Twisted tensor products of quantum affine vertex algebras and coproducts

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