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Zbl 1049.17021
Cheng, Shun-Jen; Wang, Weiqiang
Lie subalgebras of differential operators on the super circle. (English)
[J] Publ. Res. Inst. Math. Sci. 39, No. 3, 545-600 (2003). ISSN 0034-5318; ISSN 1663-4926/e

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: \par (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)], \par (ii) explore a possible superextension, by replacing the circle by super-circle, and differential operators by superdifferential operators. \par 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.
[Antun Milas (Albany)]

MSC 2000:: *17B65 Infinite-dimensional Lie algebras
17B69 Vertex operators

Keywords: differential operators; W-algebras; conformal field theory

Citations: Zbl 0826.17027; Zbl 0938.17018

Cited in: Zbl 1105.17013

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