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We define the double quantum affinization $\ddot{\mathrm{U}}_q(\mathfraka_1)$ of type $\mathfrak{a}_1$ as a topological Hopf algebra. We prove that itadmits a subalgebra $\ddot{\mathrm{U}}_q'(\mathfrak a_1)$ 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) quantumaffinization of the untwisted affine Kac-Moody Lie algebra$\dot{\mathfrak{sl}}_2$ of type $\dot{\mathfrak a}_1$, equipped with a certaintopology inherited from its natural $\mathbb Z$-grading. The isomorphism isconstructed by means of a bicontinuous action by automorphisms of an affinizedversion $\ddot{\mathfrak B}$ -- technically a split extension $\ddot{\mathfrakB} \cong \dot{\mathfrak B} \ltimes P^\vee$ by the coweight lattice $P^\vee$ --of the affine braid group $\dot{\mathfrak B}$ of type $\dot{\mathfrak a}_1$ onthat completion of $\dot{\mathrm{U}}_q(\dot{\mathfrak a}_1)$. It can beregarded as an affinized version of the Damiani-Beck isomorphism, familiar fromthe quantum affine setting. We eventually prove the corresponding triangulardecomposition of $\ddot{\mathrm{U}}_q(\mathfrak a_1)$ and briefly discuss theconsequences regarding the representation theory of quantum toroidal algebras.

On double quantum affinization: 1. Type $\mathfrak a_1$