Vertex representations of quantum affine algebras.

We construct vertex representations of quantum affine algebras of ADE type, which were first introduced in greater generality by Drinfeld and Jimbo. The limiting special case of our construction is the untwisted vertex representation of affine Lie algebras of Frenkel-Kac and Segal. Our representation is given by means of a new type of vertex operator corresponding to the simple roots and satisfying the defining relations. In the case of the quantum affine algebra of type A, we introduce vertex operators corresponding to all the roots and determine their commutation relations. This provides an analogue of a Chevalley basis of the affine Lie algebra [unk](n) in the basic representation.