Connection to PBW Bases of Nilpotent Subalgebra of $U_q$

Abstract

For a finite-dimensional simple Lie algebra $\mathfrak {g}$, let $U^+_q(\mathfrak {g})$ be the positive part of the quantized universal enveloping algebra $U_q(\mathfrak {g})$ with respect to the triangular decomposition. It has the Poincaré–Birkhoff–Witt (PBW) base labeled with the longest element of the Weyl group W of $\mathfrak {g}$. Let $A_q(\mathfrak {g})$ be the quantized coordinate ring of $\mathfrak {g}$. In this chapter, the intertwiner of the irreducible $A_q(\mathfrak {g})$ modules labeled with two different reduced expressions of W is identified with the transition matrix of the corresponding PBW bases of $U^+_q(\mathfrak {g})$. It leads to an alternative proof of the tetrahedron and 3D reflection equations within $U^+_q(\mathfrak {g})$. The boundary vectors in Sects. 3.6.1, 5.8.1 and 8.6.1 give rise to invariants of an anti-algebra involution in $U^+_q(\mathfrak {g})$ in an infinite product form.

10.1 Quantized Universal Enveloping Algebra $U_q(\mathfrak {g})$

10.1.1 Definition

In this chapter $\mathfrak {g}$ stands for a finite-dimensional simple Lie algebra. Its simple roots, simple coroots, fundamental weights are denoted by $\{\alpha _i\}_{i\in I}$,$\{h_i\}_{i\in I}$, $\{\varpi _i\}_{i\in I}$, where I is the index set of the Dynkin diagram of $\mathfrak {g}$. The weight lattice is $P = \oplus _{i \in I} {\mathbb Z}\varpi _i$ and the Cartan matrix $(a_{ij})_{i,j\in I}$ is given by $a_{ij}=\langle {h_i,\alpha _j}\rangle =2(\alpha _i, \alpha _j)/(\alpha _i,\alpha _i)$.

The quantized universal enveloping algebra $U_q(\mathfrak {g})$ is an associative algebra over ${\mathbb Q}(q)$ generated by $\{e_i, f_i, k_i^{\pm 1} \mid i\in I\}$ satisfying the relations:

$$\begin{aligned}&k_i k_j=k_j k_i, \quad k_i k_i^{-1}=k_i^{-1}k_i=1, \nonumber \\&k_i e_j k_i^{-1}=q_i^{\langle h_i,\alpha _j \rangle } e_j, \quad k_i f_j k_i^{-1}=q_i^{-\langle h_i, \alpha _j \rangle } f_j, \quad [e_i, f_j]=\delta _{ij}\frac{k_i-k_i^{-1}}{q_i-q_i^{-1}}, \nonumber \\&\sum _{r=0}^{1-a_{ij}} (-1)^r e_i^{(r)} e_j e_i^{(1-a_{ij}-r)} =\sum _{r=0}^{1-a_{ij}} (-1)^r f_i^{(r)} f_j f_i^{(1-a_{ij}-r)}=0\quad (i\ne j). \end{aligned}$$

(10.1)

Here we use the following notations: $q_i=q^{(\alpha _i,\alpha _i)/2},[m]_i=(q_i^m-q_i^{-m})/(q_i-q_i^{-1})$, $[n]_i!=\prod _{m=1}^n[m]_i$, $e_i^{(n)}=e_i^n/[n]_i!,f_i^{(n)}=f_i^n/[n]_i!$. We normalize the simple roots so that $q_i=q$ when $\alpha _i$ is a short root. The relation (10.1) is called q-Serre relation. The algebra $U_q(\mathfrak {g})$ is a Hopf algebra. For the comultiplication (or coproduct), we adopt the following¹:

$$\begin{aligned} \Delta (k_i)=k_i\otimes k_i,\quad \Delta (e_i)=e_i\otimes 1+k_i\otimes e_i,\quad \Delta (f_i)=f_i\otimes k_i^{-1}+1\otimes f_i. \end{aligned}$$

(10.2)

10.1.2 PBW Basis

Let W be the Weyl group of $\mathfrak {g}$. It is generated by simple reflections $\{s_i\mid i\in I\}$ obeying the relations: $s_i^2=1$, $(s_is_j)^{m_{ij}}=1$ ($i\ne j$), where $m_{ij}=2,3,4,6$ for $\langle {h_i,\alpha _j}\rangle \langle {h_j,\alpha _i}\rangle =0,1,2,3$, respectively. Let $w_0$ be the longest element of W and fix a reduced expression $w_0=s_{i_1}s_{i_2}\cdots s_{i_l}$. Then every positive root occurs exactly once in

$$\begin{aligned} \beta _1=\alpha _{i_1},\,\beta _2=s_{i_1}(\alpha _{i_2}),\ldots ,\, \beta _l=s_{i_1}s_{i_2}\cdots s_{i_{l-1}}(\alpha _{i_l}). \end{aligned}$$

(10.3)

Correspondingly, define elements $e_{\beta _r}\in U_q(\mathfrak {g})$ ($r=1,\ldots ,l$) by

$$\begin{aligned} e_{\beta _r}=T_{i_1}T_{i_2}\cdots T_{i_{r-1}}(e_{i_r}). \end{aligned}$$

(10.4)

Here $T_i$ is the action of the braid group on $U_q(\mathfrak {g})$. It is an algebra automorphism and is given on the generators $\{e_j\}$ by

$$\begin{aligned} T_i(e_i) = -k_if_i, \quad \; T_i(e_j)=\sum _{r=0}^{-a_{ij}}(-1)^r q_i^r e_i^{(r)}e_je_i^{(-a_{ij}-r)} \;\;\;(i\ne j). \end{aligned}$$

(10.5)

Let $U_q^+(\mathfrak {g})$ be a subalgebra of $U_q(\mathfrak {g})$ generated by $\{e_i\mid i\in I\}$. The only relation among them is the q-Serre relation (10.1) for $e_i$’s. It is known that $e_{\beta _r} \in U^+_q(\mathfrak {g})$ holds for any r. $U^+_q(\mathfrak {g})$ has the PBW basis. It depends on the reduced expression $s_{i_1}s_{i_2}\cdots s_{i_l}$ of $w_0$. Set $\textbf{i}=(i_1,i_2,\ldots ,i_l)$ and define for $A=(a_1,a_2,\ldots ,a_l)\in ({\mathbb Z}_{\ge 0})^l$

$$\begin{aligned} E_\textbf{i}^A=e_{\beta _1}^{(a_1)}e_{\beta _2}^{(a_2)}\cdots e_{\beta _l}^{(a_l)}. \end{aligned}$$

(10.6)

Then $\{E_\textbf{i}^A\mid A\in ({\mathbb Z}_{\ge 0})^l\}$ forms a basis of $U_q^+(\mathfrak {g})$. We warn that the notations $e_{i_r}$ with $i_r \in I$ and $e_{\beta _r}$ with a positive root $\beta _r$ should be distinguished properly from the context. In particular $e^{(a_r)}_{\beta _r} = (e_{\beta _r})^{a_r}/\prod _{m=1}^{a_r}\frac{p_r^m-p^{-m}_r}{p_r-p_r^{-1}}$ with $p_r = q^{(\beta _r, \beta _r)/2}$.

10.2 Quantized Coordinate Ring $A_q(\mathfrak {g})$

10.2.1 Definition

Let us give the definition of the quantized coordinate ring $A_q(\mathfrak {g})$.² The relation to the concrete realization by generators and relations in earlier chapters will be explained later.

Let $O_\textrm{int}(\mathfrak {g})$ be the category of integrable left $U_q(\mathfrak {g})$ modules M such that, for any $v\in M$, there exists $l\ge 0$ satisfying $e_{i_1}\cdots e_{i_l}v=0$ for any $i_1,\ldots ,i_l\in I$. Then $O_\textrm{int}(\mathfrak {g})$ is semisimple and any simple object is isomorphic to the irreducible module $V(\lambda )$ with dominant integral highest weight $\lambda$. Similarly, we can consider the category $O_\textrm{int}(\mathfrak {g}^\textrm{op})$ of integrable right $U_q(\mathfrak {g})$ modules $M^r$ such that, for any $u\in M^r$, there exists $l\ge 0$ satisfying $uf_{i_1}\cdots f_{i_l}=0$ for any $i_1,\ldots ,i_l\in I$. The superscript $\textrm{op}$ signifies “opposite”. $O_\textrm{int}(\mathfrak {g}^\textrm{op})$ is also semisimple and any simple object is isomorphic to the irreducible module $V^r(\lambda )$ with dominant integral highest weight $\lambda$. Let $v_\lambda$ (resp. $u_\lambda$) be a highest weight vector of $V(\lambda )$ (resp. $V^r(\lambda )$). Then there exists a unique bilinear form $(\,,\,)$

$$ V^r(\lambda )\otimes V(\lambda )\rightarrow {\mathbb Q}(q) $$

satisfying

$$\begin{aligned} (u_\lambda ,v_\lambda )&=1\quad \text { and }\\ (ug,v)&=(u,gv)\quad \text { for }u\in V^r(\lambda ),\, v\in V(\lambda ),\, g\in U_q(\mathfrak {g}). \end{aligned}$$

Let $U_q(\mathfrak {g})^*$ be $\textrm{Hom}_{{\mathbb Q}(q)}(U_q(\mathfrak {g}),{\mathbb Q}(q))$ and $\langle {\,,\,}\rangle$ be the canonical pairing between $U_q(\mathfrak {g})^*$ and $U_q(\mathfrak {g})$. The comultiplication $\Delta$ of $U_q(\mathfrak {g})$ induces a multiplication of $U_q(\mathfrak {g})^*$ by

$$\begin{aligned} \langle {\varphi \varphi ',g}\rangle =\langle {\varphi \otimes \varphi ',\Delta (g)}\rangle \qquad \text {for }g\in U_q(\mathfrak {g}), \end{aligned}$$

(10.7)

thereby giving $U_q(\mathfrak {g})^*$ the structure of ${\mathbb Q}(q)$-algebra. It also has a $U_q(\mathfrak {g})$ bimodule structure by

$$\begin{aligned} \langle {x\varphi y,g}\rangle =\langle {\varphi ,ygx}\rangle \qquad \text {for }x,y,g\in U_q(\mathfrak {g}). \end{aligned}$$

(10.8)

We define the subalgebra $A_q(\mathfrak {g})$ of $U_q(\mathfrak {g})^*$ by

$$ A_q(\mathfrak {g})=\{\varphi \in U_q(\mathfrak {g})^*;U_q(\mathfrak {g})\varphi \text { belongs to }O_\textrm{int}(\mathfrak {g})\text { and } \varphi U_q(\mathfrak {g})\text { belongs to }O_\textrm{int}(\mathfrak {g}^\textrm{op})\}, $$

and call it the quantized coordinate ring.

The following theorem is the q-analogue of the Peter–Weyl theorem.

Theorem 10.1

As a $U_q(\mathfrak {g})$ bimodule, $A_q(\mathfrak {g})$ is isomorphic to $\bigoplus _\lambda V^r(\lambda )\otimes V(\lambda )$, where $\lambda$ runs over all dominant integral weights, by the homomorphisms

$$ \Psi _\lambda :V^r(\lambda )\otimes V(\lambda )\rightarrow A_q(\mathfrak {g}) $$

given by

$$ \langle \Psi _\lambda (u\otimes v),g\rangle = (u,gv) $$

for $u\in V^r(\lambda ),v\in V(\lambda )$, and $g\in U_q(\mathfrak {g})$.³

In our case of a finite-dimensional simple Lie algebra $\mathfrak {g}$, $A_q(\mathfrak {g})$ turns out to be a Hopf algebra. See for example [66, Chap. 9]. Its comultiplication is also denoted by $\Delta$.

Let $\mathcal {R}$ be the universal R matrix for $U_q(\mathfrak {g})$. For its explicit formula see [29, p. 273] for example. For our purpose it is enough to know that

$$\begin{aligned} \mathcal {R}\in q^{(\textrm{wt}\,\cdot ,\textrm{wt}\,\cdot )} \bigoplus _{\beta \in Q^+}(U_q^+)_\beta \otimes (U_q^-)_{-\beta }, \end{aligned}$$

(10.9)

where $q^{(\textrm{wt}\,\cdot ,\textrm{wt}\,\cdot )}$ is an operator acting on the tenor product $v_\lambda \otimes v_\mu$ of weight vectors $v_\lambda ,v_\mu$ of weight $\lambda ,\mu$ by $q^{(\textrm{wt}\,\cdot ,\textrm{wt}\,\cdot )}(v_\lambda \otimes v_\mu )=q^{(\lambda ,\mu )} v_\lambda \otimes v_\mu$, $Q_+=\bigoplus _i{\mathbb Z}_{\ge 0}\alpha _i$, and $(U_q^\pm )_{\pm \beta }$ is the subspace of $U_q^\pm (\mathfrak {g})$ spanned by root vectors corresponding to $\pm \beta$.

Fix $\lambda$, let $\{u^\lambda _j\}$ and $\{v^\lambda _i\}$ be bases of $V^r(\lambda )$ and $V(\lambda )$ such that $(u^\lambda _i,v^\lambda _j)=\delta _{ij}$. Set

$$\begin{aligned} \varphi ^\lambda _{ij}=\Psi _\lambda (u^\lambda _i \otimes v^\lambda _j) \in A_q(\mathfrak {g}). \end{aligned}$$

(10.10)

Let R be the so-called constant R matrix for $V(\lambda )\otimes V(\mu )$. Denoting the homomorphism $U_q(\mathfrak {g})\rightarrow \textrm{End}(V(\lambda ))$ by $\rho _\lambda$, it is given as

$$\begin{aligned} R\propto (\rho _\lambda \otimes \rho _\mu )(P \mathcal {R}), \end{aligned}$$

(10.11)

where P stands for the exchange of the first and second components. The scalar multiple is determined appropriately depending on $\mathfrak {g}$. The reason we apply P is to fit the so-called RTT relation in (10.15). The dependence of R on $\lambda$ and $\mu$ has been suppressed in the notation. R satisfies

$$\begin{aligned} R\Delta (g)=\Delta ^{\textrm{op}}(g)R\qquad \text {for any }g\in U_q(\mathfrak {g}), \end{aligned}$$

(10.12)

where $\Delta ^{\textrm{op}}=P \circ \Delta \circ P$. Define matrix elements $R^{ij}_{kl}$ by

$$\begin{aligned} R(v^\lambda _k\otimes v^\mu _l)=\sum _{i,j}R^{ij}_{kl}v^\lambda _i\otimes v^\mu _j. \end{aligned}$$

(10.13)

Define the right action of R on $V^r(\lambda )\otimes V^r(\mu )$ in such a way that $((u^\lambda _i\otimes u^\mu _j)R,v^\lambda _k\otimes v^\mu _l)=(u^\lambda _i\otimes u^\mu _j,R(v^\lambda _k\otimes v^\mu _l))$ holds. Then we have

$$\begin{aligned} (u^\lambda _i\otimes u^\mu _j)R=\sum _{k,l}R^{ij}_{kl}u^\lambda _k\otimes u^\mu _l. \end{aligned}$$

(10.14)

Now for any $x \in U_q(\mathfrak {g})$, we have

$$\begin{aligned}&\sum _{m,p}R^{ij}_{mp}\langle {\varphi ^\lambda _{mk}\varphi ^\mu _{pl},x}\rangle = \sum _{m,p}R^{ij}_{mp}\langle {\varphi ^\lambda _{mk}\otimes \varphi ^\mu _{pl},\Delta (x)}\rangle \\&= \sum _{m,p}R^{ij}_{mp}\langle {\Psi _\lambda (u^\lambda _m \otimes v^\lambda _k)\otimes \Psi _\mu (u^\mu _p \otimes v^\mu _l),\Delta (x)}\rangle \\&=\sum _{m,p}R^{ij}_{mp}(u^\lambda _m\otimes u^\mu _p,\Delta (x)(v^\lambda _k\otimes v^\mu _l)) =((u^\lambda _i\otimes u^\mu _j)R,\Delta (x)(v^\lambda _k\otimes v^\mu _l))\\&=(u^\lambda _i\otimes u^\mu _j,R\Delta (x)(v^\lambda _k\otimes v^\mu _l)) =(u^\lambda _i\otimes u^\mu _j,\Delta ^{\textrm{op}}(x)R(v^\lambda _k\otimes v^\mu _l))\\&=\sum _{m,p}(u^\lambda _i\otimes u^\mu _j,\Delta ^{\textrm{op}}(x)(v^\lambda _m\otimes v^\mu _p))R^{mp}_{kl} =\sum _{m,p}(u^\mu _j\otimes u^\lambda _i,\Delta (x)(v^\mu _p\otimes v^\lambda _m))R^{mp}_{kl}\\&=\sum _{m,p}\langle {\varphi ^\mu _{jp}\otimes \varphi ^\lambda _{im},\Delta (x)}\rangle R^{mp}_{kl} =\sum _{m,p}\langle {\varphi ^\mu _{jp}\varphi ^\lambda _{im},x}\rangle R^{mp}_{kl}. \end{aligned}$$

Thus we get

$$\begin{aligned} \sum _{m,p}R^{ij}_{mp}\varphi ^\lambda _{mk}\varphi ^\mu _{pl} =\sum _{m,p}\varphi ^\mu _{jp}\varphi ^\lambda _{im}R^{mp}_{kl} \in A_q(\mathfrak {g}). \end{aligned}$$

(10.15)

We call such a relation an RTT relation. It forms a large family containing conventional ones as the special case where $\lambda = \mu = \varpi _r$ for some specific fundamental weight $\varpi _r$.

Example 10.2

Consider the simplest case $\mathfrak {g}=A_1$ with $\lambda = \mu =\varpi _1$. We write $u^{\varpi _1}_i, v^{\varpi _1}_i$ simply as $u_i, v_i\, (i = 1,2)$. The $U_q(sl_2)$ module structure is

$$\begin{aligned}&f_1v_1 = v_2,\;f_1v_2=0, \; e_1v_1 =0, \; e_1v_2 = v_1, \; k_1 v_1=qv_1, \; k_1 v_2 = q^{-1}v_2, \end{aligned}$$

(10.16)

$$\begin{aligned}&u_1f_1 =0, \; u_2f_1 = u_1, \; u_1e_1 = u_2, \; u_2 e_1=0, \; u_1k_1 = q u_1,\; u_2k_1 = q^{-1}u_2. \end{aligned}$$

(10.17)

The R matrix (3.3) acts as

$$\begin{aligned} R(v_1 \otimes v_1)&= qv_1 \otimes v_1, \quad \; R(v_1 \otimes v_2) = v_1 \otimes v_2 + (q-q^{-1})v_2 \otimes v_1, \end{aligned}$$

(10.18)

$$\begin{aligned} R(v_2 \otimes v_1)&= v_2 \otimes v_1, \qquad R(v_2 \otimes v_2) = qv_2 \otimes v_2, \end{aligned}$$

(10.19)

$$\begin{aligned} (u_1\otimes u_1)R&= q u_1 \otimes u_1, \quad \;\; (u_2 \otimes u_1)R = u_2 \otimes u_1 + (q-q^{-1}) u_1 \otimes u_2, \end{aligned}$$

(10.20)

$$\begin{aligned} (u_1\otimes u_2)R&= u_1 \otimes u_2, \qquad \; (u_2\otimes u_2)R = q u_2 \otimes u_2. \end{aligned}$$

(10.21)

Set $t_{ij} = \Psi _{\omega _1}(u_i \otimes v_j) \in A_q(A_1)$. Then we have

$$\begin{aligned} \begin{aligned} \langle t_{11}t_{22},x\rangle&= \langle \Psi _{\omega _1}(u_1 \otimes v_1) \otimes \Psi _{\omega _1}(u_2 \otimes v_2) , \Delta (x) \rangle = (u_1 \otimes u_2, \Delta (x)(v_1 \otimes v_2)) \\&= ( (u_1 \otimes u_2)R, \Delta (x)(v_1 \otimes v_2)) = ( u_1 \otimes u_2, \Delta ^{\textrm{op}}(x)R(v_1 \otimes v_2)) \\&=( u_1 \otimes u_2, \Delta ^{\textrm{op}}(x)(v_1 \otimes v_2 + (q-q^{-1})v_2 \otimes v_1)) \\&= ( u_2 \otimes u_1, \Delta (x)(v_2 \otimes v_1 + (q-q^{-1})v_1 \otimes v_2)) \\&= \langle \Psi _{\omega _1}(u_2\otimes v_2) \otimes \Psi _{\omega _1}(u_1\otimes v_1) \\&\qquad + (q-q^{-1}) \Psi _{\omega _1}(u_2\otimes v_1) \otimes \Psi _{\omega _1}(u_1\otimes v_2), \Delta (x)\rangle \\&= \langle t_{22} \otimes t_{11} + (q-q^{-1}) t_{21} \otimes t_{12}, \Delta (x)\rangle \\&= \langle t_{22} t_{11} + (q-q^{-1}) t_{21} t_{12}, x\rangle , \end{aligned} \end{aligned}$$

which reproduces the relation $[t_{11}, t_{22}] = (q-q^{-1}) t_{21}t_{12}$ in (3.9). Similarly, we have

$$\begin{aligned} \langle t_{11}t_{22} - q t_{12}t_{21}, x\rangle&= \langle t_{11} \otimes t_{22} - q t_{12} \otimes t_{21}, \Delta (x)\rangle \\&= (u_1\otimes u_2, \Delta (x)(v_1\otimes v_2)) - q(u_1\otimes u_2, \Delta (x)(v_2\otimes v_1)) \\&= (u_1 \otimes u_2, \Delta (x)(v_1\otimes v_2-qv_2\otimes v_1)). \end{aligned}$$

Suppose $x = e^l_1k^m_1f^n_1 \in U_q(sl_2)\, (l,m,n \in {\mathbb Z}_{\ge 0})$ without loss of generality. Since $v^0_1:=v_1\otimes v_2-qv_2\otimes v_1$ is a $U_q(sl_2)$-singlet annihilated either by $\Delta (e_1)$ and $\Delta (f_1)$, one has $\Delta (x) v^0_1 = \delta _{l0}\delta _{n0}v^0_1$. Thus the RHS of the above calculation is equal to $\delta _{l0}\delta _{n0}(u_1 \otimes u_2, v^0_1) = \delta _{l0}\delta _{n0} = \langle 1, x\rangle$. This yields $t_{11}t_{22} - q t_{12}t_{21} = 1$ in (3.9).

Let us mention the relation to the formulation of $A_q(\mathfrak {g})$ in earlier chapters using specific generators and relations. Suppose $\varpi _l$ is a fundamental weight such that any $V(\lambda )$ is included in the tensor power $V(\varpi _l)^{\otimes m}$ for some m.⁴ Denoting the base of $V^r(\varpi _l)$ and $V(\varpi _l)$ by $u_i$ and $v_i$, set

$$\begin{aligned} t_{ij} = \Psi _{\varpi _l}(u_i \otimes v_j) \in A_q(\mathfrak {g}). \end{aligned}$$

(10.22)

We know that $t_{ij}$ satisfies the RTT relation (10.15) whose structure constant is the constant R matrix for $\lambda = \mu = \varpi _l$. Any vectors $u \in V^r(\lambda )$ and $v \in V(\lambda )$ are expressed as linear combinations $u=\sum C_{i_1,\ldots , i_m} u_{i_1} \otimes \cdots \otimes u_{im}$ and $v = \sum D_{j_1, \ldots , j_m}v_{j_1} \otimes \cdots \otimes v_{j_m}$. Theorem 10.1 shows that an arbitrary element of $A_q(\mathfrak {g})$ is constructed as $\Psi _\lambda (u \otimes v)$. A calculation similar to Example 10.2 leads to $\Psi _\lambda (u \otimes v) = \sum C_{i_1,\ldots , i_m} D_{j_1, \ldots , j_m} t_{i_1 j_1} \cdots t_{i_m j_m}$, which says that $t_{ij}$’s are certainly generators. They satisfy RTT and additional relations reflecting a fine structure of the Grothendieck ring of $\mathfrak {g}$ like $V(\varpi _l)^{\otimes m} \supset V(0)$ and $V(\varpi _l)^{\otimes m} \supset V(\varpi _l)$, etc. Our individual treatment in the earlier chapters corresponds to the choice $l=1$ for $A_{n-1}, C_n, G_2$ and $l=n$ for $B_n$.⁵

10.2.2 Right Quotient Ring $A_q(\mathfrak {g})_{\mathcal {S}}$

Here we prepare the necessary ingredients for the proof of Theorem 10.6. The point is to assure the well definedness of the division in (10.39).

Recall that $w_0 \in W$ is the longest element of the Weyl group. For any $l \in I$, let $v_{w_0\varpi _l} \in V(\varpi _l)$ be a lowest weight vector. Similarly, let $u_{\varpi _l} \in V^r(\varpi _l)$ be a highest weight vector. The following element will play a key role:

$$\begin{aligned} \sigma _l=\Psi _{\varpi _l}(u_{\varpi _l}\otimes v_{w_0\varpi _l}) \in A_q(\mathfrak {g}). \end{aligned}$$

(10.23)

Example 10.3

For $\mathfrak {g}= A_1$ treated in Example 10.2, one has $\sigma _1 = \Psi _{\omega _1}(u_1 \otimes v_2) = t_{12}$.

Proposition 10.4

The commutativity $\sigma _r\sigma _s = \sigma _s\sigma _r$ holds for any $r,s \in I$.

Proof

From (10.9) and (10.11) we have

$$\begin{aligned} (u_{\varpi _r}\otimes u_{\varpi _s})R&=q^{(\varpi _r,\varpi _s)}u_{\varpi _r}\otimes u_{\varpi _s}, \end{aligned}$$

(10.24)

$$\begin{aligned} R(v_{w_0\varpi _r}\otimes v_{w_0\varpi _s})&= q^{(\varpi _r,\varpi _s)}v_{w_0\varpi _r}\otimes v_{w_0\varpi _s}, \end{aligned}$$

(10.25)

where $(w_0\varpi _r, w_0\varpi _s) = (\varpi _r,\varpi _s)$ has been used. Consider the RTT relation (10.15) with $\lambda = \varpi _r$, $\mu = \varpi _s$, and take the indices i, j, k, l so as to specify the following bases:

$$\begin{aligned} u^\lambda _i = u_{\varpi _r},\;\;u^\mu _j = u_{\varpi _s},\;\; v^\lambda _k = v_{w_0\varpi _r},\;\;v^\mu _l = v_{w_0\varpi _s}. \end{aligned}$$

(10.26)

Then (10.24) and (10.25) indicate $R^{ij}_{mp} = q^{(\varpi _r,\varpi _s)}\delta ^i_m\delta ^i_p$ and $R^{mp}_{kl} = q^{(\varpi _r,\varpi _s)}\delta ^m_k\delta ^p_l$. Thus the RTT relation (10.15) reduces to

$$\begin{aligned} \varphi ^{\varpi _r}_{ik}\varphi ^{\varpi _s}_{jl} = \varphi ^{\varpi _s}_{jl}\varphi ^{\varpi _r}_{ik}. \end{aligned}$$

(10.27)

The proof is finished by noting $\varphi ^{\varpi _r}_{ik} = \sigma _r$ and $\varphi ^{\varpi _s}_{jl} = \sigma _s$ by comparing (10.10) and (10.23). $\square$

Since $A_q(\mathfrak {g})$ is a right $U_q(\mathfrak {g})$ module, we have an element $\sigma _i e_i \in A_q(\mathfrak {g})$. Later in Sect. 10.3.2, we will need the division $(\sigma _i e_i)/\sigma _i$ for $i \in I$. The following localization is known to be possible making sense of it.

Theorem 10.5

Let n be the rank of $\mathfrak {g}$. For the multiplicatively closed subset $\mathcal {S}=\{\sigma _1^{m_1}\cdots \sigma _n^{m_n}\mid m_1,\ldots ,m_n\in {\mathbb Z}_{\ge 0}\} \subset A_q(\mathfrak {g})$, the right quotient ring $A_q(\mathfrak {g})_\mathcal {S}$ exists.

Elements of $A_q(\mathfrak {g})_\mathcal {S}$ are expressed in the form r/s with $r \in A_q(\mathfrak {g})$ and $s \in \mathcal {S}$. Theorem 10.5 guarantees the well-defined ring structure, namely, the addition and the multiplication of $r_1/s_1$ and $r_2/s_2$ in $A_q(\mathfrak {g})_\mathcal {S}$ as

$$\begin{aligned} r_1/s_1+r_2/s_2=(r_1u+r_2u')/(s_1u),\qquad (r_1/s_1)(r_2/s_2)=(r_1v')/(s_2v), \end{aligned}$$

(10.28)

where $u,u',v,v'$ are so chosen that $s_1u=s_2u'$ ($u\in \mathcal {S},u'\in A_q(\mathfrak {g})$), $r_2v=s_1v'$ ($v\in \mathcal {S},v'\in A_q(\mathfrak {g})$).

10.3 Main Theorem

In this section we fix two reduced words $\textbf{i}=(i_1,\ldots ,i_l),\, \textbf{j}=(j_1,\ldots ,j_l)$ of the longest element $w_0 = s_{i_1}\cdots s_{i_l} = s_{j_1}\cdots s_{j_l} \in W$.

10.3.1 Definitions of $\gamma ^A_B$ and $\Phi ^A_B$

In the $U_q(\mathfrak {g})$ side, we defined the PBW bases $E_\textbf{i}^A,E_\textbf{j}^B$ of $U_q^+(\mathfrak {g})$ in Sect. 10.1.2. We define their transition coefficient $\gamma ^A_B$ by

$$\begin{aligned} E_\textbf{i}^A=\sum _B \gamma ^A_B E_\textbf{j}^B. \end{aligned}$$

In the $A_q(\mathfrak {g})$ side, we have the intertwiner $\Phi : \mathcal {F}_{q_{i_1}}\otimes \cdots \otimes \mathcal {F}_{q_{i_l}} \rightarrow \mathcal {F}_{q_{j_1}}\otimes \cdots \otimes \mathcal {F}_{q_{j_l}}$ satisfying

$$\begin{aligned} \pi _\textbf{j}(g)\circ \Phi = \Phi \circ \pi _\textbf{i}(g)\qquad (\forall g \in A_q(\mathfrak {g})). \end{aligned}$$

(10.29)

We take the parameters $\mu _i$ as in (3.21) and (5.19) to be 1. The intertwiner $\Phi$ is normalized by $\Phi (|0\rangle \otimes \cdots \otimes |0\rangle ) = |0\rangle \otimes \cdots \otimes |0\rangle$. Under these conditions a matrix element $\Phi ^A_B$ of $\Phi$ is uniquely specified by

$$\begin{aligned} \Phi |B\rangle =\sum _A \Phi ^A_B |A\rangle , \end{aligned}$$

where $A=(a_1, \ldots , a_l) \in ({\mathbb Z}_{\ge 0})^l$ and $|A\rangle = |a_1\rangle \otimes \cdots \otimes |a_l\rangle \in \mathcal {F}_{q_{j_1}}\otimes \cdots \otimes \mathcal {F}_{q_{j_l}}$ and similarly for $|B\rangle \in \mathcal {F}_{q_{i_1}}\otimes \cdots \otimes \mathcal {F}_{q_{i_l}}$. The main result of this chapter is

Theorem 10.6

$$ \gamma ^A_B=\Phi ^A_B. $$

For any pair $(\textbf{i},\textbf{j})$, from $\textbf{i}$ one can reach $\textbf{j}$ by applying Coxeter relations (for indices of the simple reflections). In view of the uniqueness of $\gamma$ and $\Phi$ and the fact that the braid group action $T_i$ is an algebra homomorphism, the proof of this theorem reduces to establishing the same equality for the rank 2 case $\mathfrak {g}=A_2, C_2$ and $G_2$.⁶ This will be done in the sequel.

10.3.2 Proof of Theorem 10.6 for Rank 2 Cases

In the rank 2 cases, there are two reduced expressions $s_{i_1}\cdots s_{i_l}$ for the longest element of the Weyl group. Denote the associated sequences $\textbf{i} = (i_1,\ldots , i_l)$ by $\textbf{1}, \textbf{2}$ and set $\textbf{1}' = \textbf{2}, \textbf{2}' = \textbf{1}$. Concretely, we take them as

$$\begin{aligned} A_2: \;&\textbf{1}=(1,2,1),&\;\;&\textbf{2}=(2,1,2),&(q_1,q_2)=(q,q), \end{aligned}$$

(10.30)

$$\begin{aligned} C_2: \;&\textbf{1}=(1,2,1,2),&\;\;&\textbf{2}=(2,1,2,1),&(q_1,q_2)=(q,q^2), \end{aligned}$$

(10.31)

$$\begin{aligned} G_2: \;&\textbf{1}=(1,2,1,2,1,2),\;\;&\;\;&\textbf{2}=(2,1,2,1,2,1), \quad&(q_1,q_2)=(q,q^3), \end{aligned}$$

(10.32)

where $q_i$ defined after (10.1) is also recalled. In order to simplify the formulas in Sect. 10.4, we use the PBW bases and the Fock states in yet another normalization as follows:

$$\begin{aligned} {\tilde{E}}^A_\textbf{i}&:= ([a_1]_{i_1}!\cdots [a_l]_{i_l}!) E^A_\textbf{i} = e^{a_1}_{\beta _1}\cdots e^{a_l}_{\beta _l}, \end{aligned}$$

(10.33)

$$\begin{aligned} |A\rangle \!\rangle&:= d_{i_1,a_1}\cdots d_{i_l,a_l}|A\rangle ,\quad d_{i,a} = q^{-a(a-1)/2}_i\lambda ^a_i,\quad \lambda _i = (1-q_i^2)^{-1}, \end{aligned}$$

(10.34)

where $A=(a_1,\ldots , a_l)$. See after (10.1) for the symbol $[a]_i!$. The root vector $e_{\beta _r}$ is defined in (10.4). Accordingly, we introduce the matrix elements ${\tilde{\gamma }}^A_B$ and ${\tilde{\Phi }}^A_B$ by

$$\begin{aligned} {\tilde{E}}^A_\textbf{i} =\sum _B {\tilde{\gamma }}^A_B {\tilde{E}}^B_{\textbf{i}'},\quad \Phi |B\rangle \!\rangle =\sum _A {\tilde{\Phi }}^A_B |A\rangle \!\rangle , \quad (\textbf{i}=\textbf{1}, \textbf{2}). \end{aligned}$$

(10.35)

It follows that $\gamma ^A_B = {\tilde{\gamma }}^A_B\prod _{k=1}^l([b_k]_{i_k}!/[a_k]_{i_k}!)$ and $\Phi ^A_B= {\tilde{\Phi }}^A_B\prod _{k=1}^l(d_{i_k,a_k}/d_{i_k,b_k})$ for $B=(b_1,\ldots , b_l)$. On the other hand, we know $\Phi ^A_B=\Phi ^B_A\prod _{k=1}^l((q_{i_k}^2)_{b_k}/(q_{i_k}^2)_{a_k})$ from (3.63), (5.75) and (8.30). Due to the identity $(q_i^2)_md_{i,m}=[m]_i!$, the assertion $\gamma ^A_B=\Phi ^A_B$ of Theorem 10.6 is equivalent to

$$\begin{aligned} {\tilde{\gamma }}^A_B={\tilde{\Phi }^B_A}. \end{aligned}$$

(10.36)

Let $\rho _\textbf{i}(x)=(\rho _\textbf{i}(x)_{A B})$ be the matrix for the left multiplication of $x \in U^+_q(\mathfrak {g})$:

$$\begin{aligned} x \cdot {\tilde{E}}^{A}_\textbf{i} = \sum _B {\tilde{E}}^B_\textbf{i} \rho _\textbf{i}(x)_{BA}. \end{aligned}$$

(10.37)

Let further $\pi _\textbf{i}(g)=(\pi _\textbf{i}(g)_{A B})$ be the representation matrix of $g \in A_q(\mathfrak {g})$:

$$\begin{aligned} \pi _\textbf{i}(g) | A\rangle \!\rangle = \sum _B |B\rangle \!\rangle \pi _\textbf{i}(g)_{BA}. \end{aligned}$$

(10.38)

The following element in the right quotient ring $A_q(\mathfrak {g})_\mathcal {S}$ (see Theorem 10.5) will play a key role in our proof:

$$\begin{aligned} \xi _i = \lambda _i (\sigma _i e_i)/\sigma _i\quad (i=1,2). \end{aligned}$$

(10.39)

We recall that the general definition of $\sigma _i$ is (10.23). Its concrete form in the rank 2 case will be given in Lemmas 10.10, 10.12 and 10.14. In Sect. 10.4 we will check the following statement case by case. It says that the “conjugation” of $e_i$ by $\sigma _i$ on $A_q(\mathfrak {g})$ modules $(\sigma _i e_i)/ \sigma _i$ corresponds to $(1-q_i^2)e_i$ in $U_q^+(\mathfrak {g})$.

Proposition 10.7

For $\mathfrak {g}$ of rank 2, $\pi _\textbf{i}(\sigma _i)$ is invertible and the following equality is valid:

$$\begin{aligned} \rho _\textbf{i}(e_i)_{A B} = \pi _\textbf{i}(\xi _i)_{A B} \;\;\;(i=1,2), \end{aligned}$$

(10.40)

where the RHS means $\lambda _i\pi _\textbf{i}(\sigma _ie_i) \pi _\textbf{i}(\sigma _i)^{-1}$.

Proof of Theorem 10.6 for rank 2 case. We write both sides of (10.40) as $M^i_{AB}$ and the term for $\textbf{i}'$ instead of $\textbf{i}$ as $M^{\prime i}_{AB}$. From

$$\begin{aligned} \sum _{B,C} {\tilde{E}}^C_{\textbf{i}'} M^{\prime i}_{CB}{\tilde{\gamma }}^A_B= e_i \sum _B {\tilde{E}}^B_{\textbf{i}'}{\tilde{\gamma }}^A_B = e_i {\tilde{E}}^A_\textbf{i} = \sum _B {\tilde{E}}^{B}_\textbf{i} M^i_{BA} = \sum _{B,C}{\tilde{E}}^C_{\textbf{i}'} {\tilde{\gamma }}^B_C M^i_{BA} \end{aligned}$$

we have $\sum _{B} M^{\prime i}_{CB}{\tilde{\gamma }}^A_B = \sum _{B}{\tilde{\gamma }}^B_CM^i_{BA}$. On the other hand, the actions of the two sides of (10.29) with $g=\xi _i$ and $\textbf{j}=\textbf{i}'$ are calculated as

$$ \pi _{\textbf{i}'}(\xi _i)\circ \Phi |A\rangle \!\rangle =\pi _{\textbf{i}'}(\xi _i)\sum _B|B\rangle \!\rangle {\tilde{\Phi }}^B_A =\sum _{B,C}|C\rangle \!\rangle M_{CB}^{\prime i}{\tilde{\Phi }}_A^B $$

and

$$ \Phi \circ \pi _{\textbf{i}}(\xi _i)|A\rangle \!\rangle =\Phi \sum _B|B\rangle \!\rangle M_{BA}^i =\sum _{B,C}|C\rangle \!\rangle {\tilde{\Phi }}_B^C M_{BA}^i. $$

Hence $\sum _{B} M_{CB}^{\prime i}{\tilde{\Phi }}_A^B = \sum _{B}{\tilde{\Phi }}_B^C M_{BA}^i$. Thus ${\tilde{\gamma }}^A_B$ and ${\tilde{\Phi }}^B_A$ satisfy the same relation. Moreover, the maps $\pi _\textbf{i}$ and $\rho _\textbf{i}$ are both homomorphisms, i.e. $\pi _\textbf{i}(gh)=\pi _\textbf{i}(g)\pi _\textbf{i}(h)$ and $\rho _\textbf{i}(xy)=\rho _\textbf{i}(x)\rho _\textbf{i}(y)$. We know that ${\Phi }$ is the intertwiner of the irreducible $A_q(\mathfrak {g})$ modules and (10.36) obviously holds as $1=1$ at $A=B=(0,\ldots ,0)$. Thus it is valid for arbitrary A and B.  $\square$

Remark 10.8

The equality (10.40) is valid for any $\mathfrak {g}$.

10.4 Proof of Proposition 10.7

Here we present the explicit formulas of (10.37) with $x=e_i$ and (10.38) with $g=\sigma _i, \sigma _ie_i$ that allow one to check Proposition 10.7. In each case, there are two $\textbf{i}$-sequences, $\textbf{1}$ and $\textbf{2}=\textbf{1}'$ corresponding to the two reduced words. Define

$$\begin{aligned} \chi = \text {the anti-algebra involution of }U^+_q(\mathfrak {g})\text { such that }\chi (e_i)=e_i. \end{aligned}$$

(10.41)

Then both $E^A_\textbf{i}$ in (10.6) and ${\tilde{E}}^{A}_\textbf{i}$ in (10.33) satisfy

$$\begin{aligned} \chi (E^{A}_\textbf{i}) = E^{A^\vee }_{\textbf{i}'}, \qquad \chi ({\tilde{E}}^{A}_\textbf{i}) = {\tilde{E}}^{A^\vee }_{\textbf{i}'}, \end{aligned}$$

(10.42)

where $A^\vee = (a_l,\ldots , a_2,a_1)$ denotes the reversal of $A=(a_1,a_2,\ldots , a_l)$. Applying $\chi$ to (10.37) with $x=e_i$ yields the right multiplication formula ${\tilde{E}}^{A^\vee }_{\textbf{i}'}\cdot e_i = \sum _B {\tilde{E}}^{B^\vee }_{\textbf{i}'} \rho _\textbf{i}(e_i)_{BA}$ for the $\textbf{i}'$-sequence. In view of this fact, we shall present the left and right multiplication formulas for $\textbf{i}=\textbf{2}$ only.

As for (10.38) with $g=\xi _i$ in (10.39), explicit formulas for $\sigma _i, \sigma _i e_i \in A_q(\mathfrak {g})$ and their image by both representations $\pi _\textbf{1}$ and $\pi _\textbf{2}$ will be given. We include an exposition on how to use these data to check (10.40) along the simplest $A_2$ case. The $C_2$ and $G_2$ cases are similar.

Following (10.34), we write $|m\rangle \!\rangle :=d_{i,m}|m\rangle \in {\mathcal {F}}_{q_i}$ for each component. From the choice (10.30)–(10.32), the action of the $q_i$-oscillator on ${\mathcal {F}}_{q_i}\,(i=1,2)$ takes the form

$$\begin{aligned} \begin{aligned}&\mathrm{\textbf{a}}^+|m\rangle \!\rangle = \lambda _1^{-1}q_1^m|m+1\rangle \!\rangle ,\quad \, \mathrm{\textbf{a}}^-|m\rangle \!\rangle = [m]_1|m-1\rangle \!\rangle ,\quad \, \textbf{k}|m\rangle \!\rangle =q_1^m|m\rangle \!\rangle , \\&{\textbf{A}}^+ |m\rangle \!\rangle = \lambda _2^{-1}q_2^m|m+1\rangle \!\rangle ,\quad {\textbf{A}}^- |m\rangle \!\rangle = [m]_2|m-1\rangle \!\rangle ,\quad {\textbf{K}}|m\rangle \!\rangle =q_2^m|m\rangle \!\rangle . \end{aligned} \end{aligned}$$

(10.43)

See (10.34) and (3.13). We also use the shorthand

$$\begin{aligned} \langle m \rangle =q^m-q^{-m}. \end{aligned}$$

(10.44)

10.4.1 Explicit Formulas for $A_2$

Consider $\mathfrak {g}= A_2$.

The q-Serre relations are

$$\begin{aligned} e_1^2 e_2-[2]_1e_1 e_2 e_1+e_2 e_1^2=0,\quad e_2^2 e_1-[2]_1e_2 e_1 e_2+e_1 e_2^2=0, \end{aligned}$$

(10.45)

where $[m]_1 = \langle m \rangle /\langle 1 \rangle$. For simplicity we write the positive root vectors $e_{\beta _{i_r}}$ in (10.4) with $(i_1, i_2, i_3) = \textbf{2}$ (10.30) as

$$\begin{aligned} b_1=e_{\beta _1}= e_2,\quad b_2=e_{\beta _2}=e_1 e_2-q e_2 e_1,\quad b_3=e_{\beta _3}=e_1. \end{aligned}$$

(10.46)

The corresponding positive roots are $(\beta _1, \beta _2, \beta _3)=(\alpha _2, \alpha _1+\alpha _2, \alpha _1)$. In particular, $b_2 = T_2(e_1)$. Their commutation relations are

$$\begin{aligned} b_2 b_1 = q^{-1}{b_1 b_2},\quad b_3 b_1 = b_2+q b_1 b_3, \quad b_3 b_2 = q^{-1}{b_2 b_3}. \end{aligned}$$

(10.47)

Lemma 10.9

For ${\tilde{E}}^{a,b,c}_\textbf{2}=b_1^a b_2^b b_3^c$, we have

$$\begin{aligned} {\tilde{E}}^{a,b,c}_\textbf{2} \cdot e_1&={\tilde{E}}^{a,b,c+1}_\textbf{2},\\ {\tilde{E}}^{a,b,c}_\textbf{2} \cdot e_2&=q^{c-b} {\tilde{E}}^{a+1,b,c}_\textbf{2}+[c]_1 {\tilde{E}}^{a,b+1,c-1}_\textbf{2},\\ e_1 \cdot {\tilde{E}}^{a,b,c}_\textbf{2}&=q^{a-b} {\tilde{E}}^{a,b,c+1}_\textbf{2}+[a]_1 {\tilde{E}}^{a-1,b+1,c}_\textbf{2},\\ e_2 \cdot {\tilde{E}}^{a,b,c}_\textbf{2}&={\tilde{E}}^{a+1,b,c}_\textbf{2}. \end{aligned}$$

Proof

By induction, we have

$$\begin{aligned} {b_3 b_1^n} = q^n {b_1^n b_3}+[n]_1 {b_1^{n-1} b_2},&\quad {b_3 b_2^n} = q^{-n} {b_2^n b_3},\\ {b_3^n b_1} = q^n {b_1 b_3^n}+[n]_1 {b_2 b_3^{n-1}},&\quad {b_2^n b_1} = q^{-n} {b_1 b_2^n}. \end{aligned}$$

The lemma is a direct consequence of these formulas. $\square$

Set ${\tilde{E}}^{a,b,c}_\textbf{1} = \chi ({\tilde{E}}^{c,b,a}_\textbf{2}) = \chi (b^a_3)\chi (b^b_2)\chi (b^c_1) =b_3^ab_2^{\prime b}b_1^c$, where $b'_2 := \chi (b_2) = e_2e_1-qe_1e_2$. By applying $\chi$ to the first two relations in Lemma 10.9, we get

$$\begin{aligned} e_1\cdot {\tilde{E}}^{a,b,c}_\textbf{1} = {\tilde{E}}^{a+1,b,c}_\textbf{1}, \qquad e_2\cdot {\tilde{E}}^{a,b,c}_\textbf{1} = q^{a-b}E^{a,b,c+1}_\textbf{1} +[a]_1{\tilde{E}}^{a-1,b+1,c}_\textbf{1}. \end{aligned}$$

(10.48)

Thus we find $\rho _{\textbf{i}'}(e_i) = \rho _\textbf{i}(e_{3-i})$. This property is only valid for $A_2$ and not in $C_2$ and $G_2$.

Let $u_i\,(i=1,2,3)$ be the bases of the right $U_q(A_2)$ module $V^r(\varpi _1)$ such that $u_j = u_1e_1\cdots e_{j-1}e_j$. Similarly, let $v_i\,(i=1,2,3)$ be the bases of the left $U_q(A_2)$ module $V(\varpi _1)$ such that $v_j = f_jf_{j-1}\cdots f_1 v_1$.

$$\begin{aligned}&k_1 \quad&&k_2 \qquad&\!\!\!\!V^r(\varpi _1) &{}\qquad&\!\!\!V(\varpi _1) \\&q&&1&u_1&&v_1\\&{}&{}&&\downarrow& e_1&f_1&\downarrow \\&q^{-1}&&q&u_2&&v_2\\&{}&{}&&\downarrow& e_2&f_2&\downarrow \\&1&&q^{-1}&u_3&&v_3 \end{aligned}$$

The left two columns specify the weights for example as $u_2k_1 = q^{-1}u_2$, $k_1v_1=qv_1$. For the coproduct (10.2), the bases of $V^r(\varpi _2)$ and $V(\varpi _2)$ are similarly given as

$$\begin{aligned}&k_1 \quad&&k_2 \qquad&{}&V^r(\varpi _2) \qquad&{}&V(\varpi _2) \\&1&&q&u_1 \otimes u_2&- q u_2 \otimes u_1 \qquad&v_1 \otimes v_2&-qv_2\otimes v_1\\&&&{}&\downarrow e_2&{}&{}f_2&\downarrow \\&q&&q^{-1}&u_1 \otimes u_3&- q u_3 \otimes u_1 \quad&v_1 \otimes v_3&-qv_3\otimes v_1\\&&&{}&\downarrow e_1&{}&f_1&\downarrow \\&q^{-1}&&1&u_2 \otimes u_3&- q u_3 \otimes u_2 \quad&v_2 \otimes v_3&-qv_3\otimes v_2 \end{aligned}$$

Here $g= k_i, e_i, f_i$ are to be understood as $\Delta (g)$ in (10.2).

Following (10.22) with $l=1$ we set

$$\begin{aligned} t_{ij} = \Psi _{\varpi _1}(u_i \otimes v_j) \end{aligned}$$

(10.49)

for $1 \le i,j \le 3$. They satisfy the relations (3.5) and (3.2) of the earlier definition of $A_q(A_2)$. The formula (10.23) reads

$$\begin{aligned} \sigma _1&= \Psi _{\varpi _1}(u_1 \otimes v_3), \end{aligned}$$

(10.50)

$$\begin{aligned} \sigma _2&= \frac{1}{1+q^2}\Psi _{\varpi _2}\bigl ((u_1 \otimes u_2 - q u_2 \otimes u_1) \otimes (v_2 \otimes v_3-qv_3\otimes v_2)\bigr ), \end{aligned}$$

(10.51)

where $(1+q^2)^{-1}$ is the normalization factor.⁷ Thus we see $\sigma _1 = t_{13}$. On the other hand, from

$$\begin{aligned} \langle \sigma _2, x\rangle&= \frac{(u_1 \otimes u_2 - q u_2 \otimes u_1, \Delta (x)(v_2 \otimes v_3-qv_3\otimes v_2))}{1+q^2} \\&= \frac{\langle t_{12} \otimes t_{23} - q t_{13}\otimes t_{22} - qt_{22}\otimes t_{13} + q^2 t_{23}\otimes t_{12}, \Delta (x)\rangle }{1+q^2} \\&= \frac{\langle t_{12} t_{23} - q t_{13}t_{22} - qt_{22}t_{13} + q^2 t_{23}t_{12},x\rangle }{1+q^2} \quad (\forall x \in U_q(A_2)), \end{aligned}$$

we find $\sigma _2 = (1+q^2)^{-1} (t_{12} t_{23} - q t_{13}t_{22} - qt_{22}t_{13} + q^2 t_{23}t_{12})$.⁸ Using the relations $[t_{12},t_{23}]=(q-q^{-1})t_{22}t_{13}$ and $[t_{22},t_{13}]=0$ from (3.5), this is simplified into $\sigma _2 = t_{12}t_{23}-qt_{22}t_{13}$, which is the (3, 1)-quantum minor of $(t_{ij})_{1\le i,j \le 3}$.

Let us turn to $\sigma _ie_i$. First we note

$$\begin{aligned} \langle t_{ij}k_r, x\rangle&= (u_ik_r, xv_j) =q^{\delta _{ir}-\delta _{i,r+1}} (u_{i},xv_j) = q^{\delta _{ir}-\delta _{i,r+1}}\langle t_{ij}, x\rangle , \end{aligned}$$

(10.52)

$$\begin{aligned} \langle t_{ij}e_r, x\rangle&= (u_ie_r, xv_j) = \delta _{ir}(u_{i+1},xv_j) = \delta _{ir}\langle t_{i+1,j},x\rangle . \end{aligned}$$

(10.53)

They imply

$$\begin{aligned} t_{ij} k_r = q^{\delta _{ir}-\delta _{i,r+1}}t_{ij}, \qquad t_{ij}e_r = \delta _{ir} t_{i+1,j}. \end{aligned}$$

(10.54)

Using this and the coproduct $\Delta$ in (10.2), we see

$$\begin{aligned} \langle \sigma _1e_1, x\rangle&=\langle t_{13}e_1, x\rangle = \langle t_{23}, x \rangle ,\\ \langle \sigma _2e_2, x\rangle&= \langle (t_{12}\otimes t_{23}-qt_{22}\otimes t_{13})\Delta (e_2), \Delta (x)\rangle \\&=\langle t_{12}k_2 \otimes t_{23}e_2- q t_{22}e_2\otimes t_{13}, \Delta (x)\rangle \\&= \langle t_{12} \otimes t_{33}- q t_{32}\otimes t_{13}, \Delta (x)\rangle = \langle t_{12}t_{33}- q t_{32} t_{13}, x\rangle . \end{aligned}$$

In these calculations, one should distinctively recognize that $t_{13}e_1$ for instance is an action of $e_1 \in U_q(A_2)$ on $t_{13} \in A_q(A_2)$ viewed as an element of a right $U_q(A_2)$ module, whereas $t_{12}t_{33}$ is just a multiplication within $A_q(A_2)$. To summarize, we have shown:

Lemma 10.10

For $A_q(A_2)$, the following relations are valid:

$$\begin{aligned} \sigma _1 = t_{13}, \quad \sigma _2 = t_{12}t_{23}-qt_{22}t_{13},\quad \sigma _1 e_1 = t_{23},\quad \sigma _2e_2 = t_{12}t_{33}-qt_{32}t_{13}. \end{aligned}$$

(10.55)

From (3.35) and Lemma 10.10, we find

$$\begin{aligned} \pi _{\textbf{1}}(\sigma _1) = \textbf{k}_1\textbf{k}_2,\quad \pi _{\textbf{1}}(\sigma _1e_1) = \mathrm{\textbf{a}}^+_1\textbf{k}_2,\quad \pi _{\textbf{1}}(\sigma _2) =\textbf{k}_2\textbf{k}_3,\quad \pi _{\textbf{1}}(\sigma _2e_2) = \mathrm{\textbf{a}}^-_1\mathrm{\textbf{a}}^+_2\textbf{k}_3 +\textbf{k}_1\mathrm{\textbf{a}}^+_3, \end{aligned}$$

where a notation like $\textbf{k}_1\mathrm{\textbf{a}}^+_3 = \textbf{k}\otimes 1 \otimes \mathrm{\textbf{a}}^+$ has been used. Since $\textbf{k}\in \textrm{End}({\mathcal {F}}_q)$ is invertible, so is $\pi _\textbf{i}(\sigma _i)$ and we may write

$$\begin{aligned} \pi _\textbf{1}(\xi _1) = \lambda _1 \mathrm{\textbf{a}}^+_1\textbf{k}^{-1}_1,\quad \pi _\textbf{1}(\xi _2) = \lambda _2 (\mathrm{\textbf{a}}^-_1\mathrm{\textbf{a}}^+_2\textbf{k}^{-1}_2 +\textbf{k}_1\textbf{k}^{-1}_2\mathrm{\textbf{a}}^+_3\textbf{k}^{-1}_3), \end{aligned}$$

where $\lambda _1=\lambda _2 = (1-q^2)^{-1}$. Thus (10.43) leads to

$$\begin{aligned} \pi _\textbf{1}(\xi _1)|a,b,c\rangle \!\rangle&= |a+1,b,c\rangle \!\rangle , \end{aligned}$$

(10.56)

$$\begin{aligned} \pi _\textbf{1}(\xi _2)|a,b,c\rangle \!\rangle&= [a]_1|a-1,b+1,c\rangle \!\rangle +q^{a-b}|a,b,c+1\rangle \!\rangle . \end{aligned}$$

(10.57)

These formulas agree with (10.48) proving (10.40) for $\textbf{i}=\textbf{1}$. The other case $\textbf{i}=\textbf{2}$ also holds due to the symmetry $\pi _\textbf{2}(\xi _i) = \pi _\textbf{1}(\xi _{3-i})$. Thus Proposition 10.7 is established for $A_2$.

In terms of the 3DR in Chap. 3, Theorem 10.6 implies

$$\begin{aligned} E^{a,b,c}_\textbf{i} = \sum _{i,j,k} R^{a b c}_{i j k}\,E^{k,j,i}_{\textbf{i}'}. \end{aligned}$$

(10.58)

This is valid either for $(\textbf{i}, \textbf{i}') =(\textbf{1},\textbf{2})$ or $(\textbf{2}, \textbf{1})$ thanks to (3.62). The weight conservation (3.48) assures the equality of weights of the two sides.

10.4.2 Explicit Formulas for $C_2$

Consider $\mathfrak {g}= C_2$.

/ The q-Serre relations are

$$\begin{aligned} \begin{aligned}&e_{1}^3 e_{2}-[3]_1e_{1}^2 e_{2}e_{1}+[3]_1e_{1}e_{2}e_{1}^2 -e_{2}e_{1}^3=0,\\&e_{2}^2 e_{1}-[2]_2e_{2}e_{1}e_{2}+e_{1}e_{2}^2=0, \end{aligned} \end{aligned}$$

(10.59)

where $[m]_1 = \langle m \rangle /\langle 1 \rangle$ and $[m]_2 = \langle 2m \rangle /\langle 2 \rangle$. For simplicity we write the positive root vectors $e_{\beta _{i_r}}$ in (10.4) with $(i_1,\ldots , i_4) = \textbf{2}$ (10.31) as

$$\begin{aligned} \begin{aligned} b_1&=e_{\beta _1}=e_{2}, \quad b_2=e_{\beta _2}=e_{1}e_{2}-q^{2} e_{2}e_{1}, \\ b_3&=e_{\beta _3}=\frac{1}{[2]_1}(e_{1}b_2-b_2 e_{1}),\quad b_4=e_{\beta _4}=e_{1}. \end{aligned} \end{aligned}$$

(10.60)

Their commutation relations are

$$\begin{aligned} b_2 b_1&= q^{-2} b_1 b_2, \qquad \quad b_3 b_1 = -q^{-1} \langle 1 \rangle [2]_1^{-1} b_2^2+b_1 b_3, \end{aligned}$$

(10.61)

$$\begin{aligned} b_4 b_1&= b_2+q^{2} b_1 b_4, \quad \;\; b_3 b_2 = q^{-2} b_2 b_3, \end{aligned}$$

(10.62)

$$\begin{aligned} b_4 b_2&= [2]_1 b_3+b_2 b_4, \quad b_4 b_3 = q^{-2} b_3 b_4. \end{aligned}$$

(10.63)

Lemma 10.11

For ${\tilde{E}}^{a,b,c,d}_\textbf{2}=b_1^a b_2^b b_3^c b_4^d$, we have

$$\begin{aligned} {\tilde{E}}^{a,b,c,d}_\textbf{2} \cdot e_{1}&={\tilde{E}}^{a,b,c,d+1}_\textbf{2},\\ {\tilde{E}}^{a,b,c,d}_\textbf{2} \cdot e_{2}&=[d]_1 q^{d-2 c-1} {\tilde{E}}^{a,b+1,c,d-1}_\textbf{2}+q^{2 (d-b)} {\tilde{E}}^{a+1,b,c,d}_\textbf{2}\\&-\langle 1 \rangle q^{2 d-2 c+1}[c]_2 [2]_1^{-1} {\tilde{E}}^{a,b+2,c-1,d}_\textbf{2}+[d-1]_1 [d]_1 {\tilde{E}}^{a,b,c+1,d-2}_\textbf{2},\\ e_{1}\cdot {\tilde{E}}^{a,b,c,d}_\textbf{2}&= [2]_1 [b]_1 q^{2 a-b+1} {\tilde{E}}^{a,b-1,c+1,d}_\textbf{2} +q^{2 a-2 c} {\tilde{E}}^{a,b,c,d+1}_\textbf{2}+[a]_2 {\tilde{E}}^{a-1,b+1,c,d}_\textbf{2},\\ e_{2}\cdot {\tilde{E}}^{a,b,c,d}_\textbf{2}&={\tilde{E}}^{a+1,b,c,d}_\textbf{2}. \end{aligned}$$

Proof

By induction, we have

$$\begin{aligned} {b_4 b_1^n}&= {b_1^n b_4} q^{2 n}+[n]_2 {b_1^{n-1},b_2},\\ {b_4 b_2^n}&= [2]_1 [n]_1 {b_2^{n-1} b_3} q^{-n+1}+{b_2^n b_4},\\ {b_4 b_3^n}&= q^{-2 n} {b_3^n b_4},\\ {b_4^n b_1}&= [n]_1 {b_2 b_4^{n-1}} q^{n-1}+{b_1 b_4^n} q^{2 n}+[n-1]_1 [n]_1 {b_3 b_4^{n-2}},\\ {b_3^n b_1}&= -q^{1-2 n}\langle 1 \rangle [n]_2 [2]_1^{-1} {b_2^2 b_3^{n-1}}+{b_1 b_3^n},\\ {b_3^n b_2}&= q^{-2 n} {b_2 b_3^n},\\ {b_2^n b_1}&= q^{-2 n} {b_1 b_2^n}. \end{aligned}$$

The lemma is a direct consequence of these formulas. $\square$

Set ${\tilde{E}}_\textbf{1}^{a,b,c,d} = \chi ({\tilde{E}}^{d,c,b,a}_\textbf{2})$. The left multiplication formula for this basis is deduced from the above lemma by applying $\chi$.

Let $u_i$ and $v_i\,(i=1,2,3,4)$ be bases of $V^r(\varpi _1)$ and $V(\varpi _1)$ such that $u_j = u_1e_1\cdots e_{j-1}e_j$ and $v_j = f_jf_{j-1}\cdots f_1 v_1$, where $e_3=e_1, f_3=f_1$ just temporarily.

$$\begin{aligned}&k_1 \quad &{}& k_2 \quad&&V^r(\varpi _1) &{}\quad&V(\varpi _1) \\&q&&1&&u_1&{}&v_1\\&&{}&{}&{}&\downarrow e_1&{}& f_1\downarrow \\&q^{-1}&&q&&u_2&{}&v_2\\&&{}&{}&{}&\downarrow e_2&{}&f_2\downarrow \\&q&&q^{-1}&&u_3&{}&v_3 \\&&{}&{}&{}&\downarrow e_1&{}&f_1\downarrow \\&q^{-1}&&1&&u_4&{}&v_4 \end{aligned}$$

The left two columns specify the weights as in the $A_2$ case. For the coproduct (10.2), the bases of $V(\varpi _2)$ and $V^r(\varpi _2)$ are similarly given as

$$\begin{aligned}&k_1 \quad&&k_2 &V^r(\varpi _2) &{}&V(\varpi _2) \\&1&&q&u_1 \otimes u_2&- q u_2 \otimes u_1 \qquad&v_1 \otimes v_2&-qv_2\otimes v_1\\&{}&{}&{}&\downarrow e_2&{}&f_2\downarrow \\&q^2&&q^{-1}&u_1 \otimes u_3&- q u_3 \otimes u_1 \quad&v_1 \otimes v_3&-qv_3\otimes v_1\\&{}&{}&{}&\downarrow e_1&{}&f_1\downarrow \\&1&&1&u_2 \otimes u_3&+ q u_1 \otimes u_4 \qquad&v_2 \otimes v_3&+qv_1\otimes v_4\\ &{}&&{}&-qu_4 \otimes u_1&-q^2 u_3 \otimes u_2 \qquad&-q v_4 \otimes v_1&-q^2 v_3\otimes v_2\\&{}&{}&{}&\downarrow e_1&{}&f_1\downarrow \\&q^{-2}&&q&u_2 \otimes u_4&- q u_4 \otimes u_2 \quad&v_2 \otimes v_4&-qv_4\otimes v_2\\&{}&{}&{}&\downarrow e_2&{}&f_2\downarrow \\&1&&q^{-1}&u_3 \otimes u_4&- q u_4 \otimes u_3 \quad&v_3 \otimes v_4&-qv_4\otimes v_3 \end{aligned}$$

Arrows here indicate the images only up to overall normalization.

We adopt the definition of $t_{ij}$ in (10.22) with $l=1$ for $1 \le i,j \le 4$. Then $t_{ij}$’s satisfy the relations (5.1), (5.2) of the earlier definition of $A_q(C_2)$. The formula (10.23) reads as

$$\begin{aligned} \sigma _1&= \Psi _{\varpi _1}(u_1 \otimes v_4), \end{aligned}$$

(10.64)

$$\begin{aligned} \sigma _2&= \frac{1}{1+q^2}\Psi _{\varpi _2}\bigl ((u_1 \otimes u_2 - q u_2 \otimes u_1) \otimes (v_3 \otimes v_4-qv_4\otimes v_3)\bigr ). \end{aligned}$$

(10.65)

By a calculation similar to $A_q(A_2)$ using the commutation relations

$$\begin{aligned}{}[t_{24},t_{13}]=(q-q^{-1})t_{23}t_{14},\quad [t_{14},t_{23}] = 0, \end{aligned}$$

(10.66)

we get:

Lemma 10.12

For $A_q(C_2)$, the following relations are valid:

$$\begin{aligned} \sigma _1 = t_{14}, \quad \sigma _2 = t_{13}t_{24}-qt_{23}t_{14},\quad \sigma _1 e_1 = t_{24},\quad \sigma _2e_2 = t_{13}t_{34}-qt_{33}t_{14}. \end{aligned}$$

(10.67)

Images of the generators $t_{ij}$ by the representations $\pi _\textbf{1}$ and $\pi _\textbf{2}$ in (10.31) are available in Sect. 5.4 as $\pi _\textbf{1}(t_{ij}) = P_{14}P_{23}\pi _{2121}(\tilde{\Delta }(t_{ij}))P_{14}P_{23}$ and $\pi _\textbf{2}(t_{ij}) = \pi _{2121}(\Delta (t_{ij}))$, where the conjugation by $P_{14}P_{23}$ reverses the order of the four-fold tensor product. See (5.39) and (5.40). From (5.37), the relations (5.41)–(5.56) are displaying the concrete form of $\pi _\textbf{2}(t_{ij}) K = K (P_{14}P_{23}\pi _\textbf{1}(t_{ij})P_{14}P_{23})$. For convenience, we pick those generators appearing in Lemma 10.12:

$$\begin{aligned} \pi _\textbf{1}(t_{13})&= {\textbf{a}}^- _1{\textbf{k}}_3{\textbf{K}}_4 + {\textbf{k}}_1 {\textbf{A}}^- _2{\textbf{a}}^+ _3{\textbf{K}}_4+{\textbf{k}}_1{\textbf{K}}_2{\textbf{a}}^- _3{\textbf{A}}^+ _4, \end{aligned}$$

(10.68)

$$\begin{aligned} \pi _\textbf{1}(t_{14})&= - {\textbf{k}}_1{\textbf{K}}_2 {\textbf{k}}_3, \end{aligned}$$

(10.69)

$$\begin{aligned} \pi _\textbf{1}(t_{23})&= {\textbf{a}}^+ _1{\textbf{A}}^- _2 {\textbf{a}}^+ _3 {\textbf{K}}_4 + {\textbf{a}}^+ _1 {\textbf{K}}_2 {\textbf{a}}^- _3 {\textbf{A}}^+ _4-q {\textbf{k}}_1 {\textbf{k}}_3{\textbf{K}}_4, \end{aligned}$$

(10.70)

$$\begin{aligned} \pi _\textbf{1}(t_{24})&= - {\textbf{a}}^+ _1{\textbf{K}}_2 {\textbf{k}}_3, \end{aligned}$$

(10.71)

$$\begin{aligned} \pi _\textbf{1}(t_{33})&= {\textbf{a}}^- _1{\textbf{A}}^+ _2{\textbf{a}}^- _3{\textbf{A}}^+ _4-q^2{\textbf{a}}^- _1{\textbf{K}}_2{\textbf{a}}^+ _3{\textbf{K}}_4-q{\textbf{k}}_1{\textbf{k}}_3{\textbf{A}}^+ _4, \end{aligned}$$

(10.72)

$$\begin{aligned} \pi _\textbf{1}(t_{34})&= -{\textbf{a}}^- _1{\textbf{A}}^+ _2{\textbf{k}}_3- {\textbf{k}}_1{\textbf{a}}^+ _3, \end{aligned}$$

(10.73)

$$\begin{aligned} \pi _\textbf{2}(t_{13})&= {\textbf{k}}_2{\textbf{K}}_3{\textbf{a}}^- _4, \end{aligned}$$

(10.74)

$$\begin{aligned} \pi _\textbf{2}(t_{14})&= -{\textbf{k}}_2{\textbf{K}}_3{\textbf{k}}_4, \end{aligned}$$

(10.75)

$$\begin{aligned} \pi _\textbf{2}(t_{23})&= {\textbf{A}}^- _1{\textbf{a}}^+ _2{\textbf{K}}_3{\textbf{a}}^- _4+{\textbf{K}}_1{\textbf{a}}^- _2{\textbf{A}}^+ _3{\textbf{a}}^- _4-q {\textbf{K}}_1{\textbf{k}}_2{\textbf{k}}_4, \end{aligned}$$

(10.76)

$$\begin{aligned} \pi _\textbf{2}(t_{24})&= -{\textbf{A}}^- _1{\textbf{a}}^+ _2{\textbf{K}}_3{\textbf{k}}_4-{\textbf{K}}_1{\textbf{a}}^- _2{\textbf{A}}^+ _3{\textbf{k}}_4-{\textbf{K}}_1{\textbf{k}}_2{\textbf{a}}^+ _4, \end{aligned}$$

(10.77)

$$\begin{aligned} \pi _\textbf{2}(t_{33})&= {\textbf{A}}^+ _1{\textbf{a}}^- _2{\textbf{A}}^+ _3{\textbf{a}}^- _4-q {\textbf{A}}^+ _1{\textbf{k}}_2{\textbf{k}}_4-q^2{\textbf{K}}_1{\textbf{a}}^+ _2{\textbf{K}}_3{\textbf{a}}^- _4, \end{aligned}$$

(10.78)

$$\begin{aligned} \pi _\textbf{2}(t_{34})&= -{\textbf{A}}^+ _1{\textbf{a}}^- _2{\textbf{A}}^+ _3{\textbf{k}}_4-{\textbf{A}}^+ _1{\textbf{k}}_2{\textbf{a}}^+ _4+q^2{\textbf{K}}_1{\textbf{a}}^+ _2{\textbf{K}}_3{\textbf{k}}_4. \end{aligned}$$

(10.79)

From this and Lemma 10.12 we get

$$ \begin{array}{rl} \pi _\textbf{1}(\sigma _1)&{}=- {\textbf{k}_1} {\textbf{K}_2} {\textbf{k}_3},\\ \pi _\textbf{1}(\sigma _1e_1)&{}=- {\textbf{a}^+_1} {\textbf{K}_2} {\textbf{k}_3},\\ \pi _\textbf{1}(\sigma _2)&{}=- {\textbf{K}_2} {\textbf{k}_3}^{2} {\textbf{K}_4},\\ \pi _\textbf{1}(\sigma _2e_2)&{}=- {\textbf{a}^-_1}^2 {\textbf{A}^+_2} {\textbf{k}_3}^{2} {\textbf{K}_4} -[2]_1 {\textbf{a}^-_1} {\textbf{k}_1} {\textbf{a}^+_3} {\textbf{k}_3} {\textbf{K}_4} - {\textbf{k}_1}^{2} {\textbf{A}^-_2} {\textbf{a}^+_3}^2 {\textbf{K}_4} - {\textbf{A}^+_4} {\textbf{k}_1}^{2} {\textbf{K}_2},\\ \lambda _1^{-1}\pi _\textbf{1}(\xi _1)&{}={\textbf{a}^ +_1} {\textbf{k}_1}^{-1},\\ \lambda _2^{-1}\pi _\textbf{1}(\xi _2)&{}={\textbf{a}^- _1}^2 {\textbf{A}^+_2} {\textbf{K}_2}^{-1} +{\textbf{k}_1}^{2} {\textbf{A}^-_2} {\textbf{K}_2}^{-1} {\textbf{a}^+_3}^2 {\textbf{k}_3}^{-2} +[2]_1 {\textbf{a}^-_1} {\textbf{k}_1} {\textbf{K}_2}^{-1} {\textbf{a}^+_3} {\textbf{k}_3}^{-1}\\ &{}\quad +\, {\textbf{k}_1}^{2} {\textbf{k}_3}^{-2} {\textbf{A}^+_4} {\textbf{K}_4}^{-1},\\ \pi _\textbf{2}(\sigma _1)&{}=- \textbf{k}_2 \textbf{K}_3 \textbf{k}_4,\\ \pi _\textbf{2}(\sigma _1e_1)&{}=- {\textbf{K}_1} \textbf{k}_2 {\textbf{a}^+_4} - {\textbf{K}_1} {\textbf{a}^-_2} {\textbf{A}^+_3} \textbf{k}_4 - {\textbf{A}^-_1} {\textbf{a}^+_2} \textbf{K}_3 \textbf{k} _4,\\ \pi _\textbf{2}(\sigma _2)&{}=- {\textbf{K}_1} \textbf{k}_2^2 \textbf{K}_3,\\ \pi _\textbf{2}(\sigma _2e_2)&{}=- {\textbf{A}^+_1} \textbf{k}_2^2 \textbf{K}_3,\\ \lambda _1^{-1}\pi _\textbf{2}(\xi _1)&{}={\textbf{A}^- _1} {\textbf{a}^+_2} \textbf{k}_2^{-1} +{\textbf{K}_1} {\textbf{a}^-_2} \textbf{k}_2^{-1} {\textbf{A}^+_3} \textbf{K}_ 3^{-1} +{\textbf{K}_1} \textbf{K}_3^{-1} {\textbf{a}^+_4} \textbf{k}_4^{-1},\\ \lambda _2^{-1}\pi _\textbf{2}(\xi _2)&{}={\textbf{A}^ +_1} {\textbf{K}_1}^{-1}. \end{array} $$

Note that ${\pi _\textbf{i}(\sigma _i)}$ is invertible. Comparing these formulas with Lemma 10.11 by using (10.43), the equality (10.40) is directly checked. Thus Proposition 10.7 is established for ${C_2}$.

In terms of the 3D K in Chap. 5, Theorem 10.6 implies

$$\begin{aligned} E^{a,b,c,d}_\textbf{2} = \sum _{i,j,k,l} K^{a b c d}_{i j k l}\,E^{l,k,j,i}_\textbf{1}. \end{aligned}$$

(10.80)

The weight conservation (5.65) assures the equality of weights of the two sides.

10.4.3 Explicit Formulas for $G_2$

Consider $\mathfrak {g}= G_2$.

The q-Serre relations are

$$\begin{aligned} \begin{aligned}&e_{1}^4 e_{2}-[4]_1e_{1}^3 e_{2}e_{1}+[4]_1[3]_1/[2]_1^{-1}e_{1}^2 e_{2}e_{1}^2 -[4]_1e_{1}e_{2}e_{1}^3 +e_{2}e_{1}^4=0,\\&e_{2}^2 e_{1}-[2]_2e_{2}e_{1}e_{2}+e_{1}e_{2}^2=0, \end{aligned} \end{aligned}$$

(10.81)

where ${[m]_1 = \langle m \rangle /\langle 1 \rangle }$ and ${[m]_2 = \langle 3m \rangle /\langle 3 \rangle }$. For simplicity we write the positive root vectors ${e_{\beta _{i_r}}}$ in (10.4) with ${(i_1,\ldots , i_6) = \textbf{2}}$ (10.32) as

$$\begin{aligned} \begin{aligned} b_1&=e_{\beta _1}=e_{2}, \quad b_2=e_{\beta _2}=e_{1}e_{2}-q^3 e_{2}e_{1}, \\ b_4&=e_{\beta _3}=\dfrac{1}{[2]_1}(e_{1}b_2-q b_2 e_{1}),\quad b_5=e_{\beta _4}=\dfrac{1}{[3]_1}(e_{1}b_4-q^{-1} b_4 e_{1}), \\ b_3&=e_{\beta _5} =\dfrac{1}{[3]_1}(b_4 b_2-q^{-1} b_2 b_4), \quad b_6=e_{\beta _6}=e_{1}. \end{aligned} \end{aligned}$$

(10.82)

Their commutation relations are

$$\begin{aligned} b_2 b_1&= b_1 b_2 q^{-3}, \qquad \qquad \qquad b_3 b_1 = \langle 1 \rangle ^2 b_2^3 q^{-3} [3]_1^{-1}+b_1 b_3 q^{-3}, \end{aligned}$$

(10.83)

$$\begin{aligned} b_4 b_1&= b_1 b_4-b_2^2 \langle 1 \rangle q^{-1}, \end{aligned}$$

(10.84)

$$\begin{aligned} b_5 b_1&= b_1 b_5 q^3-b_2 b_4 \langle 1 \rangle q^{-1}-(q^4+q^2-1) b_3 q^{-3}, \end{aligned}$$

(10.85)

$$\begin{aligned} b_6 b_1&= b_1 b_6 q^3+b_2, \qquad \qquad \;\;\; b_3 b_2 = b_2 b_3 q^{-3}, \end{aligned}$$

(10.86)

$$\begin{aligned} b_4 b_2&= b_2 b_4 q^{-1}+b_3 [3]_1, \qquad \;\;\; b_5 b_2 = b_2 b_5-b_4^2 \langle 1 \rangle q^{-1}, \end{aligned}$$

(10.87)

$$\begin{aligned} b_6 b_2&= q b_2 b_6+b_4 [2]_1, \qquad \qquad b_4 b_3 = b_3 b_4 q^{-3}, \end{aligned}$$

(10.88)

$$\begin{aligned} b_5 b_3&= \langle 1 \rangle ^2 b_4^3 q^{-3} [3]_1^{-1}+b_3 b_5 q^{-3}, \end{aligned}$$

(10.89)

$$\begin{aligned} b_6 b_3&= b_3 b_6-b_4^2 \langle 1 \rangle q^{-1}, \qquad \quad \; b_5 b_4 = b_4 b_5 q^{-3}, \end{aligned}$$

(10.90)

$$\begin{aligned} b_6 b_4&= [3]_1 b_5+b_4 b_6 q^{-1}, \qquad \quad b_6 b_5 = b_5 b_6 q^{-3}. \end{aligned}$$

(10.91)

Lemma 10.13

For ${{\tilde{E}}^{a,b,c,d,e,f}_\textbf{2}=b_1^a b_2^b b_3^cb_4^db_5^e b_6^f}$, we have

$$\begin{aligned} {\tilde{E}}^{a,b,c,d,e,f}_\textbf{2} \cdot e_{1}= \;&{\tilde{E}}^{a,b,c,d,e,f+1}_\textbf{2}.\\ {\tilde{E}}^{a,b,c,d,e,f}_\textbf{2} \cdot e_{2}= \;&-\langle 1 \rangle [e]_2 q^{-3 c-d+3 f-1} {\tilde{E}}^{a,b+1,c,d+1,e-1,f}_\textbf{2}\\&+\langle 1 \rangle ^2 [e-1]_2 [e]_2 [3]_1^{-1}q^{-3 e+3 f+3} {\tilde{E}}^{a,b,c,d+3,e-2,f}_\textbf{2}\\&-\langle 3 \rangle [d-1]_1 [d]_1 q^{-3 c-2 d+3 e+3 f+1} {\tilde{E}}^{a,b+1,c+1,d-2,e,f}_\textbf{2}\\&-\langle 1 \rangle [d]_1 q^{-6 c-d+3 (e+f)} {\tilde{E}}^{a,b+2,c,d-1,e,f}_\textbf{2}\\&+[f-1]_1 [f]_1 q^{-3 e+f-2} {\tilde{E}}^{a,b,c,d+1,e,f-2}_\textbf{2}\\&+[3]_1 [d]_1 [f]_1 q^{2 f-2 d} {\tilde{E}}^{a,b,c+1,d-1,e,f-1}_\textbf{2}\\&+[f]_1 q^{-3 c-d+2 f-2} {\tilde{E}}^{a,b+1,c,d,e,f-1}_\textbf{2}\\&+q^{-3 (b+c-e-f)} {\tilde{E}}^{a+1,b,c,d,e,f}_\textbf{2}\\&+\langle 1 \rangle ^2 [c]_2 [3]_1^{-1}q^{3 (-2 c+e+f+1)} {\tilde{E}}^{a,b+3,c-1,d,e,f}_\textbf{2}\\&-\langle 3 \rangle [d-2]_1 [d-1]_1 [d]_1 q^{3 (-d+e+f+2)} {\tilde{E}}^{a,b,c+2,d-3,e,f}_\textbf{2}\\&-\langle 1 \rangle [e]_2 [f]_1 q^{-3 e+2 f} {\tilde{E}}^{a,b,c,d+2,e-1,f-1}_\textbf{2}\\&-[e]_2 q^{-3 d+3 f} (q^{2d+1} [3]_1-[2]_2) {\tilde{E}}^{a,b,c+1,d,e-1,f}_\textbf{2}\\&+[f-2]_1 [f-1]_1 [f]_1 {\tilde{E}}^{a,b,c,d,e+1,f-3}_\textbf{2}.\,\\ e_{1}\cdot {\tilde{E}}^{a,b,c,d,e,f}_\textbf{2}= \;&-\langle 1 \rangle [c]_2 q^{3 a+b-3 c+2} {\tilde{E}}^{a,b,c-1,d+2,e,f}_\textbf{2}\\&+[3]_1 [b-1]_1 [b]_1 q^{3 a-b+2} {\tilde{E}}^{a,b-2,c+1,d,e,f}_\textbf{2}\\&+[3]_1 [d]_1 q^{3 a+b-2 d+2} {\tilde{E}}^{a,b,c,d-1,e+1,f}_\textbf{2}\\&+q^{3 a+b-d-3 e} {\tilde{E}}^{a,b,c,d,e,f+1}_\textbf{2}\\&+[2]_1 [b]_1 q^{3 (a-c)} {\tilde{E}}^{a,b-1,c,d+1,e,f}_\textbf{2}\\&+[a]_2 {\tilde{E}}^{a-1,b+1,c,d,e,f}_\textbf{2}.\\ e_{2}\cdot {\tilde{E}}^{a,b,c,d,e,f}_\textbf{2}= \;&{\tilde{E}}^{a+1,b,c,d,e,f}_\textbf{2}. \end{aligned}$$

Proof

By induction, we have

$$\begin{aligned} b_6 b_1^n&= q^{3 n} b_1^n b_6+[n]_2 b_1^{n-1} b_2,\\ b_6 b_2^n&= [3]_1 q^{2-n} [n-1]_1 [n]_1 b_2^{n-2} b_3+q^n b_2^n b_6+[2]_1 [n]_1 b_2^{n-1} b_4,\\ b_4 b_3^n&= q^{-3 n} b_3^n b_4,\\ b_6 b_3^n&= b_3^n b_6-\langle 1 \rangle q^{2-3 n} [n]_2 b_3^{n-1} b_4 b_4,\\ b_6 b_4^n&= [3]_1 q^{2-2 n} [n]_1 b_4^{n-1} b_5+q^{-n} b_4^n b_6, \\ b_6 b_5^n&= q^{-3 n} b_5^n b_6,\\ \textrm{and}&\\ b_6^n b_1&= q^{n-2} [n-1]_1 [n]_1 b_4 b_6^{n-2}+q^{3 n} b_1 b_6^n \\&\quad +q^{2 (n-1)} [n]_1 b_2 b_6^{n-1}+[n-2]_1 [n-1]_1 [n]_1 b_5 b_6^{n-3},\\ b_5^n b_1&= \langle 1 \rangle ^2 q^{-3 (n-1)} [n-1]_2 [n]_2 [3]_1^{-1}b_4^3 b_5^{n-2}+q^{3 n} b_1 b_5^n\\&\quad -q^{-3}(q^4+q^2-1) [n]_2 b_3 b_5^{n-1}-q^{-1}\langle 1 \rangle [n]_2 b_2 b_4 b_5^{n-1},\\ b_5^n b_2&= b_2 b_5^n-\langle 1 \rangle q^{2-3 n} [n]_2 b_4 b_4 b_5^{n-1},\\ b_5^n b_4&= q^{-3 n} b_4 b_5^n,\\ b_4^n b_1&= -\langle 3 \rangle q^{6-3 n} [n-2]_1 [n-1]_1 [n]_1 b_3^2 b_4^{n-3}-\langle 1 \rangle q^{-n} [n]_1b_2^2 b_4^{n-1}\\&\quad -\langle 3 \rangle q^{1-2 n} [n-1]_1 [n]_1 b_2 b_3 b_4^{n-2}+b_1 b_4^n,\\ b_4^n b_2&= [3]_1 q^{2-2 n} [n]_1 b_3 b_4^{n-1}+q^{-n} b_2 b_4^n,\\ b_4^n b_3&= q^{-3 n} b_3 b_4^n,\\ b_3^n b_1&= q^{-3 n} b_1 b_3^n+\langle 1 \rangle ^2 q^{3-6 n} [n]_2 [3]_1^{-1}b_2^3 b_3^{n-1},\\ b_3^n b_2&= q^{-3 n} b_2 b_3^n,\\ b_2^n b_1&= q^{-3 n} b_1 b_2^n. \end{aligned}$$

The lemma is a direct consequence of these formulas. $\square$

Let ${v_i\,(i=1,\ldots , 7)}$ be the basis of ${V(\varpi _1)}$ for which the representation matrix is given by (8.79)–(8.81). Its highest and lowest weight vectors are ${v_1}$ and ${v_7}$, respectively. Let ${u_i \in V^r(\varpi _1)}$ be the dual base of ${v_i}$.

The representation ${V(\varpi _2)}$ is the adjoint representation with dimension 14. Its lowest weight vector is ${v^{(14)}_{14}}$ in (8.84), which is ${v_6\otimes v_7-q v_7\otimes v_6}$ in the notation here. The highest weight vector of ${V^r(\varpi _2)}$ is ${u_1\otimes u_2 - q u_2 \otimes u_1}$. From these facts we have

$$\begin{aligned} \sigma _1&= \Psi _{\varpi _1}(u_1 \otimes v_7), \end{aligned}$$

(10.92)

$$\begin{aligned} \sigma _2&= \frac{1}{1+q^2}\Psi _{\varpi _2}\bigl ((u_1 \otimes u_2 - q u_2 \otimes u_1) \otimes (v_6 \otimes v_7-qv_7\otimes v_6)\bigr ). \end{aligned}$$

(10.93)

We define ${t_{ij}}$ by the formula (10.22) with ${l=1}$ for ${1\le i,j \le 7}$. They satisfy the relations (8.3) and (8.4) of the earlier definition of ${A_q(G_2)}$. By a calculation similar to ${A_q(A_2)}$ using the commutation relations

$$\begin{aligned}{}[t_{16},t_{27}]=(q-q^{-1})t_{26}t_{17},\quad [t_{17},t_{26}] = 0, \end{aligned}$$

(10.94)

we get⁹

Lemma 10.14

For ${A_q(G_2)}$, the following relations are valid:

$$\begin{aligned} \sigma _1 = t_{17}, \quad \sigma _2 = t_{16}t_{27}-qt_{27}t_{16},\quad \sigma _1 e_1 = t_{27},\quad \sigma _2e_2 = t_{16}t_{37}-qt_{36}t_{17}. \end{aligned}$$

(10.95)

Images of the generators ${t_{ij}}$ by the representations ${\pi _\textbf{1}}$ and ${\pi _\textbf{2}}$ in (10.31) are available from (8.11) and (8.12). For convenience, we present explicit formulas for those appearing in Lemma 10.14:

$$\begin{aligned} \pi _\textbf{1}(t_{16})&= {\textbf{a}}^- _1{\textbf{k}}_3{\textbf{K}}_4{\textbf{k}}_5^2{\textbf{K}}_6+ {\textbf{k}}_1{\textbf{A}}^- _2{\textbf{a}}^+ _3{\textbf{K}}_4{\textbf{k}}_5^2{\textbf{K}}_6+ {\textbf{k}}_1{\textbf{K}}_2\textbf{a}_3^{-2} {\textbf{A}}^+ _4{\textbf{k}}_5^2{\textbf{K}}_6 \\ {}&+ [2]_1{\textbf{k}}_1{\textbf{K}}_2{\textbf{a}}^- _3{\textbf{k}}_3 {\textbf{a}}^+ _5{\textbf{k}}_5{\textbf{K}}_6+ {\textbf{k}}_1{\textbf{K}}_2{\textbf{k}}_3^2{\textbf{A}}^- _4\textbf{a}_5^{+2}{\textbf{K}}_6 + {\textbf{k}}_1{\textbf{K}}_2{\textbf{k}}_3^2{\textbf{K}}_4{\textbf{a}}^- _5{\textbf{A}}^+ _6, \\ \pi _\textbf{1}(t_{17})&= {\textbf{k}}_1{\textbf{K}}_2{\textbf{k}}_3^2{\textbf{K}}_4{\textbf{k}}_5, \\ \pi _\textbf{1}(t_{27})&= {\textbf{a}}^+ _1{\textbf{K}}_2{\textbf{k}}_3^2{\textbf{K}}_4{\textbf{k}}_5, \\ \pi _\textbf{1}(t_{36})&= \textbf{a}_1^{-2}{\textbf{A}}^+ _2\textbf{a}_2^{-2}{\textbf{A}}^+ _4{\textbf{k}}_5^2{\textbf{K}}_6+ [2]_1\textbf{a}_1^{-2}{\textbf{A}}^+ _2{\textbf{a}}^- _3{\textbf{k}}_3{\textbf{a}}^+ _5{\textbf{k}}_5{\textbf{K}}_6+ \textbf{a}_1^{-2}{\textbf{A}}^+ _2{\textbf{k}}_3^2{\textbf{A}}^- _4 \textbf{a}_5^{+2}{\textbf{K}}_6 \\ {}&+ \textbf{a}_1^{-2}{\textbf{A}}^+ _2{\textbf{k}}_3^2{\textbf{K}}_4{\textbf{a}}^- _5{\textbf{A}}^+ _6- q^3 \textbf{a}_1^{-2} {\textbf{K}}_2{\textbf{a}}^+ _3{\textbf{K}}_4{\textbf{k}}_5^2{\textbf{K}}_6+ [2]_1{\textbf{a}}^- _1{\textbf{k}}_1 {\textbf{a}}^+ _5{\textbf{k}}_5{\textbf{K}}_6 \\ {}&- [2]_1{\textbf{a}}^- _1{\textbf{k}}_1 {\textbf{a}}^- _3{\textbf{k}}_3{\textbf{A}}^+ _4{\textbf{k}}_5^2{\textbf{K}}_6+ [2]_1{\textbf{a}}^- _1{\textbf{k}}_1 {\textbf{a}}^+ _3{\textbf{k}}_3 {\textbf{A}}^- _4\textbf{a}_5^{+2}{\textbf{K}}_6+ [2]_1{\textbf{a}}^- _1{\textbf{k}}_1 {\textbf{a}}^+ _3{\textbf{k}}_3 {\textbf{K}}_4{\textbf{a}}^- _5{\textbf{A}}^+ _6 \\ {}&- q[2]_1^2 {\textbf{a}}^- _1{\textbf{k}}_1{\textbf{k}}_3^2{\textbf{a}}^+ _5{\textbf{k}}_5{\textbf{K}}_6+ {\textbf{k}}_1^2{\textbf{A}}^- _2\textbf{a}_3^{+2}{\textbf{A}}^- _4\textbf{a}_5^{+2}{\textbf{K}}_6+ {\textbf{k}}_1^2{\textbf{A}}^- _2\textbf{a}_3^{+2}{\textbf{K}}_4{\textbf{a}}^- _5{\textbf{A}}^+ _6 \\ {}&- q^2[2]_1{\textbf{k}}_1^2{\textbf{A}}^- _2{\textbf{a}}^+ _3{\textbf{k}}_3{\textbf{a}}^+ _5{\textbf{k}}_5{\textbf{K}}_6+ q^2 {\textbf{k}}_1^2{\textbf{A}}^- _2{\textbf{k}}_3^2{\textbf{A}}^+ _4{\textbf{k}}_5^2{\textbf{K}}_6+ {\textbf{k}}_1^2{\textbf{K}}_2{\textbf{a}}^- _3{\textbf{A}}^+ _4{\textbf{a}}^- _5{\textbf{A}}^+ _6 \\ {}&- q^3{\textbf{k}}_1^2{\textbf{K}}_2{\textbf{a}}^- _3{\textbf{K}}_4\textbf{a}_5^{+2}{\textbf{K}}_6- q {\textbf{k}}_1^2{\textbf{K}}_2{\textbf{k}}_3{\textbf{k}}_5{\textbf{A}}^+ _6, \\ \pi _\textbf{1}(t_{37})&= \textbf{a}_1^{-2}{\textbf{A}}^+ _2{\textbf{k}}_3^2{\textbf{K}}_4{\textbf{k}}_5+ [2]_1{\textbf{a}}^- _1{\textbf{k}}_1 {\textbf{a}}^+ _3{\textbf{k}}_3{\textbf{K}}_4{\textbf{k}}_5+ {\textbf{k}}_1^2{\textbf{A}}^- _2\textbf{a}_3^{+2}{\textbf{K}}_4{\textbf{k}}_5 \\ {}&+ {\textbf{k}}_1^2{\textbf{K}}_2{\textbf{a}}^- _3{\textbf{A}}^+ _4{\textbf{k}}_5+ {\textbf{k}}_1^2{\textbf{K}}_2{\textbf{k}}_3{\textbf{a}}^+ _5, \\ \pi _\textbf{2}(t_{16})&= {\textbf{k}}_2{\textbf{K}}_3{\textbf{k}}_4^2{\textbf{K}}_5{\textbf{a}}^- _6, \\ \pi _\textbf{2}(t_{17})&= {\textbf{k}}_2{\textbf{K}}_3{\textbf{k}}_4^2{\textbf{K}}_5{\textbf{k}}_6, \\ \pi _\textbf{2}(t_{27})&= {\textbf{A}}^- _1{\textbf{a}}^+ _2{\textbf{K}}_3{\textbf{k}}_4^2{\textbf{K}}_5{\textbf{k}}_6+ {\textbf{K}}_1\textbf{a}_2^{-2}{\textbf{A}}^+ _3{\textbf{k}}_4^2{\textbf{K}}_5{\textbf{k}}_6+ [2]_1{\textbf{K}}_1{\textbf{a}}^- _2{\textbf{k}}_2{\textbf{a}}^+ _4{\textbf{k}}_4{\textbf{K}}_5{\textbf{k}}_6 \\ {}&+ {\textbf{K}}_1{\textbf{k}}_2^2{\textbf{A}}^- _3\textbf{a}_4^{+2}{\textbf{K}}_5{\textbf{k}}_6+ {\textbf{K}}_1{\textbf{k}}_2^2{\textbf{K}}_3{\textbf{a}}^- _4{\textbf{A}}^+ _5{\textbf{k}}_6+ {\textbf{K}}_1{\textbf{k}}_2^2{\textbf{K}}_3{\textbf{k}}_4{\textbf{a}}^+ _6, \\ \pi _\textbf{2}(t_{36})&= {\textbf{A}}^+ _1\textbf{a}_2^{-2}{\textbf{A}}^+ _3{\textbf{k}}_4^2{\textbf{K}}_5{\textbf{a}}^- _6+ [2]_1{\textbf{A}}^+ _1{\textbf{a}}^- _2{\textbf{k}}_2{\textbf{a}}^+ _4{\textbf{k}}_4{\textbf{K}}_5{\textbf{a}}^- _6+ {\textbf{A}}^+ _1{\textbf{k}}_2^2{\textbf{A}}^- _3\textbf{a}_4^{+2}{\textbf{K}}_5{\textbf{a}}^- _6 \\ {}&+ {\textbf{A}}^+ _1{\textbf{k}}_2^2{\textbf{K}}_3{\textbf{a}}^- _4{\textbf{A}}^+ _5{\textbf{a}}^- _6- q{\textbf{A}}^+ _1{\textbf{k}}_2^2{\textbf{K}}_3{\textbf{k}}_4{\textbf{k}}_6- q^3 {\textbf{K}}_1{\textbf{a}}^+ _2{\textbf{K}}_3{\textbf{k}}_4^2{\textbf{K}}_5{\textbf{a}}^- _6, \\ \pi _\textbf{2}(t_{37})&= {\textbf{A}}^+ _1\textbf{a}_2^{-2}{\textbf{A}}^+ _3{\textbf{k}}_4^2{\textbf{K}}_5{\textbf{k}}_6+ [2]_1{\textbf{A}}^+ _1{\textbf{a}}^- _2{\textbf{k}}_2{\textbf{a}}^+ _4{\textbf{k}}_4{\textbf{K}}_5{\textbf{k}}_6+ {\textbf{A}}^+ _1{\textbf{k}}_2^2{\textbf{A}}^- _3\textbf{a}_4^{+2}{\textbf{K}}_5{\textbf{k}}_6 \\ {}&+ {\textbf{A}}^+ _1{\textbf{k}}_2^2{\textbf{K}}_3{\textbf{a}}^- _4{\textbf{A}}^+ _5{\textbf{k}}_6+ {\textbf{A}}^+ _1{\textbf{k}}_2^2{\textbf{K}}_3{\textbf{k}}_4{\textbf{a}}^+ _6- q^3{\textbf{K}}_1{\textbf{a}}^+ _2{\textbf{K}}_3{\textbf{k}}_4^2{\textbf{K}}_5{\textbf{k}}_6. \end{aligned}$$

From this and Lemma 10.14 we get

$$ \begin{array}{rl} \pi _\textbf{1}(\sigma _1)&{}=\textbf{k}_1 \textbf{K}_2 \textbf{k}^{2}_3 \textbf{K}_4 \textbf{k}_5,\\ \pi _\textbf{1}(\sigma _2)&{}=\textbf{K}_2 \textbf{k}^{3}_3 \textbf{K}^{2}_4 \textbf{k}^{3}_5 \textbf{K}_6,\\ \pi _\textbf{1}(\sigma _1e_1)&{}=\textbf{a}^+_1 \textbf{K}_2 \textbf{k}^{2}_3 \textbf{K}_4 \textbf{k}_5,\\ \pi _\textbf{1}(\sigma _2e_2)&{}=\textbf{k}^{3}_1 \textbf{K}^{2}_2 \textbf{k}^{3}_3 \textbf{K}_4 \textbf{A}^+_6 + [2]_2 \textbf{k}^{3}_1 \textbf{A}^-_2 \textbf{K}_2 \textbf{A}^+_4 \textbf{K}_4 \textbf{k}^{3}_5 \textbf{K}_6 + \textbf{a}_1^{-3} \textbf{A}^+_2 \textbf{k}^{3}_3 \textbf{K}^{2}_4 \textbf{k}^{3}_5 \textbf{K}_6\\ {} &{} + \; [3]_1 \textbf{a}_1^{-2} \textbf{k}_1 \textbf{a}^+_3 \textbf{k}^{2}_3 \textbf{K}^{2}_4 \textbf{k}^{3}_5 \textbf{K}_6 + [3]_1 \textbf{a}^-_1 \textbf{k}^{2}_1 \textbf{K}_2 \textbf{k}^{2}_3 \textbf{K}_4 \textbf{a}^+_5 \textbf{k}^{2}_5 \textbf{K}_6\\ {} &{} - \; q [3]_1 \textbf{k}^{3}_1 \textbf{A}^-_2 \textbf{K}_2 \textbf{k}^{2}_3 \textbf{A}^+_4 \textbf{K}_4 \textbf{k}^{3}_5 \textbf{K}_6 + [3]_1 \textbf{k}^{3}_1 \textbf{K}^{2}_2 \textbf{a}^-_3 \textbf{k}^{2}_3 \textbf{a}_5^{+2} \textbf{k}_5 \textbf{K}_6\\ {} &{} + \; \textbf{k}^{3}_1 \textbf{K}^{2}_2 \textbf{k}^{3}_3 \textbf{A}^-_4 \textbf{a}_5^{+3} \textbf{K}_6 + [3]_1 \textbf{a}^-_1 \textbf{k}^{2}_1 \textbf{A}^-_2 \textbf{a}_3^{+2} \textbf{k}_3 \textbf{K}^{2}_4 \textbf{k}^{3}_5 \textbf{K}_6\\ {} &{} + \; [3]_1 \textbf{a}^-_1 \textbf{k}^{2}_1 \textbf{K}_2 \textbf{a}^-_3 \textbf{k}_3 \textbf{A}^+_4 \textbf{K}_4 \textbf{k}^{3}_5 \textbf{K}_6 + \textbf{k}^{3}_1 \textbf{A}_2^{-2} \textbf{a}_3^{+3} \textbf{K}^{2}_4 \textbf{k}^{3}_5 \textbf{K}_6\\ {} &{} + \; [3]_1 \textbf{k}^{3}_1 \textbf{A}^-_2 \textbf{K}_2 \textbf{a}^+_3 \textbf{k}_3 \textbf{K}_4 \textbf{a}^+_5 \textbf{k}^{2}_5 \textbf{K}_6 + \textbf{k}^{3}_1 \textbf{K}^{2}_2 \textbf{a}_3^{-3} \textbf{A}_4^{+2} \textbf{k}^{3}_5 \textbf{K}_6\\ {} &{} + \; [3]_1 \textbf{k}^{3}_1 \textbf{K}^{2}_2 \textbf{a}_3^{-2} \textbf{k}_3 \textbf{A}^+_4 \textbf{a}^+_5 \textbf{k}^{2}_5 \textbf{K}_6,\\ \lambda _1^{-1}\pi _\textbf{1}(\xi _1)&{}=\textbf{a}^+_1 \textbf{k}^{-1}_1,\\ \lambda _2^{-1}\pi _\textbf{1}(\xi _2)&{}=\textbf{a}_1^{-3} \textbf{A}^+_2 \textbf{K}^{-1}_2 + [2]_2 \textbf{k}^{3}_1 \textbf{A}^-_2 \textbf{k}^{-3}_3 \textbf{A}^+_4 \textbf{K}^{-1}_4 - q [3]_1 \textbf{k}^{3}_1 \textbf{A}^-_2 \textbf{k}^{-1}_3 \textbf{A}^+_4 \textbf{K}^{-1}_4\\ {} &{} +\; [3]_1 \textbf{a}_1^{-2} \textbf{k}_1 \textbf{K}^{-1}_2 \textbf{a}^+_3 \textbf{k}^{-1}_3 + [3]_1 \textbf{a}^-_1 \textbf{k}^{2}_1 \textbf{a}^-_3 \textbf{k}^{-2}_3 \textbf{A}^+_4 \textbf{K}^{-1}_4\\ {} &{} + \; [3]_1 \textbf{a}^-_1 \textbf{k}^{2}_1 \textbf{k}^{-1}_3 \textbf{K}^{-1}_4 \textbf{a}^+_5 \textbf{k}^{-1}_5 + \textbf{k}^{3}_1 \textbf{K}_2 \textbf{K}^{-1}_4 \textbf{k}^{-3}_5 \textbf{A}^+_6 \textbf{K}^{-1}_6\\ {} &{} +\; [3]_1 \textbf{a}^-_1 \textbf{k}^{2}_1 \textbf{A}^-_2 \textbf{K}^{-1}_2 \textbf{a}_3^{+2} \textbf{k}^{-2}_3 + [3]_1 \textbf{k}^{3}_1 \textbf{A}^-_2 \textbf{a}^+_3 \textbf{k}^{-2}_3 \textbf{K}^{-1}_4 \textbf{a}^+_5 \textbf{k}^{-1}_5\\ {} &{} + \; \textbf{k}^{3}_1 \textbf{A}_2^{-2} \textbf{K}^{-1}_2 \textbf{a}_3^{+3} \textbf{k}^{-3}_3 + [3]_1 \textbf{k}^{3}_1 \textbf{K}_2 \textbf{a}^-_3 \textbf{k}^{-1}_3 \textbf{K}^{-2}_4 \textbf{a}_5^{+2} \textbf{k}^{-2}_5\\ {} &{} + \; \textbf{k}^{3}_1 \textbf{K}_2 \textbf{A}^-_4 \textbf{K}^{-2}_4 \textbf{a}_5^{+3} \textbf{k}^{-3}_5 + \textbf{k}^{3}_1 \textbf{K}_2 \textbf{a}_3^{-3} \textbf{k}^{-3}_3 \textbf{A}_4^{+2} \textbf{K}^{-2}_4\\ {} &{} + \; [3]_1 \textbf{k}^{3}_1 \textbf{K}_2 \textbf{a}_3^{-2} \textbf{k}^{-2}_3 \textbf{A}^+_4 \textbf{K}^{-2}_4 \textbf{a}^+_5 \textbf{k}^{-1}_5,\\ \\ \pi _\textbf{2}(\sigma _1)&{}=\textbf{k}_2 \textbf{K}_3 \textbf{k}^{2}_4 \textbf{K}_5 \textbf{k}_6,\\ \pi _\textbf{2}(\sigma _2)&{}=\textbf{K}_1 \textbf{k}^{3}_2 \textbf{K}^{2}_3 \textbf{k}^{3}_4 \textbf{K}_5,\\ \pi _\textbf{2}(\sigma _1e_1)&{}=\textbf{K}_1 \textbf{k}^{2}_2 \textbf{K}_3 \textbf{k}_4 \textbf{a}^+_6 + \textbf{A}^-_1 \textbf{a}^+_2 \textbf{K}_3 \textbf{k}^{2}_4 \textbf{K}_5 \textbf{k}_6 + \textbf{K}_1 \textbf{k}^{2}_2 \textbf{K}_3 \textbf{a}^-_4 \textbf{A}^+_5 \textbf{k}_6\\ {} &{} + \; \textbf{K}_1 \textbf{a}_2^{-2} \textbf{A}^+_3 \textbf{k}^{2}_4 \textbf{K}_5 \textbf{k}_6 + [2]_1 \textbf{K}_1 \textbf{a}^-_2 \textbf{k}_2 \textbf{a}^+_4 \textbf{k}_4 \textbf{K}_5 \textbf{k}_6 + \textbf{K}_1 \textbf{k}^{2}_2 \textbf{A}^-_3 \textbf{a}_4^{+2} \textbf{K}_5 \textbf{k}_6,\\ \pi _\textbf{2}(\sigma _2e_2)&{}=\textbf{A}^+_1 \textbf{k}^{3}_2 \textbf{K}^{2}_3 \textbf{k}^{3}_4 \textbf{K}_5,\\ \lambda _1^{-1}\pi _\textbf{2}(\xi _1)&{}=\textbf{A}^-_1 \textbf{a}^+_2 \textbf{k}^{-1}_2 + [2]_1 \textbf{K}_1 \textbf{a}^-_2 \textbf{K}^{-1}_3 \textbf{a}^+_4 \textbf{k}^{-1}_4 + \textbf{K}_1 \textbf{a}_2^{-2} \textbf{k}^{-1}_2 \textbf{A}^+_3 \textbf{K}^{-1}_3\\ {} &{} + \; \textbf{K}_1 \textbf{k}_2 \textbf{a}^-_4 \textbf{k}^{-2}_4 \textbf{A}^+_5 \textbf{K}^{-1}_5 + \textbf{K}_1 \textbf{k}_2 \textbf{k}^{-1}_4 \textbf{K}^{-1}_5 \textbf{a}^+_6 \textbf{k}^{-1}_6 + \textbf{K}_1 \textbf{k}_2 \textbf{A}^-_3 \textbf{K}^{-1}_3 \textbf{a}_4^{+2} \textbf{k}^{-2}_4,\\ \lambda _2^{-1}\pi _\textbf{2}(\xi _2)&{}=\textbf{A}^+_1 \textbf{K}^{-1}_1. \end{array} $$

Note that ${\pi _\textbf{i}(\sigma _i)}$ is invertible. Comparing these formulas with Lemma 10.13 by using (10.43), the equality (10.40) is directly checked. Thus Proposition 10.7 is established for ${G_2}$.

In terms of the intertwiner F in Chap. 8, Theorem 10.6 implies

$$\begin{aligned} E^{a,b,c,d,e,f}_\textbf{2} = \sum _{i,j,k,l,m,n} F^{a b c d e f}_{i j k l m n}\,E^{n,m,l,k,j,i}_\textbf{1}. \end{aligned}$$

(10.96)

The weight conservation (8.29) assures the equality of weights of the two sides.

10.5 Tetrahedron and 3D Reflection Equations from PBW Bases

The relation (10.58) serves as an auxiliary linear system by which the tetrahedron equation (2.6) is established as the non-linear consistency condition. To see this, consider a PBW basis (10.6) of ${U_q^+(A_3)}$ having the form ${E^{a,b,c,d,e}_{1,2,3,1,2,1}}$. In addition to $E^{...ab...}_{...13...} = E^{...ba...}_{...31...}$, we may apply (10.58) as

$$\begin{aligned} E^{...abc...}_{...121...} = \sum R^{abc}_{ijk}E^{...kji...}_{...212...}, \quad E^{...abc...}_{...212...} = \sum R^{abc}_{ijk}E^{...kji...}_{...121...} \end{aligned}$$

(10.97)

reflecting the $U_q^+(A_2)$ subalgebra structure. Then we have

$$\begin{aligned}&E^{a,b,c,d,e,f}_{1,2,3,1,2,1} = E^{a,b,d,c,e,f}_{1,2,1,3,2,1} = \sum R^{abd}_{a_1b_1d_1}E^{d_1,b_1,a_1,c,e,f}_{2,1,2,3,2,1} \\&= \sum R^{abd}_{a_1b_1d_1}R^{a_1ce}_{a_2c_1e_1}E^{d_1,b_1,e_1,c_1,a_2,f}_{2,1,3,2,3,1} \\&= \sum R^{abd}_{a_1b_1d_1}R^{a_1ce}_{a_2c_1e_1}E^{d_1,e_1,b_1,c_1,f,a_2}_{2,3,1,2,1,3} \\&= \sum R^{abd}_{a_1b_1d_1}R^{a_1ce}_{a_2c_1e_1}R^{b_1c_1f}_{b_2c_2f_1} E^{d_1,e_1,f_1,c_2,b_2,a_2}_{2,3,2,1,2,3} \\&= \sum R^{abd}_{a_1b_1d_1}R^{a_1ce}_{a_2c_1e_1}R^{b_1c_1f}_{b_2c_2f_1} R^{d_1e_1f_1}_{d_2e_2f_2}E^{f_2,e_2,d_2,c_2,b_2,a_2}_{3,2,3,1,2,3}. \end{aligned}$$

There is another route going from $E^{a,b,c,d,e,f}_{1,2,3,1,2,1}$ to $E^{f_2,e_2,d_2,c_2,b_2,a_2}_{3,2,3,1,2,3}$ as

$$\begin{aligned}&E^{a,b,c,d,e,f}_{1,2,3,1,2,1} = \sum R^{def}_{d_1e_1f_1}E^{a,b,c,f_1,e_1,d_1}_{1,2,3,2,1,2} \\&= \sum R^{def}_{d_1e_1f_1}R^{bcf_1}_{b_1c_1f_2}E^{a,f_2,c_1,b_1,e_1,d_1}_{1,3,2,3,1,2} \\&= \sum R^{def}_{d_1e_1f_1}R^{bcf_1}_{b_1c_1f_2}E^{f_2,a,c_1,e_1,b_1,d_1}_{3,1,2,1,3,2} \\&= \sum R^{def}_{d_1e_1f_1}R^{bcf_1}_{b_1c_1f_2}R^{a c_1e_1}_{a_1 c_2 e_2} E^{f_2,e_2,c_2,a_1,b_1,d_1}_{3,2,1,2,3,2} \\&= \sum R^{def}_{d_1e_1f_1}R^{bcf_1}_{b_1c_1f_2}R^{a c_1e_1}_{a_1 c_2 e_2} R^{a_1b_1d_1}_{a_2b_2d_2} E^{f_2,e_2,c_2,d_2,b_2,a_2}_{3,2,1,3,2,3} \\&= \sum R^{def}_{d_1e_1f_1}R^{bcf_1}_{b_1c_1f_2}R^{a c_1e_1}_{a_1 c_2 e_2} R^{a_1b_1d_1}_{a_2b_2d_2} E^{f_2,e_2,d_2,c_2,b_2,a_2}_{3,2,3,1,2,3}. \end{aligned}$$

Comparison of them leads to

$$\begin{aligned} \sum R^{abd}_{a_1b_1d_1}R^{a_1ce}_{a_2c_1e_1}R^{b_1c_1f}_{b_2c_2f_1} R^{d_1e_1f_1}_{d_2e_2f_2} = \sum R^{def}_{d_1e_1f_1}R^{bcf_1}_{b_1c_1f_2}R^{a c_1e_1}_{a_1 c_2 e_2} R^{a_1b_1d_1}_{a_2b_2d_2} \end{aligned}$$

(10.98)

for arbitrary a, b, c, d, e, f and $a_2,b_2,c_2,d_2,e_2,f_2$. The sums are over $a_1,b_1,c_1,d_1,e_1,f_1\in {\mathbb Z}_{\ge 0}$ on both sides. They are finite sums due to the weight conservation (3.48). The identity (10.98) reproduces the tetrahedron equation (2.9).

A similar proof of the 3D reflection equation (4.3) is possible based on (10.80). We now start from a PBW basis (10.6) of $U_q^+(C_3)$ having the form $E^{a,b,c,d,e,f,g,h,i}_{3,2,3,1,2,1,3,2,1}$ and apply (10.97) and $E^{...abcd...}_{...3232...} = \sum K^{abcd}_{ijkl}E^{...lkji...}_{...2323...}$. The two routes are as follows:

$$\begin{aligned}&E^{a,b,c,d,e,f,g,h,i}_{3,2,3,1,2,1,3,2,1} = E^{a,b,d,c,e,g,f,h,i}_{3,2,1,3,2,3,1,2,1} = \sum R^{fhi}_{f_1h_1i_1}E^{a,b,d,c,e,g,i_1,h_1,f_1}_{3,2,1,3,2,3,2,1,2} \\&= \sum R^{fhi}_{f_1h_1i_1}K^{cegi_1}_{c_1e_1g_1i_2} E^{a,b,d,i_2,g_1,e_1,c_1,h_1,f_1}_{3,2,1,2,3,2,3,1,2} \\&= \sum R^{fhi}_{f_1h_1i_1}K^{cegi_1}_{c_1e_1g_1i_2}R^{bdi_2}_{b_1d_1i_3} E^{a,i_3,d_1,b_1,g_1,e_1,c_1,h_1,f_1}_{3,1,2,1,3,2,3,1,2} \\&= \sum R^{fhi}_{f_1h_1i_1}K^{cegi_1}_{c_1e_1g_1i_2}R^{bdi_2}_{b_1d_1i_3} E^{i_3,a,d_1,g_1,b_1,e_1,h_1,c_1,f_1}_{1,3,2,3,1,2,1,3,2} \\&= \sum R^{fhi}_{f_1h_1i_1}K^{cegi_1}_{c_1e_1g_1i_2}R^{bdi_2}_{b_1d_1i_3}R^{b_1e_1h_1}_{b_2e_2h_2} E^{i_3,a,d_1,g_1,h_2,e_2,b_2,c_1,f_1}_{1,3,2,3,2,1,2,3,2} \\&= \sum R^{fhi}_{f_1h_1i_1}K^{cegi_1}_{c_1e_1g_1i_2}R^{bdi_2}_{b_1d_1i_3}R^{b_1e_1h_1}_{b_2e_2h_2} K^{ad_1g_1h_2}_{a_1d_2g_2h_3} E^{i_3,h_3,g_2,d_2,a_1,e_2,b_2,c_1,f_1}_{1,2,3,2,3,1,2,3,2} \\&= \sum R^{fhi}_{f_1h_1i_1}K^{cegi_1}_{c_1e_1g_1i_2}R^{bdi_2}_{b_1d_1i_3}R^{b_1e_1h_1}_{b_2e_2h_2} K^{ad_1g_1h_2}_{a_1d_2g_2h_3} E^{i_3,h_3,g_2,d_2,e_2,a_1,b_2,c_1,f_1}_{1,2,3,2,1,3,2,3,2} \\&= \sum R^{fhi}_{f_1h_1i_1}K^{cegi_1}_{c_1e_1g_1i_2}R^{bdi_2}_{b_1d_1i_3}R^{b_1e_1h_1}_{b_2e_2h_2} K^{ad_1g_1h_2}_{a_1d_2g_2h_3}K^{a_1b_2c_1f_1}_{a_2b_3c_2f_2} E^{i_3,h_3,g_2,d_2,e_2,f_2,c_2,b_3,a_2}_{1,2,3,2,1,2,3,2,3} \\&= \sum R^{fhi}_{f_1h_1i_1}K^{cegi_1}_{c_1e_1g_1i_2}R^{bdi_2}_{b_1d_1i_3}R^{b_1e_1h_1}_{b_2e_2h_2} K^{ad_1g_1h_2}_{a_1d_2g_2h_3}K^{a_1b_2c_1f_1}_{a_2b_3c_2f_2}R^{d_2e_2f_2}_{d_3e_3f_3} E^{i_3,h_3,g_2,f_3,e_3,d_3,c_2,b_3,a_2}_{1,2,3,1,2,1,3,2,3} \\ \text {and} \\&E^{a,b,c,d,e,f,g,h,i}_{3,2,3,1,2,1,3,2,1} = \sum R^{def}_{d_1e_1f_1}E^{a,b,c,f_1,e_1,d_1,g,h,i}_{3,2,3,2,1,2,3,2,1}\\&= \sum R^{def}_{d_1e_1f_1}K^{abcf_1}_{a_1b_1c_1 f_2} E^{f_2,c_1,b_1,a_1,e_1,d_1,g,h,i}_{2,3,2,3,1,2,3,2,1}\\&= \sum R^{def}_{d_1e_1f_1}K^{abcf_1}_{a_1b_1c_1 f_2} E^{f_2,c_1,b_1,e_1,a_1,d_1,g,h,i}_{2,3,2,1,3,2,3,2,1}\\&= \sum R^{def}_{d_1e_1f_1}K^{abcf_1}_{a_1b_1c_1 f_2}K^{a_1d_1gh}_{a_2d_2g_1h_1} E^{f_2,c_1,b_1,e_1,h_1,g_1,d_2,a_2,i}_{2,3,2,1,2,3,2,3,1}\\&= \sum R^{def}_{d_1e_1f_1}K^{abcf_1}_{a_1b_1c_1 f_2}K^{a_1d_1gh}_{a_2d_2g_1h_1} R^{b_1e_1h_1}_{b_2e_2h_2} E^{f_2,c_1,h_2,e_2,b_2,g_1,d_2,a_2,i}_{2,3,1,2,1,3,2,3,1}\\&= \sum R^{def}_{d_1e_1f_1}K^{abcf_1}_{a_1b_1c_1 f_2}K^{a_1d_1gh}_{a_2d_2g_1h_1} R^{b_1e_1h_1}_{b_2e_2h_2} E^{f_2,h_2,c_1,e_2,g_1,b_2,d_2,i,a_2}_{2,1,3,2,3,1,2,1,3}\\&= \sum R^{def}_{d_1e_1f_1}K^{abcf_1}_{a_1b_1c_1 f_2}K^{a_1d_1gh}_{a_2d_2g_1h_1} R^{b_1e_1h_1}_{b_2e_2h_2}R^{b_2d_2i}_{b_3d_3i_1} E^{f_2,h_2,c_1,e_2,g_1,i_1,d_3,b_3,a_2}_{2,1,3,2,3,2,1,2,3}\\&= \sum R^{def}_{d_1e_1f_1}K^{abcf_1}_{a_1b_1c_1 f_2}K^{a_1d_1gh}_{a_2d_2g_1h_1} R^{b_1e_1h_1}_{b_2e_2h_2}R^{b_2d_2i}_{b_3d_3i_1}K^{c_1e_2g_1i_1}_{c_2e_3g_2i_2} E^{f_2,h_2,i_2,g_2,e_3,c_2,d_3,b_3,a_2}_{2,1,2,3,2,3,1,2,3}\\&= \sum R^{def}_{d_1e_1f_1}K^{abcf_1}_{a_1b_1c_1 f_2}K^{a_1d_1gh}_{a_2d_2g_1h_1} R^{b_1e_1h_1}_{b_2e_2h_2}R^{b_2d_2i}_{b_3d_3i_1}K^{c_1e_2g_1i_1}_{c_2e_3g_2i_2} R^{f_2h_2i_2}_{f_3h_3i_3} E^{i_3,h_3,f_3,g_2,e_3,c_2,d_3,b_3,a_2}_{1,2,1,3,2,3,1,2,3}\\&= \sum R^{def}_{d_1e_1f_1}K^{abcf_1}_{a_1b_1c_1 f_2}K^{a_1d_1gh}_{a_2d_2g_1h_1} R^{b_1e_1h_1}_{b_2e_2h_2}R^{b_2d_2i}_{b_3d_3i_1}K^{c_1e_2g_1i_1}_{c_2e_3g_2i_2} R^{f_2h_2i_2}_{f_3h_3i_3} E^{i_3,h_3,g_2,f_3,e_3,d_3,c_2,b_3,a_2}_{1,2,3,1,2,1,3,2,3}. \end{aligned}$$

Thus we get

$$\begin{aligned} \begin{aligned}&\sum R^{fhi}_{f_1h_1i_1}K^{cegi_1}_{c_1e_1g_1i_2}R^{bdi_2}_{b_1d_1i_3}R^{b_1e_1h_1}_{b_2e_2h_2} K^{ad_1g_1h_2}_{a_1d_2g_2h_3}K^{a_1b_2c_1f_1}_{a_2b_3c_2f_2}R^{d_2e_2f_2}_{d_3e_3f_3} \\&= \sum R^{def}_{d_1e_1f_1}K^{abcf_1}_{a_1b_1c_1 f_2}K^{a_1d_1gh}_{a_2d_2g_1h_1} R^{b_1e_1h_1}_{b_2e_2h_2}R^{b_2d_2i}_{b_3d_3i_1}K^{c_1e_2g_1i_1}_{c_2e_3g_2i_2} R^{f_2h_2i_2}_{f_3h_3i_3} \end{aligned} \end{aligned}$$

(10.99)

for any a, b, c, d, e, f, g, h, i and $a_2, b_3, c_2, d_3, e_3, f_3, g_2, h_3, i_3$. The sums are over $a_1, b_1, b_2, c_1, d_1, d_2, e_1, e_2, f_1, f_2, g_1, h_1, h_2, i_1, i_2 \in {\mathbb Z}_{\ge 0}$ on both sides. They are finite sums due to the weight conservation (3.48) and (5.65). The identity (10.99) reproduces the 3D reflection equation (4.5). By a parallel argument for $U^+_q(B_3)$, the 3D reflection equation of type B (6.31) can also be derived.

10.6 $\chi$-Invariants

Theorem 10.6 implies non-trivial identities in (a completion of) $U^+_q(\mathfrak {g})$. They are stated as invariance of some infinite products under the anti-involution $\chi$ introduced in (10.42). Here we illustrate the derivation along $\mathfrak {g}=A_2$ and present the results for $C_2$ and $G_2$. The point is to translate the boundary vectors in Sects. 3.6.1, 5.8.1 and 8.6.1 in terms of the PBW basis.

Let us write the boundary vectors (3.132) as

$$\begin{aligned} |\eta _s\rangle = \sum _{m\ge 0}\eta _{s,m} |m\rangle \quad (s=1,2). \end{aligned}$$

(10.100)

By comparing the coefficient of $|a\rangle \otimes |b\rangle \otimes |c\rangle$ on the two sides of (3.143) using (3.47), we get

$$\begin{aligned} \sum _{i,j,k}\eta _{s,i}\eta _{s,j}\eta _{s,k}R^{abc}_{ijk} = \eta _{s,a}\eta _{s,b}\eta _{s,c}. \end{aligned}$$

(10.101)

In view of (3.63), this is equivalent to

$$\begin{aligned} \sum _{a,b,c}\hat{\eta }_{s,a}\hat{\eta }_{s,b}\hat{\eta }_{s,c}R^{abc}_{ijk} = \hat{\eta }_{s,i}\hat{\eta }_{s,j}\hat{\eta }_{s,k},\qquad \hat{\eta }_{s,a} = (q^2)_a\eta _{s,a}. \end{aligned}$$

(10.102)

Multiply this by $E_\textbf{2}^{k,j,i}$ and sum over $i,j,k\in {\mathbb Z}_{\ge 0}$. From (10.46) and (10.6), the RHS gives

$$\begin{aligned} \sum _{i,j,k}E^{k,j,i}_\textbf{2} \hat{\eta }_{s,i}\hat{\eta }_{s,j}\hat{\eta }_{s,k} = \Bigl (\sum _{k}\hat{\eta }_{s,k}\frac{(b_1)^k}{[k]_1!}\Bigr ) \Bigl (\sum _{j}\hat{\eta }_{s,j}\frac{(b_2)^j}{[j]_1!}\Bigr ) \Bigl (\sum _{i}\hat{\eta }_{s,i}\frac{(b_3)^i}{[i]_1!}\Bigr ). \end{aligned}$$

(10.103)

As for the LHS, we have

$$\begin{aligned} \sum _{a,b,c}\Bigl (\sum _{i,j,k}R^{abc}_{ijk} E^{k,j,i}_\textbf{2}\Bigr ) \hat{\eta }_{s,a}\hat{\eta }_{s,b}\hat{\eta }_{s,c}&= \sum _{a,b,c}E^{a,b,c}_\textbf{1} \hat{\eta }_{s,a}\hat{\eta }_{s,b}\hat{\eta }_{s,c} \nonumber \\&= \chi \Bigl (\sum _{a,b,c}E^{c,b,a}_\textbf{2} \hat{\eta }_{s,a}\hat{\eta }_{s,b}\hat{\eta }_{s,c}\Bigr ). \end{aligned}$$

(10.104)

The first equality is due to (10.58) which is the $A_2$ case of the main theorem of this chapter. The second equality is (10.42). The quantity within $\chi$ in (10.104) is equal to (10.103). Thus we find that (10.103) is $\chi$-invariant. To describe the result neatly we introduce a quantum-dilogarithm-type infinite product:

$$\begin{aligned} \Theta _q(z) = \sum _m \frac{q^{m(m-1)/2}z^m}{(q)_m} = (-z;q)_\infty . \end{aligned}$$

(10.105)

Then a direct calculation using (3.132) yields

$$\begin{aligned} \sum _{m}\hat{\eta }_{s,m}\frac{z^m}{[m]_1!} = {\left\{ \begin{array}{ll} \Theta _q((1-q^2)z) &{} s=1, \\ \Theta _{q^4}(q(1-q^2)^2z^2) &{} s=2. \end{array}\right. } \end{aligned}$$

(10.106)

Thus we get a corollary of Theorem 10.6 and Proposition 3.28.

Corollary 10.15

Set $c_i = (1-q^2)b_i, c'_i = \chi (c_i) \in U^+_q(A_2)\, (i=1,2,3)$ using $b_i$ in (10.46) and the anti-algebra involution $\chi$ in (10.41). Then the following equalities are valid:

$$\begin{aligned} \Theta _q(c_1)\Theta _q(c_2)\Theta _q(c_3)&= \Theta _q(c'_3)\Theta _q(c'_2)\Theta _q(c'_1), \end{aligned}$$

(10.107)

$$\begin{aligned} \Theta _{q^4}(qc_1^2)\Theta _{q^4}(qc_2^2)\Theta _{q^4}(qc_3^2)&= \Theta _{q^4}(q{c'_3}^2)\Theta _{q^4}(q{c'_2}^2)\Theta _{q^4}(q{c'_1}^2). \end{aligned}$$

(10.108)

Remark 10.16

By the rescaling $e_1\rightarrow x e_1, e_2 \rightarrow ye_2$ with parameters x, y, the identity (10.107) is seemingly generalized to

$$ \Theta _q(xc_1)\Theta _q(xyc_2)\Theta _q(yc_3) = \Theta _q(yc'_3)\Theta _q(xyc'_2)\Theta _q(xc'_1) $$

containing x, y in the same manner as spectral parameters in the Yang–Baxter equation. The same holds for (10.108). Similar remarks apply to the $C_2$ and $G_2$ cases in the sequel where the parameters arranged along the positive roots fit the spectral parameters in the reflection and the $G_2$ reflection equations.

The product (10.107) is expanded as

$$\begin{aligned} \begin{aligned}&\Theta _q(c_1)\Theta _q(c_2)\Theta _q(c_3) \\&= 1+ (1+q)(e_1 + e_2) + q(1+q)(e_1^2+e_2^2) + (1+q)(e_1e_2+e_2e_1) \\&+ (1+q)^2(e_1e_2e_1+e_2e_1e_2) + \frac{q^3(1-q^2)^2(e_1^3+e_2^3)}{(1-q)(1-q^3)} + \frac{q^6(1-q^2)^3(e_1^4+e_2^4)}{(1-q)(1-q^3)(1-q^4)} \\&+\frac{q^2(1-q^2)^2(e_1e_2e_1^2+e_1^2e_2e_1+e_2e_1e_2^2+e_2^2e_1e_2)}{(1-q)(1-q^3)} \\&+ \frac{q(1-q^2)^2\bigl ( q(e_1^2e_2^2+e_2^2e_1^2) +(1+q)^2 e_1e_2^2e_1-q(1+q^2)e_2e_1^2e_2\bigr )}{(1-q)(1-q^4)} + \cdots , \end{aligned} \end{aligned}$$

(10.109)

where the q-Serre relation (10.45) has been used to make it manifestly invariant under $\chi$. Similarly, (10.108) is expanded as

$$\begin{aligned} \begin{aligned}&\Theta _{q^4}(qc_1^2)\Theta _{q^4}(qc_2^2)\Theta _{q^4}(qc_3^2) \\&= 1+ \frac{q(1-q^2)^2(e_1^2 + e_2^2)}{1-q^4} +\frac{q^6(1-q^2)^4(e_1^4+e_2^4)}{(1-q^4)(1-q^8)} \\&+ \frac{q^2(1-q^2)^3(e_1^2e_2^2+e_2^2e_1^2-(1+q^2)e_2e_1^2e_2)}{(1-q^4)^2} + \cdots . \end{aligned} \end{aligned}$$

(10.110)

For $C_2$, the relevant results are (10.80) and Proposition 5.21 concerning the boundary vectors in (5.118)–(5.120). There are three identities corresponding to the choices of (r, k) in (5.136).

Corollary 10.17

Set $c_i = (1-q^4)b_i\,(i=1,3), c_i= (1-q^2)b_i\,(i=2,4)$ and $c'_i = \chi (c_i) \in U^+_q(C_2)\, (i=1,2,3,4)$ using $b_i$ in (10.60) and the anti-algebra involution $\chi$ in (10.41). Then the following equalities are valid:

$$\begin{aligned} \Theta _{q^2}(c_1)\Theta _q(c_2)\Theta _{q^2}(c_3)\Theta _q(c_4)&= \Theta _{q}(c'_4)\Theta _{q^2}(c'_3)\Theta _{q}(c'_2)\Theta _{q^2}(c'_1), \end{aligned}$$

(10.111)

$$\begin{aligned} \Theta _{q^2}(c_1)\Theta _{q^4}(qc_2^2)\Theta _{q^2}(c_3)\Theta _{q^4}(qc_4^2)&=\Theta _{q^4}(q{c'_4}^2)\Theta _{q^2}(c'_3)\Theta _{q^4}(q{c'_2}^2)\Theta _{q^2}(c'_1), \end{aligned}$$

(10.112)

$$\begin{aligned} \Theta _{q^8}(q^2c_1^2)\Theta _{q^4}(qc_2^2)\Theta _{q^8}(q^2c_3^2)\Theta _{q^4}(qc_4^2)&=\Theta _{q^4}(q{c'_4}^2)\Theta _{q^8}(q^2{c'_3}^2) \Theta _{q^4}(q{c'_2}^2)\Theta _{q^8}(q^2{c'_1}^2). \end{aligned}$$

(10.113)

For $G_2$, the relevant result is Conjecture 8.9 for the boundary vector (8.61) and (10.96).

Corollary 10.18

Set $c_i = (1-q^6)b_i\,(i=1,3,5), c_i= (1-q^2)b_i\,(i=2,4,6)$ and $c'_i = \chi (c_i) \in U^+_q(G_2)\, (i=1,\ldots , 6)$ using $b_i$ in (10.82) and the anti-algebra involution $\chi$ in (10.41). If Conjecture 8.9 holds, the following equality is valid:

$$\begin{aligned} \begin{aligned}&\Theta _{q^3}(c_1)\Theta _{q}(c_2) \Theta _{q^3}(c_3)\Theta _{q}(c_4) \Theta _{q^3}(c_5)\Theta _{q}(c_6) \\&=\Theta _{q}(c'_6)\Theta _{q^3}(c'_5) \Theta _{q}(c'_4)\Theta _{q^3}(c'_3) \Theta _{q}(c'_2)\Theta _{q^3}(c'_1). \end{aligned} \end{aligned}$$

(10.114)

10.7 Bibliographical Notes and Comments

This chapter is an extended exposition of [102]. The braid group action (10.5) is introduced in [111]. The formulation of quantized coordinate ring in this chapter follows [76, 139]. See also [43] and [29, Chap. 7]. For quantum cluster algebra structure of quantized coordinate rings, see [52].

The Peter–Weyl-type Theorem 10.1 is taken from [76, Proposition 7.2.2]. Proposition 10.4 is a special case of [66, Corollary 9.1.4]. In [149, Theorem 7], $U^+_q(\mathfrak {g})$ has been identified with an explicit subalgebra of $A_q(\mathfrak {g})_{\mathcal {S}}$. A proof of Theorem 10.5 adapted to the present setting has been given in [102, Sect. 3.2]. The main result, Theorem 10.6, is due to [102, Theorem 5]. The case $\mathfrak {g}=A_2$ was obtained earlier in the pioneering work [131]. Remark 10.8 is due to [141], where a unified conceptual proof of Theorem 10.6 has been attained. See also [128] for yet another proof using the representation theory of q-boson algebra and the Drinfeld pairing of $U_q(\mathfrak {g})$. The multiplication rule on the PBW bases like Lemmas 10.9, 10.11 and 10.13 plays an important role also in the study of the positive principal series representations and modular double [61]. For type $C_2$, one can adjust the definition of $E^A_\textbf{i}$ in (10.6) with that in [148] by setting $v=q^{-1}$. Some of the results like Lemma 10.13 have also been obtained in [147]. An analogue of Sect. 10.5 for quantum superalgebras has been argued in [151].