Free field representation of level-\(k\) Yangian double \(DY(\text{sl}_2)_k\) and deformation of Wakimoto modules (Q1359681)

The Yangian is a Hopf algebra associated with the rational solutions of the quantum Yang-Baxter equation (YBE). Recently, \textit{K. Iohara} and \textit{M. Kohno} [Lett. Math. Phys. 37, 319-328 (1996; Zbl 0867.17013)], \textit{K. Iohara} [J. Phys. A, Math. Gen. 29, 4593-4621 (1996; Zbl 0899.17009)], \textit{S. Khoroshkin} [Sémin. Congr. 2, 119-135 (1995; Zbl 0887.17007)] and \textit{S. Khoroshkin, D. Lebedev} and \textit{S. Pakuliak} [Phys. Lett. A 222, 381-392 (1996)] have succeeded in defining a central extension of the Yangian double \({\mathcal D}Y(\text{sl}_n)\) by means of the Drinfeld currents and in obtaining infinite-dimensional representations which are analogous to the level-1 highest weight representations of the affine Lie algebra \(\widehat{\text{sl}}_n\) and the quantum affine algebra \(U_q(\widehat {\text{sl}}_n)\). Here the author extends this to the higher level representation for the central extension of \({\mathcal D}Y (\text{sl}_2)\). He considers a representation of the level-\(k\) \((\neq 0,-2)\) Drinfeld currents in terms of the three free bosonic fields \(\Phi\), \(\phi\) and \(\chi\). This can be regarded as a deformation of the so-called Wakimoto construction of the affine Lie algebras. As a consequence, an analogue of the level-\(k\) Wakimoto modules is obtained as a certain restriction of the Fock spaces of these bosons. He also gives a boson representation of vertex operators intertwining the resultant modules.