Classical limit of quantum Borcherds-Bozec algebras

Abstract

Let $\mathfrak{g}$ be a Borcherds-Bozec algebra, $U(\mathfrak{g})$ be its universal enveloping algebra and $U_{q}(\mathfrak{g})$ be the corresponding quantum Borcherds-Bozec algebra. We show that the classical limit of $U_{q}(\mathfrak{g})$ is isomorphic to $U(\mathfrak{g})$ as Hopf algebras. Thus $U_{q}(\mathfrak{g})$ can be regarded as a quantum deformation of $U(\mathfrak{g})$. We also give explicit formulas for the commutation relations among the generators of $U_{q}(\mathfrak{g})$.