Twisted quantum affinizations and quantization of extended affine lie algebras

Abstract

In this paper, for an arbitrary Kac-Moody Lie algebra g {\mathfrak g} and a diagram automorphism μ \mu of g {\mathfrak g} satisfying certain natural linking conditions, we introduce and study a μ \mu -twisted quantum affinization algebra U ℏ ( g ^ μ ) {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right ) of g {\mathfrak g} . When g {\mathfrak g} is of finite type, U ℏ ( g ^ μ ) {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right ) is Drinfeld’s current algebra realization of the twisted quantum affine algebra. When μ = i d \mu =\mathrm {id} and g {\mathfrak g} in affine type, U ℏ ( g ^ μ ) {\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 U ℏ ( g ^ μ ) {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right ) . Second, we give a simple characterization of the affine quantum Serre relations on restricted U ℏ ( g ^ μ ) {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right ) -modules in terms of “normal order products”. Third, we prove that the category of restricted U ℏ ( g ^ μ ) {\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 U ℏ ( g ^ μ ) {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right ) . Last, we study the classical limit of U ℏ ( g ^ μ ) {\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 ℏ \hbar -deformation of all nullity 2 2 extended affine Lie algebras.