Double Yangian and the universal R-matrix

Abstract

We describe the double Yangian of the general linear Lie algebra ${\mathfrak{gl}}_N$ by following a general scheme of Drinfeld. This description is based on the construction of the universal $R$-matrix for the Yangian. To make the exposition self contained, we include the proofs of all necessary facts about the Yangian itself. In particular, we describe the centre of the Yangian by using its Hopf algebra structure, and provide a proof of the analogue of the Poincaré-Birkhoff-Witt theorem for the Yangian based on its representation theory. This proof extends to the double Yangian, thus giving a description of its underlying vector space.

40 pages, proof of Lemma 7.1 amended

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• Principal subspaces for the quantum affine vertex algebra in type $A_1^{(1)}$
• A note on constructing quasi modules for quantum vertex algebras from twisted Yangians
• Semi-infinite construction for the double Yangian of type $A_1^{(1)}$