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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, 𝒲 1+ , and the Lie algebra of inifinite matrices gl . They were able to describe all interesting representations of 𝒲 1+ by using a convenient series of embeddings of 𝒲 1+ into gl . Since then several generalizations have been obtained. In particular, it is of interest to:

(i) study classical subalgebras of 𝒲 1+ 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:
17B65Infinite-dimensional Lie (super)algebras
17B69Vertex operators; vertex operator algebras and related structures