A quantum shuffle approach to admissible quantum affine (super-)algebra of types $A_1^{(1)}$ and $C(2)^{(2)}$ and their equitable presentations

Abstract: In this study, we focus on the positive part $U_q^{+}$ of the admissible quantum affine algebra $\mathcal{U}q(\widehat{\mathfrak{s l}_2})$, newly defined in \cite{HZ}, and the quantum affine superalgebra $U_q(C(2)^{(2)})$. Both of these algebras have presentations involving two generators, $e{\alpha}$ and $e_{\delta-\alpha}$, which satisfy the cubic $q$-Serre relations. According to the works of Hu-Zhuang and Khoroshkin-Lukierski-Tolstoy, there exist the Damiani and the Beck $PBW$ bases for these two (super)algebras. In this paper, we employ the $q$-shuffle (super)algebra and Catalan words to present these two bases in a closed-form expression. Ultimately, the equitable presentations of $\mathcal{U}_q(\widehat{\mathfrak{sl}_2})$ and the bosonization of $U_q(C(2)^{(2)})$ are presented.