Twisted quantum affinizations and quantization of extended affine Lie algebras

Abstract

In this paper, for an arbitrary Kac-Moody Lie algebra $\mathfrak g$ and a diagram automorphism $\mu$ of $\mathfrak g$ satisfying certain natural linking conditions, we introduce and study a $\mu$-twisted quantum affinization algebra $\mathcal U_\hbar\left(\hat{\mathfrak g}_\mu\right)$ of $\mathfrak g$. When $\mathfrak g$ is of finite type, $\mathcal U_\hbar\left(\hat{\mathfrak g}_\mu\right)$ is Drinfeld's current algebra realization of the twisted quantum affine algebra. When $\mu=\mathrm{id}$ and $\mathfrak g$ in affine type, $\mathcal U_\hbar\left(\hat{\mathfrak g}_\mu\right)$ is the quantum toroidal algebra introduced by Ginzburg, Kapranov and Vasserot. As the main results of this paper, we first prove a triangular decomposition for $\mathcal U_\hbar\left(\hat{\mathfrak g}_\mu\right)$. Second, we give a simple characterization of the affine quantum Serre relations on restricted $\mathcal U_\hbar\left(\hat{\mathfrak g}_\mu\right)$-modules in terms of "normal order products". Third, we prove that the category of restricted $\mathcal U_\hbar\left(\hat{\mathfrak g}_\mu\right)$-modules is a monoidal category and hence obtain a topological Hopf algebra structure on the "restricted completion" of $\mathcal U_\hbar\left(\hat{\mathfrak g}_\mu\right)$. Last, we study the classical limit of $\mathcal U_\hbar\left(\hat{\mathfrak g}_\mu\right)$ and abridge it to the quantization theory of extended affine Lie algebras. In particular, based on a classification result of Allison-Berman-Pianzola, we obtain the $\hbar$-deformation of all nullity $2$ extended affine Lie algebras.