Twisted tensor products of quantum affine vertex algebras and coproducts

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