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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.

17B65Infinite-dimensional Lie (super)algebras
17B69Vertex operators; vertex operator algebras and related structures
Full Text: DOI arXiv
[1] Awata, H., Fukuma, M., Matsuo, Y. and Odake, S., Quasifinite highest weight modules over the super W1+\infty algebra, Comm. Math. Phys., 170 (1995), 151-179. · Zbl 0832.17026 · doi:10.1007/BF02099443
[2] , Character and determinant formulae of quasifinite representations of the W algebra, Comm. Math. Phys., 172 (1995), 377-400. · Zbl 0853.17023 · doi:10.1007/BF02099433
[3] , Representation theory of the W1+\infty algebra, Quantum field theory, integrable models and beyond (Kyoto, 1994). Progr. Theor. Phys. Suppl., 118 (1995), 343-373.
[4] Bouwknegt, P. and Schoutens, K., W-symmetry in conformal field theory, Phys. Rep., 223 (1993), 183-276.
[5] Cheng, S.-J. and Lam, N., Infinite-dimensional Lie Superalgebras and Hook Schur Functions, Comm. Math. Phys., 238 (2003), 95-118. · Zbl 1042.17019 · doi:10.1007/s00220-003-0819-3 · arxiv:math/0206034
[6] Cheng, S.-J. and Wang, W., Howe Duality for Lie Superalgebras, Compositio Math., 128 (2001), 55-94. · Zbl 1023.17017 · doi:10.1023/A:1017594504827 · arxiv:math/0008093
[7] , Remarks on the Schur-Howe-Sergeev Duality, Lett. Math. Phys., 52 (2000), 143-153.
[8] Frenkel, E., Kac, V., Radul, A. and Wang, W., W1+\infty and W(glN ) with central charge N , Comm. Math. Phys., 170 (1995), 337-357. · Zbl 0838.17028 · doi:10.1007/BF02108332
[9] Feigin, B. and Frenkel, E., Integrals of motions and quantum groups, Lect. Notes in Math., 1620 Berlin-Heidelberg-New York, Springer Verlag, 1996. · Zbl 0885.58034 · arxiv:hep-th/9310022
[10] Feingold, A. and Frenkel, I., Classical affine algebras, Adv. Math., 56 (1985), 117-172. · Zbl 0601.17012 · doi:10.1016/0001-8708(85)90027-1
[11] Howe, R., Remarks on classical invariant theory, Trans. Amer. Math. Soc., 313 (1989), 539-570. · Zbl 0674.15021 · doi:10.2307/2001418