\(RLL\)-realization of two-parameter quantum affine algebra in type \(D_n^{(1)}\) (Q6568722)

A natural open question is how to work out the \(RLL\)-realizations for the quantum affine algebras \(U_{r,s}(\widehat {\mathfrak g})\) of types \(B_n^{(1)}, C^{(1)}_n, D^{(1)}_n\). The difficulty lies in that there has been no information about two-parameter basic \(R\)-matrices in the corresponding cases for many years, let alone their Yang-Baxterizations.\N\NThe paper under review is the first to give the \(RLL\)-realization of the two-parameter quantum affine algebra \(U_{r,s}(\widehat{\mathfrak{so}_{2n}})\) by the Reshetikhin and Semenov-Tian-Shanski method. The authors firstly obtain the basic \(R\)-matrix of the two-parameter quantum group \(U=U_{r,s}\mathcal(\mathfrak{so}_{2n})\) via its weight representation theory and determine its \(R\)-matrix with spectral parameters for the two-parameter quantum affine algebra \(U=U_{r,s}\mathcal(\widehat{\mathfrak{so}_{2n}})\). Then they give the isomorphism between Faddeev-Reshetikhin-Takhtajan and Drinfeld-Jimbo definitions of \(U_{r,s}\mathcal(\mathfrak{so}_{2n})\), and further the spectral parameter dependent \(R\)-matrix \(\hat{R}(z)\) as the Yang-Baxterzation of the basic \(R\)-matrix. Finally, they obtain the commutation relations between Gaussian generators, and arrive at its \(RLL\)-formalism of the Drinfeld realization of two-parameter quantum affine algebra \(U=U_{r,s}\mathcal(\widehat{\mathfrak{so}_{2n}})\).