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.