On the Structure of Quantum Affine Superalgebra $U_v(\mathrm{A}(0,2{)}^{(4)})$

Abstract

We study the quantum affine superalgebra $U_v(\mathrm{A}(0,2{)}^{(4)})$ defined abstractly by Yamane, when $v$ is not a root of unity. We prove that the defining relations satisfied by the Chevalley generators of $U_v^{+}(\mathrm{A}(0,2{)}^{(4)})$, or equivalently the corresponding Nichols algebra $B(\mathbb{V})$, are the quantum Serre relations, which means the Drinfeld–Jimbo-type realization of $U_v(\mathrm{A}(0,2{)}^{(4)})$ is obtained. Based on Lyndon word theory, we construct distinguished imaginary root vectors and give an estimation of their comultiplication in the form of generating functions. With these results, we determine all the root multiplicities and give a PBW basis of $U_v^{+}(\mathrm{A}(0,2{)}^{(4)})$. We also describe concretely all the commutation relations among real root vectors and distinguished imaginary root vectors using generating functions, i.e., the Drinfeld second realization is obtained for $U_v^{+}(\mathrm{A}(0,2{)}^{(4)})$.