Higher level representation of the elliptic quantum group $U_{q,p}(\widehat{\mathfrak{sl}}_2)$ and its integrability

Abstract

By using an elliptic analogue of the Drinfeld coproduct, we construct the level-$(k+1)$ representation of the elliptic quantum group $U_{q,p}(\widehat{\mathfrak{sl}}_2)$ from the level-1 highest weight representation. The quantum Z-algebra of level-$(k+1)$ is realized. We also find the elliptic analogue of the condition of integrability for higher level modules constructed by the Drinfeld coproduct. This also enables us to express $\Delta^k(e(z))\Delta^k(e(zq^2))\dots\Delta^k(e(zq^{2(N-1)}))$ and $\Delta^k(f(z))\Delta^k(f(zq^2))\Delta^k(f(zq^{-2}))\dots\Delta^k(f(zq^{-2(N-1)}))$ as vertex operators of the level-$(k+1)$ bosons.