The Heisenberg double of the quantum Euclidean group and its representations

The Heisenberg double D q ( E 2 ) of the quantum Euclidean group O q ( E 2 ) is the smash product of O q ( E 2 ) with its Hopf dual U q ( e 2 ) . For the algebra D q ( E 2 ), explicit descriptions of its prime, primitive and maximal spectra are obtained. All the prime factors of D q ( E 2 ) are pres...

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Bibliographic Details
Published in:Science China. Mathematics 2023-08, Vol.66 (8), p.1713-1736
Main Author: Tao, Wen-Qing
Format: Article
Language:English
Subjects:
Algebra
Applications of Mathematics
Automorphisms
Group theory
Mathematics
Mathematics and Statistics
Modules
Oil field equipment
Representations
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Summary:The Heisenberg double D q ( E 2 ) of the quantum Euclidean group O q ( E 2 ) is the smash product of O q ( E 2 ) with its Hopf dual U q ( e 2 ) . For the algebra D q ( E 2 ), explicit descriptions of its prime, primitive and maximal spectra are obtained. All the prime factors of D q ( E 2 ) are presented as generalized Weyl algebras. As a result, we obtain that the algebra D q ( E 2 ) has no finite-dimensional representations, and D q ( E 2 ) 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 D q ( E 2 ).
ISSN:1674-7283
1869-1862
DOI:10.1007/s11425-021-2043-7