A quantum shuffle approach to quantum affine super algebra of type $C(2)^{(2)}$ and its equitable presentation

In this study, we focus on the positive part $U_q^{+}$ of the quantum affine superalgebra $U_q(C(2)^{(2)})$. This algebra admits a presentation with two two generators $e_{\alpha}$ and $e_{\delta-\alpha}$, which satisfy the cubic $q$-Serre relations. According to the work of Khoroshkin-Lukierski-Tolstoy, the Damiani and the Beck $PBW$ bases exist for this superalgebra. In this paper, we utilize the $q$-shuffle superalgebra and Catalan words to present these two bases in a closed-form expression. Ultimately, we present the bosonization of $U_q(C(2)^{(2)})$.