TWISTED QUANTUM AFFINIZATIONS AND QUANTIZATION OF EXTENDED AFFINE LIE ALGEBRAS

In this paper, for an arbitrary Kac-Moody Lie algebra g and a diagram automorphism mu of g satisfying certain natural linking conditions, we introduce and study a mu-twisted quantum affinization algebra U-(h) over bar((g) over cap (mu)) of g. When g is of finite type, U-(h) over bar((g) over cap (mu)) is Drinfeld's current algebra realization of the twisted quantum affine algebra. When mu = id and g in affine type, U-(h) over bar((g) over cap (mu)) 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-(h) over bar((g) over cap (mu)). Second, we give a simple characterization of the affine quantum Serre relations on restricted U-(h) over bar((g) over cap (mu))-modules in terms of "normal order products". Third, we prove that the category of restricted U-(h) over bar((g) over cap (mu))-modules is a monoidal category and hence obtain a topological Hopf algebra structure on the "restricted completion" of U-(h) over bar((g) over cap (mu)). Last, we study the classical limit of U-(h) over bar((g) over cap (mu)) 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 (h) over bar -deformation of all nullity 2 extended affine Lie algebras.