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The Heisenberg double of the quantum Euclidean group and its representations

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Abstract

The Heisenberg double Dq(E2) of the quantum Euclidean group \({{\cal O}_q}({E_2})\) is the smash product of \({{\cal O}_q}({E_2})\) with its Hopf dual \({U_q}({e_2})\). For the algebra Dq(E2), explicit descriptions of its prime, primitive and maximal spectra are obtained. All the 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 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).

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11601167)

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  1. School of Mathematical Science, Yangzhou University, Yangzhou, 225002, China

    Wen-Qing Tao

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  1. Wen-Qing Tao
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Correspondence to Wen-Qing Tao.

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Tao, WQ. The Heisenberg double of the quantum Euclidean group and its representations. Sci. China Math. 66, 1713–1736 (2023). https://doi.org/10.1007/s11425-021-2043-7

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  • DOI: https://doi.org/10.1007/s11425-021-2043-7

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