k) \in \sigma_0(E^{(1)}_{6}) \mid \mathrm{d}_\circ(1,i)-14 \leqslant_2 k \leqslant_2 -\mathrm{d}_\circ(1,i) + 2\delta_{i,2} \}, \\ & \sigma_Q({E_{7}^{(1)}}) \mathbin{:=} \{ (i,({-}q)^k) \in \sigma_0(E^{(1)}_{7}) \mid{-}\mathrm{d}_\circ(1,i)-16 + 2\delta_{i,2} \leqslant_2 k \leqslant_2-\mathrm{d}_\circ(1,i) + 2\delta_{i,2} \}, \\ & \sigma_Q({E_{8}^{(1)}}) \mathbin{:=} \{ (i,({-}q)^k) \in \sigma_0(E^{(1)}_{8}) \mid{-}\mathrm{d}_\circ(1,i)-28 + 2\delta_{i,2} \leqslant_2 k \leqslant_2 -\mathrm{d}_\circ(1,i) + 2\delta_{i,2} \}, \\ & \sigma_Q({F_{4}^{(1)}}) \mathbin{:=} \biggl\{ (i, ({-}1)^iq^{k}) \in \sigma_0(F^{(1)}_{4}) \biggm| \mathrm{d}_\circ(i,3)-10 + \frac{\delta_{i,3}}{2} \leqslant k \leqslant \mathrm{d}_\circ(i,3)-2 + \frac{\delta_{i,3}}{2} \biggr\}, \\ & \sigma_Q({G_{2}^{(1)}}) \mathbin{:=} \{ (i, ({-}q_t)^{k}) \in \sigma_0(G^{(1)}_{2}) \mid{-}\mathrm{d}_\circ(2,i)-10 \leqslant_2 k \leqslant_2 -\mathrm{d}_\circ(2,i) \} , \end{align*}

where $\mathrm {d}_\circ (i,j)$ denotes the distance between $i$ and $j$ in the Dynkin diagram of ${\mathfrak {g}}_0$.

We now assume that ${\mathfrak {g}}$ is of twisted type. Then one can define

\begin{align*} &\sigma_Q(A^{(2)}_{N}) \mathbin{:=} \{(i, ({-}q)^{k} )^\star \mid (i,({-}q)^{k}) \in \sigma_Q(A^{(1)}_{N}) \} \quad (N=2n-1 \text{ or } 2n) , \\ &\sigma_Q(D^{(2)}_{n+1}) \mathbin{:=} \{(i, ({-}q)^{k})^\star\mid (i,({-}q)^{k}) \in \sigma_Q(D^{(1)}_{n+1}) \}, \\ &\sigma_Q(E^{(2)}_{6}) \mathbin{:=} \{(i,({-}q)^{k} )^\star\mid (i,({-}q)^{k}) \in \sigma_Q(E^{(1)}_{6}) \}, \\ &\sigma_Q(D^{(3)}_{4}) \mathbin{:=} \{(i,({-}q)^{k})^{{\dagger}}\mid (i,({-}q)^{k}) \in \sigma_Q(D^{(1)}_{4}) \}, \end{align*}

where for $(i,a) \in \sigma _0({\mathfrak {g}}^{(1)}_{N})$ we set

\[ \begin{aligned} (i,a)^\star{=} \begin{cases} (i,a) & \text{if } \mathfrak{g}=A^{(1)}_{N}, \, i \leqslant \lfloor (N+1)/2 \rfloor \:\text{ or }\: \mathfrak{g}=E^{(1)}_{6}, \, i =1, \\ (N+1-i,({-}1)^Na) & \text{if } \mathfrak{g}=A^{(1)}_{N}, \: i > \lfloor (N+1)/2 \rfloor, \\ (i,\sqrt{-1}^{n+1-i}a) & \text{if } \mathfrak{g}=D^{(1)}_{n+1}, \: i \leqslant n-1, \\ (n,({-}1)^ia) & \text{if } \mathfrak{g}=D^{(1)}_{n+1}, \: i\in \{n,n+1\}, \\ (2,a) & \text{if } \mathfrak{g}=E^{(1)}_{6}, \: i=3, \\ (2,-a) & \text{if } \mathfrak{g}=E^{(1)}_{6}, \: i=5, \\ (1,-a) & \text{if } \mathfrak{g}=E^{(1)}_{6}, \: i=6 , \\ (3,\sqrt{-1}a) & \text{if } \mathfrak{g}=E^{(1)}_{6}, \: i=4 , \\ (4,\sqrt{-1}a) & \text{if } \mathfrak{g}=E^{(1)}_{6}, \: i=2 \end{cases} \end{aligned} \]

and

\begin{align*} (i,a)^{{\dagger}}= \begin{cases} (2,a) & \text{if } i=2, \\ (1,(\delta_{i,1}+\delta_{i,3} \omega+\delta_{i,4} \omega^2)a) & \text{if } i \ne 2 \end{cases} \end{align*}

(see [Reference Kang, Kashiwara, Kim and OhKKKO16, Proposition 4.3] and [Reference Oh and ScrimshawOS19a, Proposition 6.5] for details of $\star$ and ${{\dagger}}$). The bijection $\phi _Q\colon {\Delta ^+_{Q}} \mathop {\xrightarrow {\sim}}{\sigma _Q({\mathfrak {g}})}$ is defined by composing the bijection for untwisted type with the maps $\star$ and ${{\dagger}}$.

Comparing the above descriptions of ${\sigma _Q({\mathfrak {g}})}$ with the descriptions of ${\sigma _0({\mathfrak {g}})}$ given in § 2.3, one can easily show that

(2.12)\begin{equation} \begin{aligned} & {\sigma_0({\mathfrak{g}})} = \bigsqcup_{k \in \mathbb{\mspace{1mu}Z\mspace{1mu}} } {\sigma_Q({\mathfrak{g}})} ^{*k}, \\ & {\sigma_Q({\mathfrak{g}})} ^{*k} \cap {\sigma_Q({\mathfrak{g}})} ^{*k'} = \emptyset \quad \text{for } k,k' \in \mathbb{\mspace{1mu}Z\mspace{1mu}} \text{ with } k \ne k', \end{aligned} \end{equation}

where ${\sigma _Q({\mathfrak {g}})} ^{*k} \mathbin {:=} \{ (i^{*k},(p^*)^ka) \mid (i,a) \in {\sigma _Q({\mathfrak {g}})} \}$ with $i^{*k} =i$ if $k$ is even and $i^{*k} =i^*$ if $k$ is odd (see [Reference Fujita and OhFO21, Proposition 5.9]). Note that $p^*$ is given in (2.5).

Let ${\mathscr {C}_{\mathfrak {g}}^{Q}}$ be the smallest full subcategory of $\mathscr {C}_{\mathfrak {g}}^0$ with the following properties:

1. (a) ${\mathscr {C}_{\mathfrak {g}}^{Q}}$ contains $\mathbf {1}$ and $V(\varpi _i)_x$ for all $(i,x) \in {\sigma _Q({\mathfrak {g}})}$;

2. (b) ${\mathscr {C}_{\mathfrak {g}}^{Q}}$ is stable by taking subquotients, extensions and tensor products.

It was shown in [Reference Hernandez and LeclercHL15, Theorem 6.1], [Reference Kang, Kashiwara, Kim and OhKKKO16, Corollary 5.6], [Reference Kashiwara and OhKO19, Corollary 6.6] and [Reference Oh and ScrimshawOS19a, § 6] that the Grothendieck ring $K({\mathscr {C}_{\mathfrak {g}}^{Q}})$ of the monoidal category ${\mathscr {C}_{\mathfrak {g}}^{Q}}$ is isomorphic to the coordinate ring $\mathbb {C}[N]$ of the maximal unipotent group $N$ associated with ${{\mathfrak {g}}_{\mathrm {fin}}}$. The set ${\Delta ^+_{Q}}$ has a convex order $\prec _Q$ arising from $Q$.

Let $\beta \in {\Delta ^+_{Q}}$ and write $(i,a) = \phi _Q(\beta )$. Then set

\[ V_Q(\beta) \mathbin{:=} V(\varpi_i)_a \in {\mathscr{C}_{\mathfrak{g}}^{Q}}. \]

Under the categorification, the modules $V_Q(\beta )$ correspond to the dual PBW vectors of $\mathbb {C}[N]$ with respect to the convex order $\prec _Q$ on ${\Delta ^+_{Q}}$.

The proposition below follows from [Reference Kang, Kashiwara and KimKKK15, § 4.3], [Reference Kang, Kashiwara, Kim and OhKKKO16, Proposition 4.9 and Theorem 5.1], [Reference Kashiwara and OhKO19, § 4.3] and [Reference Oh and ScrimshawOS19a, § 6].

5.1 Blocks

We recall the notion of blocks. Let $\mathcal {C}$ be an abelian category such that any object of $\mathcal {C}$ has finite length.

The following lemma is obvious.

Proof. Let $\ell$ and $\ell '$ be the lengths of $X$ and $X'$, respectively. We use induction on $\ell +\ell '$. If $X$ and $X'$ are simple, then the claimed result is clear by the assumption.

Suppose that $X'$ is not simple. Then there exists an exact sequence $0 \rightarrow M \rightarrow X' \rightarrow N \rightarrow 0$ with a simple $M$. It in turn gives the exact sequence

\[ \operatorname{Ext}^1_\mathcal{C}(X, M) \rightarrow \operatorname{Ext}^1_\mathcal{C}(X, X') \rightarrow \operatorname{Ext}^1_\mathcal{C}(X, N). \]

By the induction hypothesis we have $\operatorname {Ext}^1_\mathcal {C}(X, M) = \operatorname {Ext}^1_\mathcal {C}(X, N) = 0$, which tells us that $\operatorname {Ext}^{1}_\mathcal {C}( X,X' )=0$.

The case where $X$ is not simple can be proved in the same manner.

Lemma 5.4 Let $\mathfrak {c}$ be the set of isomorphism classes of simple objects of $\mathcal {C}$, and let $\mathfrak {c}=\bigsqcup _{a\in A}\mathfrak {c}_a$ be a partition of $\mathfrak {c}$. We assume that

For $a\in A$, let $\mathcal {C}_a$ be the full subcategory of $\mathcal {C}$ consisting of objects $X$ such that any simple subquotient of $X$ belongs to $\mathfrak {c}_a$. Then $\mathcal {C}=\bigoplus _{a\in A}\mathcal {C}_a$.

Proof. It is enough to show that any object $X$ of $\mathcal {C}$ has a decomposition $X\simeq \bigoplus _{a\in A}X_a$ with $X_a\in \mathcal {C}_a$. In order to prove this, we shall argue by induction on the length of $X$. We may assume that X is non-zero. Let us take a subobject $Y$ of $X$ such that $X/Y$ is simple. Then the induction hypothesis implies that $Y = \bigoplus _{a\in A}Y_a$ with $Y_a\in \mathcal {C}_a$.

Take $a_0\in A$ such that $X/Y$ belongs to $\mathfrak {c}_{a_0}$. Then define $Z\in \mathcal {C}$ by the exact sequence

(5.1)\begin{equation} 0 \rightarrow \bigoplus_{a\not=a_0}Y_a \rightarrow X \rightarrow Z \rightarrow 0. \end{equation}

Since we have an exact sequence $0\to Y_{a_0}\to Z\to X/Y\to 0$, $Z$ belongs to $\mathcal {C}_{a_0}$. Then Lemma 5.3 tells us that $\operatorname {Ext}^1(Z,\bigoplus _{a\not =a_0}Y_a )=0$. Hence the exact sequence (5.1) splits, i.e. $X \simeq Z\oplus \bigoplus _{a\not =a_0}Y_a$.

Let $\approx$ be the equivalence relation on the set of isomorphism classes of simple objects of $\mathcal {C}$ generated by the relation $\approx '$ defined as follows: for simple objects $S,S' \in \mathcal {C}$,

\[ [S] \approx' [S'] \quad \text{if and only if } \quad \operatorname{Ext}^1_\mathcal{C}(S,S') \ne 0. \]

Proof. Lemma 5.4 implies the decomposition

\[ \mathcal{C} = \bigoplus_{a \in A} \mathcal{C}_a. \]

Moreover, since $a$ is a $\approx$-equivalence class, there is no non-trivial decomposition of $\mathcal {C}_a$ for any $a \in A$.

The next corollary follows directly from Theorem 5.5.

5.2 Direct decomposition of $\mathscr {C}_{\mathfrak {g}}$

In this subsection, we shall prove that $\mathscr {C}_{\mathfrak {g}}$ has a decomposition parameterized by elements of $\mathcal {W}$.

Lemma 5.7 For modules $M,N \in \mathscr {C}_{\mathfrak {g}}$, there exists an isomorphism

(5.2)\begin{equation} \Psi\colon \mathbf{k}[z^{\pm1}] \otimes \operatorname{Hom}_{U'_q({\mathfrak{g}})}(N, \mathbf{1}) \otimes \operatorname{Hom}_{U'_q({\mathfrak{g}})}( \mathbf{1}, M) \buildrel \sim \over \longrightarrow \operatorname{Hom}_{ U'_q({\mathfrak{g}})}(N, M_z)\end{equation}

defined by $\Psi (a(z) \otimes f \otimes g ) = a(z) ( g \circ f )$ for $a(z) \in \mathbf {k}[z^{\pm 1}]$, $f \in \operatorname {Hom}_{U'_q({\mathfrak {g}})}(N, \mathbf {1})$ and $g \in \operatorname {Hom}_{U'_q({\mathfrak {g}})}(\mathbf {1}, M)$.

Proof. Note that $\mathbf {k}[z^{\pm 1}] \otimes \operatorname {Hom}_{U'_q({\mathfrak {g}})}( \mathbf {1}, M)\mathop {\xrightarrow {\sim}}\operatorname {Hom}_{U'_q({\mathfrak {g}})}( \mathbf {1}, M_z)$. There is a quotient $N'$ of $N$ which is a direct sum of copies of ${\bf {1}}$ and $\operatorname {Hom}(N',{\bf {1}})\mathop {\xrightarrow {\sim}}\operatorname {Hom}(N,{\bf {1}})$. Since (5.2) for $N'$ is obviously an isomorphism, $\Psi$ is injective.

To prove that $\Psi$ is surjective, we shall decompose a given non-zero $f\colon N \to M_z$ into $N \to {\bf {1}}^{\oplus \ell } \to M_z$ for some $\ell \in \mathbb {\mspace {1mu}Z\mspace {1mu}}_{>0}$. Here $\mathbf {1}^{\oplus \ell }$ is the direct sum of $\ell$ copies of the trivial module $\mathbf {1}$. Without loss of generality, we may assume that $f$ is injective. We set $\operatorname {wt}(N) \mathbin {:=} \{ \lambda \in \mathsf {P}_{\mathrm {cl}\mspace {1mu}} \mid N_\lambda \ne 0 \}$.

If $\operatorname {wt}(N) =\{0\}$, then $N$ should be isomorphic to $\mathbf {1} ^{\oplus \ell }$ for some $\ell \in \mathbb {\mspace {1mu}Z\mspace {1mu}}_{>0}$, which is the desired result.

Now suppose that $\operatorname {wt}(N)\not =\{0\}$. We choose a non-zero weight $\lambda \in \operatorname {wt}(N)$.

Note that the $U'_q({\mathfrak {g}})$-module structure on $M_z$ extends to a $U_q({\mathfrak {g}})$-module structure and we have a weight decomposition $M_z=\bigoplus _{\mu \in \mathsf {P}}(M_z)_\mu$. Then

\[ f(N_\lambda)\subset \bigoplus_{\mu\in\mathsf{P},\,{\mathrm{cl}\mspace{1mu}}(\mu)=\lambda}(M_z)_\mu, \]

where ${\mathrm {cl}\mspace {1mu}}\colon \mathsf {P}\to \mathsf {P}_{\mathrm {cl}\mspace {1mu}}$ is the classical projection. There exist $w \in \mathsf {W}$ and a non-zero integer $n$ such that $w( \mu ) = \mu +n\delta$ for any $\mu \in {\mathrm {cl}\mspace {1mu}}^{-1}(\lambda )$. We now consider the braid group action $T_w$ defined by $w$ on an integral module (see [Reference LusztigLus90, Reference SaitoSai94]). Then the $\mathbf {k}$-linear automorphism $T_w$ sends $(M_z)_\mu$ to $(M_z)_{w\mu }$. The space $f(N_\lambda )$ is invariant under the automorphism $T_w$, but any non-zero finite-dimensional subspace of $\bigoplus _{\mu \in \mathsf {P},\,{\mathrm {cl}\mspace {1mu}}(\mu )=\lambda } (M_z)_{\mu }$ cannot be invariant under $T_w$. This is a contradiction.

Proposition 5.8 For modules $M,N \in \mathscr {C}_{\mathfrak {g}}$ and a simple module $L \in \mathscr {C}_{\mathfrak {g}}$, we have the isomorphisms

\[ \mathbf{k}[z^{\pm1}] \otimes \operatorname{Hom}_{U_q'({\mathfrak{g}})}(M,N) \buildrel \sim \over \longrightarrow \operatorname{Hom}_{ \mathbf{k}[z^{\pm1}] \otimes U_q'({\mathfrak{g}})}(M\otimes L_z,N \otimes L_z). \]

Proof. By Lemma 5.7, we obtain that

\begin{align*} \operatorname{Hom}_{ \mathbf{k}[z^{\pm1}] \otimes U_q'({\mathfrak{g}}) }(M\otimes L_z,N \otimes L_z) & \simeq \operatorname{Hom}_{U_q'({\mathfrak{g}}) }(N^* \otimes M , (L \otimes L^*)_z) \\ & \simeq {{\mathbf{k}}}[z^{\pm1}] \otimes \operatorname{Hom}_{U_q'({\mathfrak{g}})}(N^* \otimes M ,{\bf{1}}) \mathop\otimes \operatorname{Hom}_{U_q'({\mathfrak{g}})}({\bf{1}}, L\otimes L^*)\\ & \simeq {{\mathbf{k}}}[z^{\pm1}] \otimes \operatorname{Hom}_{U_q'({\mathfrak{g}})}(M ,N). \end{align*}

Lemma 5.9 Let $M$ and $N$ be simple modules in $\mathscr {C}_{\mathfrak {g}}$. If

\[ \frac{c_{M,L}(z)}{c_{N,L}(z)} \notin\mathbf{k}(z) \quad\text{for some simple module } L \in \mathscr{C}_{\mathfrak{g}}, \]

then we have

\[ \operatorname{Ext}^1_{U_q'({\mathfrak{g}})}(M,N)=0. \]

Proof. Let $L \in \mathscr {C}_{\mathfrak {g}}$ be a simple module such that ${c_{M,L}(z)}/{c_{N,L}(z)} \notin \mathbf {k}(z)$.

We shall prove that any exact sequence

\[ 0 \rightarrow N \rightarrow X \rightarrow M \rightarrow 0 \]

splits. We set $\hat {L}_z \mathbin {:=} \mathbf {k}((z)) \otimes _{\mathbf {k} [z^{\pm 1}] } L_z$, where $L_z$ is the affinization of $L$. Then the following diagram commutes.

We set

\[ f(z) \mathbin{:=} \frac{ c_{M, L}(z) }{ c_{N, L}(z) } \notin \mathbf{k}(z) \quad\text{{and}}\quad R \mathbin{:=} c_{M, L}(z) R^{\mspace{1mu}\mathrm{univ}}_{X, \hat{L}_z}\colon X\otimes \hat{L}_z \rightarrow \hat{L}_z \otimes X. \]

It follows from

\[ c_{M,L}(z) R^{\mspace{1mu}\mathrm{univ}}_{M, \hat{L}_z}( M \otimes L_z) \subset L_z \otimes M \quad\text{{and}}\quad c_{N,L}(z) R^{\mspace{1mu}\mathrm{univ}}_{N, \hat{L}_z}( N \otimes L_z) \subset L_z \otimes N \]

that

\[ R( X \otimes L_z) \subset L_z \otimes X + \hat{L}_z \otimes N \quad\text{{and}}\quad R(N \otimes L_z ) \subset f(z) ( L_z \otimes N ). \]

Therefore $R$ induces the $\mathbf {k}[z^{\pm 1}]\mathop \otimes U'_q({\mathfrak {g}})$-linear homomorphism

\[ \mathcal{R} \colon M \otimes L_z\simeq \frac{X\mathop\otimes L_z}{N\mathop\otimes L_z} \longrightarrow \frac{\mathbf{k}(z) \otimes L_z \otimes X + \hat{L}_z \otimes N}{\mathbf{k}(z) \otimes L_z \otimes X + f(z) \mathbf{k}(z) \otimes L_z \otimes N}. \]

We set $\mathcal {P} \mathbin {:=} {\mathbf {k}((z))}/({\mathbf {k}(z) + f(z) \mathbf {k}(z)})$. Since

\begin{align*} \frac{ \mathbf{k}(z) \otimes L_z \otimes X + \hat{L}_z \otimes N}{ \mathbf{k}(z) \otimes L_z \otimes X + f(z) \mathbf{k}(z) \otimes L_z \otimes N } & \simeq \frac{\hat{L}_z \otimes N}{\mathbf{k}(z) \otimes L_z \otimes N + f(z) \mathbf{k}(z) \otimes L_z \otimes N }\\ & \simeq \mathcal{P} \otimes_{\mathbf{k}[z^{\pm1}]} L_z \otimes N, \end{align*}

we have the homomorphism of $\mathbf {k}[z^{\pm 1}] \mathop \otimes U'_q({\mathfrak {g}})$-modules

\[ \mathcal{R} \colon M \otimes L_z \longrightarrow \mathcal{P} \otimes_{\mathbf{k}[z^{\pm1}]} L_z \otimes N. \]

Let us show that $\mathcal {R}$ vanishes.

Assume that $\mathcal {R}\not =0$. Then

\[ \operatorname{Hom}_{\mathbf{k}[z^{\pm1}]\mathop \otimes U'_q({\mathfrak{g}})}(M\mathop\otimes L_z,\mathcal{P} \otimes_{\mathbf{k}[z^{\pm1}]} L_z \otimes N) \simeq \mathcal{P} \otimes_{\mathbf{k}[z^{\pm1}]} \operatorname{Hom}_{\mathbf{k}[z^{\pm1}]\mathop \otimes U'_q({\mathfrak{g}})} (M\mathop\otimes L_z,L_z \otimes N) \]

implies that $\operatorname {Hom}_{\mathbf {k}[z^{\pm 1}]\mathop \otimes U'_q({\mathfrak {g}})}(M\mathop \otimes L_z,L_z \otimes N)\not \simeq 0$.

Since $\mathbf {k}(z)\mathop \otimes _{\mathbf {k}[z^{\pm 1}]}(M\mathop \otimes L_z)$ and $\mathbf {k}(z)\mathop \otimes _{\mathbf {k}[z^{\pm 1}]}(L_z \otimes N)$ are simple $\mathbf {k}(z)\mathop \otimes U'_q({\mathfrak {g}})$-modules, they are isomorphic. Since $\mathbf {k}(z)\mathop \otimes _{\mathbf {k}[z^{\pm 1}]}(L_z \otimes N)$ and $\mathbf {k}(z)\mathop \otimes _{\mathbf {k}[z^{\pm 1}]}(N\mathop \otimes L_z)$ are isomorphic, we conclude that $\mathbf {k}(z)\mathop \otimes _{\mathbf {k}[z^{\pm 1}]}(M\mathop \otimes L_z)\simeq \mathbf {k}(z)\mathop \otimes _{\mathbf {k}[z^{\pm 1}]}(N\mathop \otimes L_z)$. On the other hand, Proposition 5.8 implies that

\[ \mathbf{k}(z) \otimes \operatorname{Hom}_{U_q'({\mathfrak{g}})}(M,N) \mathop{\xrightarrow{\sim }} \operatorname{Hom}_{\mathbf{k}(z)\otimes U_q'({\mathfrak{g}})}(\mathbf{k}(z) \otimes_{\mathbf{k}[z^{\pm1}]}M\otimes L_z, \mathbf{k}(z) \otimes_{\mathbf{k}[z^{\pm1}]} N \otimes L_z). \]

Hence $\operatorname {Hom}_{U_q'({\mathfrak {g}})}(M,N) \not =0$, and we obtain that $M$ and $N$ are isomorphic, which is a contradiction. Therefore $\mathcal {R}=0$, which means that

\[ R\bigl(\mathbf{k}(z)\mathop\otimes (X\mathop\otimes L_z)\bigr)\subset \mathbf{k}(z) \otimes L_z \otimes X + f(z) \mathbf{k}(z) \otimes L_z \otimes N. \]

Let us consider the composition

We have

\[ R\bigl(\mathbf{k}(z)\mathop\otimes (X\mathop\otimes L_z)\bigr)\cap \hat{L}_z \mathop\otimes N =R\bigl( \mathbf{k}(z)\mathop\otimes (N\mathop\otimes L_z)\bigr)=f(z)\mathbf{k}(z)\mathop\otimes L_z\mathop\otimes N. \]

Hence $\ker (\Phi )=K\cap \bigl (\mathbf {k}(z) \otimes L_z \otimes N\bigr ) =\bigl ( f(z)\mathbf {k}(z) \otimes L_z \otimes N)\cap \bigl (\mathbf {k}(z) \otimes L_z \otimes N\bigr )$ vanishes, which means that $\Phi$ is a monomorphism.

Since $\mathbf {k}(z) \otimes L_z\mathop \otimes M$ and $\mathbf {k}(z) \otimes L_z\mathop \otimes N$ are simple $\mathbf {k}(z)\mathop \otimes U'_q({\mathfrak {g}})$-modules, $\mathbf {k}(z) \otimes L_z \otimes X$ has length $2$. Similarly, $R\bigl (\mathbf {k}(z)\mathop \otimes (X\mathop \otimes L_z)\bigr )$ also has length $2$. On the other hand, $\mathbf {k}(z) \otimes L_z \otimes X + f(z) \mathbf {k}(z) \otimes L_z \otimes N$ has length no more than $3$, which implies that $K$ does not vanish. Hence $\Phi$ is an isomorphism. Thus we conclude that the homomorphism

\begin{align*} \operatorname{Hom}(\mathbf{k}(z)\mathop\otimes L_z\mathop\otimes M,\mathbf{k}(z)\mathop\otimes L_z\mathop\otimes X)\to& \operatorname{Hom}(\mathbf{k}(z)\mathop\otimes L_z\mathop\otimes M,\mathbf{k}(z)\mathop\otimes L_z\mathop\otimes M)\\ &=\mathbf{k}(z)\operatorname{id}_{\mathbf{k}(z)\mathop\otimes L_z\mathop\otimes M} \end{align*}

is surjective. Then Proposition 5.8 implies that this homomorphism is isomorphic to

\[ \mathbf{k}(z)\mathop\otimes \operatorname{Hom}(M,X)\twoheadrightarrow \mathbf{k}(z)\mathop\otimes \operatorname{Hom}(M,M). \]

Thus we conclude that $\operatorname {Hom}(M,X)\to \operatorname {Hom}(M,M)$ is surjective, that is,

\[ 0\xrightarrow{\,{\hspace{2ex}}\,} N\xrightarrow{\,{\hspace{2ex}}\,} X\xrightarrow{\,{\hspace{2ex}}\,} M\xrightarrow{\,{\hspace{2ex}}\,}0 \]

splits.

For $\alpha \in \mathcal {W}$, let $\mathscr {C}_{{\mathfrak {g}}, \alpha }$ be the full subcategory of $\mathscr {C}_{\mathfrak {g}}$ consisting of objects $X$ such that $\mathsf {E} (S)=\alpha$ for any simple subquotient $S$ of $X$.