# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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, ${𝒲}_{1+\infty }$, and the Lie algebra of inifinite matrices $\phantom{\rule{4.pt}{0ex}}{\text{gl}}_{\infty }$. They were able to describe all interesting representations of ${𝒲}_{1+\infty }$ by using a convenient series of embeddings of ${𝒲}_{1+\infty }$ 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 ${𝒲}_{1+\infty }$ 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
##### Keywords:
differential operators; W-algebras; conformal field theory