Twisted tensor products of quantum affine vertex algebras and coproducts

Abstract

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

arXiv commentarXiv admin note: substantial text overlap with arXiv:2310.03571