2022/09/29 by Tao, Wenqing
#16D25 #16D60 #16S40 #16W20 #20G42 #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2209.14713
The Heisenberg double Dq(E2) of the quantum Euclidean group Oq(E2) is the smash product of Oq(E2) with its Hopf dual Uq(\mathfrake2). For the algebra Dq(E2), explicit descriptions of its prime, primitive, and maximal spectra are obtained. All prime factors of Dq(E2) are presented as generalized Weyl algebras. As a result, we obtain that the algebra Dq(E2) has no finite-dimensional representations, and that Dq(E2) cannot have a Hopf algebra structure. The automorphism groups of the quantum Euclidean group and its Heisenberg double are determined. Some centralizers are explicitly described via generators and defining relations. This enables us to give a classification of simple weight modules, and the so-called a-weight modules, over the algebra Dq(E2).