On the second realization for the positive part of $U_q(\widehat{sl_2})$ of equitable type

Abstract

The equitable presentation of the quantum algebra $U_q(\widehat{sl_2})$ is considered. This presentation was originally introduced by Ito and Terwilliger. In this paper, following Terwilliger’s recent works the (nonstandard) positive part of $U_q(\widehat{sl_2})$ of equitable type $U_q^{IT,+}$ and its second realization (current algebra) $U_q^{T,+}$ are introduced and studied. A presentation for $U_q^{T,+}$ is given in terms of a K-operator satisfying a Freidel–Maillet-type equation and a condition on its quantum determinant. Realizations of the K-operator in terms of Ding–Frenkel L-operators are considered, from which an explicit injective homomorphism from $U_q^{T,+}$ to a subalgebra of Drinfeld’s second realization (current algebra) of $U_q(\widehat{sl_2})$ is derived, and the comodule algebra structure of $U_q^{T,+}$ is characterized. The central extension of $U_q^{T,+}$ and its relation with Drinfeld’s second realization of $U_q(\widehat{gl_2})$ is also described using the framework of Freidel–Maillet algebras.

Appendix A. Drinfeld–Jimbo and Drinfeld (second realization) presentation of \(U_q(\widehat{sl_2})\)

For the quantum affine Kac–Moody algebra $U_q(\widehat{sl_2})$, two standard presentations are recalled. The Drinfeld–Jimbo presentation $U_q^{DJ}$ and the Drinfeld (second) presentation $U_q^{Dr}$, see, e.g., [7, p.392].

1.1 A.1. Drinfeld–Jimbo presentation $U_q^{DJ}$

Define the extended Cartan matrix $\{a_{ij}\}$ ($a_{ii}=2$, $a_{ij}=-2$ for $i\ne j$). The quantum affine algebra $U_{q}(\widehat{sl_2})$ over ${{\mathbb {C}}}(q)$ is generated by $\{E_j,F_j,K_j^{\pm 1}\}$, $j\in \{0,1\}$ which satisfy the defining relations

$$\begin{aligned}&K_iK_j=K_jK_i, \, K_iK_i^{-1}=K_i^{-1}K_i=1, \, K_iE_jK_i^{-1}= q^{a_{ij}}E_j ,\, K_iF_jK_i^{-1}= q^{-a_{ij}}F_j\,\\&[E_i,F_j]=\delta _{ij}\frac{K_i-K_i^{-1}}{q-q^{-1}} \end{aligned}$$

together with the $q-$Serre relations ($i\ne j$)

$$\begin{aligned} \big [E_i, \big [E_i, \big [E_i,E_j \big ]_{q} \big ]_{q^{-1}} \big ]= & {} 0\ , \end{aligned}$$

(A.1)

$$\begin{aligned} \big [F_i, \big [F_i, \big [F_i,F_j \big ]_{q} \big ]_{q^{-1}} \big ]= & {} 0\ . \end{aligned}$$

(A.2)

The product $C=K_0K_1$ is the central element of the algebra.

The Hopf algebra structure is ensured by the existence of a comultiplication $\Delta$ , antipode $\mathcal{S}$ and a counit $\mathcal{E}$ with

$$\begin{aligned} \Delta (E_i)= & {} 1 \otimes E_i + E_i \otimes K_i \ , \nonumber \\ \Delta (F_i)= & {} F_i \otimes 1 + K_i^{-1}\otimes F_i\ ,\nonumber \\ \Delta (K_i)= & {} K_i\otimes K_i\ , \end{aligned}$$

(A.3)

$$\begin{aligned} \mathcal{S}(E_i)=-E_iK_i^{-1} ,\, \mathcal{S}(F_i)=-K_iF_i ,\, \mathcal{S}(K_i)=K_i^{-1}, \, \mathcal{S}({1})=1\ \end{aligned}$$

and

$$\begin{aligned} \mathcal{E}(E_i)=\mathcal{E}(F_i)=0 ,\, \mathcal{E}(K_i)=1,\, \mathcal{E}(1)=1. \end{aligned}$$

1.2 A.2. Drinfeld’s second realization $U_q^{Dr}$

A second presentation for the quantum affine algebra $U_q(\widehat{sl_2})$, known as the Drinfeld’s second realization, is now recalled. In [11], it is shown that $U_q(\widehat{sl_2})$ is isomorphic to the associative algebra over ${{\mathbb {C}}}(q)$ with generators $\{{{\textsf {x}}}_k^{\pm }, {\textsf {h}}_{\ell }, {\textsf {K}}^{\pm 1} |k\in {{\mathbb {Z}}},\ell \in {{\mathbb {Z}}}\backslash \{0\} \}$, central elements $C^{\pm 1/2}$ and the following relations (see, e.g., [7, Theorem 12.2.1]):

$$\begin{aligned}&C^{1/2}C^{-1/2}=1\ ,\quad {\textsf {K}}{\textsf {K}}^{-1}={\textsf {K}}^{-1}{\textsf {K}}=1 \ , \end{aligned}$$

(A.4)

$$\begin{aligned}&\big [ {\textsf {h}}_k,{\textsf {h}}_\ell \big ] = \delta _{k+\ell ,0}\frac{1}{k}\big [ 2k \big ]_q \frac{C^k-C^{-k}}{q-q^{-1}} \ , \qquad \big [{\textsf {h}}_k, {\textsf {K}}^{\pm 1} \big ] =0\ , \end{aligned}$$

(A.5)

$$\begin{aligned}&\big [ {\textsf {h}}_k, {\textsf {x}}^\pm _\ell \big ] = \pm \frac{1}{k}\big [ 2k\big ]_q C^{\mp |k|/2} {\textsf {x}}^\pm _{k+\ell }\ , \end{aligned}$$

(A.6)

$$\begin{aligned}&{\textsf {K}}{\textsf {x}}^\pm _k {\textsf {K}}^{-1} = q^{\pm 2} {\textsf {x}}^\pm _{k} \ , \end{aligned}$$

(A.7)

$$\begin{aligned}&{\textsf {x}}^\pm _{k+1} {\textsf {x}}^\pm _\ell -q^{\pm 2} {\textsf {x}}^\pm _{\ell } {\textsf {x}}^\pm _{k+1} = q^{\pm 2} {\textsf {x}}^\pm _{k} {\textsf {x}}^\pm _{\ell +1} - {\textsf {x}}^\pm _{\ell +1} {\textsf {x}}^\pm _k \ , \end{aligned}$$

(A.8)

$$\begin{aligned}&\big [ {\textsf {x}}^+_k, {\textsf {x}}^-_\ell \big ] = \frac{(C^{(k-\ell )/2} \psi _{k+\ell } - C^{-(k-\ell )/2}\phi _{k+\ell })}{q-q^{-1}}\ , \end{aligned}$$

(A.9)

where the $\psi _{k}$ and $\phi _{k}$ are defined by the following equalities of formal power series in the indeterminate z:

$$\begin{aligned} \psi (z)= & {} \sum _{k=0}^\infty \psi _{k} z^{-k} = {\textsf {K}}\exp \left( (q-q^{-1}) \sum _{k=1}^{\infty } {\textsf {h}}_k z^{-k} \right) , \end{aligned}$$

(A.10)

$$\begin{aligned} \phi (z)= & {} \sum _{k=0}^\infty \phi _{-k} z = {\textsf {K}}^{-1} \exp \left( - (q-q^{-1}) \sum _{k=1}^{\infty } {\textsf {h}}_{-k} z \right) . \end{aligned}$$

(A.11)

Note that there exists an automorphism such that:

$$\begin{aligned} \theta :&{\textsf {x}}^\pm _k \mapsto {\textsf {x}}^\mp _k ,\, {\textsf {h}}_k \mapsto -{\textsf {h}}_k ,\, {\textsf {K}}\mapsto {\textsf {K}},\, C \mapsto C^{-1},\, q \mapsto q^{-1}. \end{aligned}$$

(A.12)

An isomorphism $U_q^{DJ}\rightarrow U_q^{Dr}$ is given by (see, e.g., [7, p. 393]:

$$\begin{aligned} K_0 \mapsto C{\textsf {K}}^{-1} , \, K_1 \mapsto {\textsf {K}},\, E_1 \mapsto {\textsf {x}}_0^+ , \, E_0 \mapsto {\textsf {x}}_1^-{\textsf {K}}^{-1} ,\, F_1 \mapsto {\textsf {x}}_0^- , \, F_0 \mapsto {\textsf {K}}{\textsf {x}}_{-1}^+\ .\nonumber \\ \end{aligned}$$

(A.13)

Note that it is still an open problem to find the complete Hopf algebra isomorphism between $U_q^{DJ}$ and $U_q^{Dr}$. Only partial information is known, see, e.g., [6, Section 4.4].