Non-weight modules over the mirror Heisenberg-Virasoro algebra

• Published in: Science China. Mathematics Vol. 65; no. 11; pp. 2243 - 2254
• Main Authors: Gao, Dongfang, Ma, Yao, Zhao, Kaiming
• Format: Journal Article
• Language: English
• Published: Beijing Science China Press 01.11.2022; Springer Nature B.V
• Subjects: Applications of Mathematics; Mathematical analysis; Mathematics; Mathematics and Statistics; Modules; Tensors; Whittaker module; 17B66; 17B68; 17B20; 17B10; 17B65; mirror Heisenberg-Virasoro algebra; free module; tensor product; irreducible module
• ISSN: 1674-7283; 1869-1862
• DOI: 10.1007/s11425-021-1939-5

In this paper, we study irreducible non-weight modules over the mirror Heisenberg-Virasoro algebra D , including Whittaker modules, U ( ℂ d 0 ) -free modules and their tensor products. More precisely, we give the necessary and sufficient conditions for the Whittaker modules to be irreducible. We determine all the D -module structures on U ( ℂ d 0 ) , and find the necessary and sufficient conditions for these modules to be irreducible. At last, we determine the necessary and sufficient conditions for the tensor products of Whittaker modules and U ( ℂ d 0 ) -free modules to be irreducible, and obtain that any two such tensor products are isomorphic if and only if the corresponding Whittaker modules and U ( ℂ d 0 ) -free modules are isomorphic. These lead to many new irreducible non-weight modules over D .