Twisted tensor products of quantum affine vertex algebras and coproducts

Let $g$ be a symmetrizable Kac-Moody Lie algebra, and let $V_{\hat{g},ħ}^{\ell }$, $L_{\hat{g},ħ}^{\ell }$ be the quantum affine vertex algebras constructed in [11]. For any complex numbers ℓ and ${\ell }^{\prime }$, we present an ħ-adic quantum vertex algebra homomorphism Δ from $V_{\hat{g},ħ}^{\ell +{\ell }^{\prime }}$ to the twisted tensor product ħ-adic quantum vertex algebra $V_{\hat{g},ħ}^{\ell }\hat{\otimes }{V}_{\hat{g},ħ}^{{\ell }^{\prime }}$. In addition, if both ℓ and ${\ell }^{\prime }$ are positive integers, we show that Δ induces an ħ-adic quantum vertex algebra homomorphism from $L_{\hat{g},ħ}^{\ell +{\ell }^{\prime }}$ to the twisted tensor product ħ-adic quantum vertex algebra $L_{\hat{g},ħ}^{\ell }\hat{\otimes }{L}_{\hat{g},ħ}^{{\ell }^{\prime }}$. Moreover, we prove the coassociativity of Δ.