## 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.

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 Monday, March 4, 2019