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Lie subalgebras of differential operators on the super circle. (English) Zbl 1049.17021
In mid 90s, V. Kac and A. Radul [Commun. Math. Phys. 157, 429–457 (1993; Zbl 0826.17027)] discovered a nice relationship between the Lie algebra of differential operators on the circle, \(\mathcal{W}_{1+\infty}\), and the Lie algebra of inifinite matrices \(\text{ gl}_\infty\). They were able to describe all interesting representations of \(\mathcal{W}_{1+\infty}\) by using a convenient series of embeddings of \(\mathcal{W}_{1+\infty}\) into \(\text{ gl}_\infty\). Since then several generalizations have been obtained. In particular, it is of interest to:
(i) study classical subalgebras of \(\mathcal{W}_{1+\infty}\) and their relationship with classical Lie algebras of infinite matrices [see V. Kac, W. Wang and C. Yan, Adv. Math. 139, 56–140 (1998; Zbl 0938.17018)],
(ii) explore a possible superextension, by replacing the circle by super-circle, and differential operators by superdifferential operators.
In this paper Wang and Cheng pursue the latter direction. Even though it requires an effort to obtain all results in parallel to Kac-Radul’s and Kac-Wang-Yan’s papers, all the results are expected. The exposition is concise and nicely written.

MSC:
17B65 Infinite-dimensional Lie (super)algebras
17B69 Vertex operators; vertex operator algebras and related structures
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