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Lie subalgebras of differential operators on the super circle. (English) Zbl 1049.17021
In mid 90s, {\it V. Kac} and {\it 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, $\cal{W}_{1+\infty}$, and the Lie algebra of inifinite matrices $\text{ gl}_\infty$. They were able to describe all interesting representations of $\cal{W}_{1+\infty}$ by using a convenient series of embeddings of $\cal{W}_{1+\infty}$ into $\text{ gl}_\infty$. Since then several generalizations have been obtained. In particular, it is of interest to: (i) study classical subalgebras of $\cal{W}_{1+\infty}$ and their relationship with classical Lie algebras of infinite matrices [see {\it V. Kac}, {\it W. Wang} and {\it 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
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References:
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