On double quantum affinization: 1. Type ${\mathfrak{a}}_1$

Abstract

We define the double quantum affinization ${\ddot{\mathrm{U}}}_q({\mathfrak{a}}_1)$ of type ${\mathfrak{a}}_1$ as a topological Hopf algebra. We prove that it admits a subalgebra ${\ddot{\mathrm{U}}}_q^{\prime }({\mathfrak{a}}_1)$ whose completion is (bicontinuously) isomorphic to the completion of the quantum toroidal algebra ${\dot{\mathrm{U}}}_q({\dot{\mathfrak{a}}}_1)$, defined as the (simple) quantum affinization of the untwisted affine Kac-Moody Lie algebra ${\dot{\mathfrak{sl}}}_2$ of type ${\dot{\mathfrak{a}}}_1$, equipped with a certain topology inherited from its natural $\mathbb{Z}$-grading. The isomorphism is constructed by means of a bicontinuous action by automorphisms of an affinized version $\ddot{\mathfrak{B}}$ -- technically a split extension $\ddot{\mathfrak{B}}\cong \dot{\mathfrak{B}}⋉P^{\vee }$ by the coweight lattice $P^{\vee }$ -- of the affine braid group $\dot{\mathfrak{B}}$ of type ${\dot{\mathfrak{a}}}_1$ on that completion of ${\dot{\mathrm{U}}}_q({\dot{\mathfrak{a}}}_1)$. It can be regarded as an affinized version of the Damiani-Beck isomorphism, familiar from the quantum affine setting. We eventually prove the corresponding triangular decomposition of ${\ddot{\mathrm{U}}}_q({\mathfrak{a}}_1)$ and briefly discuss the consequences regarding the representation theory of quantum toroidal algebras.

40 pages

Cited by in corpus (1)

• Weight-finite modules over the quantum affine and double quantum affine algebras of type ${\mathfrak{a}}_1$