*(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 $\phantom{\rule{4.pt}{0ex}}{\text{gl}}_{\infty}$. 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 $\phantom{\rule{4.pt}{0ex}}{\text{gl}}_{\infty}$. 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.