Isomorphism between the R-matrix and Drinfeld presentations of quantum affine algebra: type C

Naihuan Jing, Ming Liu and Alexander Molev

Abstract

An explicit isomorphism between the \(R\)-matrix and Drinfeld presentations of the quantum affine algebra in type \(A\) was given by Ding and I. Frenkel (1993). We show that this result can be extended to types \(B\), \(C\) and \(D\) and give a detailed construction for type \(C\) in this paper. In all classical types the Gauss decomposition of the generator matrix in the \(R\)-matrix presentation yields the Drinfeld generators. To prove that the resulting map is an isomorphism we follow the work of E. Frenkel and Mukhin (2002) in type \(A\) and employ the universal \(R\)-matrix to construct the inverse map. A key role in our construction is played by an embedding theorem which allows us to consider the quantum affine algebra of rank \(n-1\) in the \(R\)-matrix presentation as a subalgebra of the corresponding algebra of rank \(n\) of the same type.