2019/03/01 by Mounzer, Elie, Zegers, Robin
#FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1903.00418
We define the double quantum affinization Uq(\mathfrak a1) of type \mathfraka1 as a topological Hopf algebra. We prove that it admits a subalgebra Uq'(\mathfrak a1) whose completion is (bicontinuously) isomorphic to the completion of the quantum toroidal algebra Uq(\mathfrak a1), defined as the (simple) quantum affinization of the untwisted affine Kac-Moody Lie algebra \mathfraksl2 of type \mathfrak a1, 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 \mathfrak B -- technically a split extension \mathfrak B ≅ \mathfrak B \ltimes P^\vee by the coweight lattice P^\vee -- of the affine braid group \mathfrak B of type \mathfrak a1 on that completion of Uq(\mathfrak a1). 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 Uq(\mathfrak a1) and briefly discuss the consequences regarding the representation theory of quantum toroidal algebras.