$U(\frak h)$-free modules over the Lie algebras of differential operators

Munayim Dilxat; Shoulan Gao; Dong Liu; Limeng Xia

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arXiv · 2212.02106

$U(\frak h)$-free modules over the Lie algebras of differential operators

Munayim Dilxat·Shoulan Gao·Dong Liu·Limeng Xia

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Abstract

In this paper, we consider some non-weight modules over the Lie algebra of Weyl type. First, we determine the modules whose restriction to $U(\frak h)$ are free of rank $1$ over the Lie algebra of differential operators on the circle. Then we determine the necessary and sufficient conditions for the tensor products of quasi-finite highest weight modules and $U(\frak h)$-free modules to be irreducible, and obtain that any two such tensor products are isomorphic if and only if the corresponding highest weight modules and $U(\frak h)$-free modules are isomorphic. Finally, we extend such results to the Lie algebras of differential operators in the general case.

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BibTeX RIS

Munayim Dilxat, Shoulan Gao, Dong Liu, Limeng Xia. 2022-12-05. $U(\frak h)$-free modules over the Lie algebras of differential operators. https://arxiv.org/abs/2212.02106

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