Bakas, Ioannis; Khesin, Boris; Kiritsis, Elias The logarithm of the derivative operator and higher spin algebras of \(W_{\infty{}}\) type. (English) Zbl 0765.35049 Commun. Math. Phys. 151, No. 2, 233-243 (1993). Summary: The authors use the notion of the logarithm of the derivative operator to describe \(W_ \infty\) type algebras as central extensions of the algebra of differential operators. They also provide closed formulae for the truncations of \(W_{1+\infty}\) to higher spin algebras with \(s\geq M\), for all \(M\geq 2\). The results are extended to matrix valued differential operators, introducing a logarithmic generalization of the Maurer-Cartan cocycle. Cited in 17 Documents MSC: 35Q53 KdV equations (Korteweg-de Vries equations) 58J70 Invariance and symmetry properties for PDEs on manifolds 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations 35S05 Pseudodifferential operators as generalizations of partial differential operators 17B68 Virasoro and related algebras Keywords:Virasoro algebra; algebra of differential operators; Maurer-Cartan cocycle PDFBibTeX XMLCite \textit{I. Bakas} et al., Commun. Math. Phys. 151, No. 2, 233--243 (1993; Zbl 0765.35049) Full Text: DOI References: [1] Moya, J.: Proc. Camb. Phil. Soc.45, 99 (1949); Baker, G.: Phys. Rev.109, 2198 (1958); Fairlie, D.B.: Proc. Camb. Phil. Soc.60, 581 (1964) · doi:10.1017/S0305004100000487 [2] Bakas, I.: Phys. LettB228, 57 (1989); Commun. Math. Phys.134, 487 (1990) · doi:10.1016/0370-2693(89)90525-X [3] Zamolodchikov, A.B.: Theor. Math. Phys.65, 1205 (1985); Zamolodchikov, A.B., Fateev, V.A.: Nucl. Phys.B280 [FS18], 644 (1987); Fateev, V.A., Lykyanov, S.L.: Int. J. Mod. Phys.A3, 507 (1988) · doi:10.1007/BF01036128 [4] Pope, C.N., Romans, L.J., Shen, X.: Phys. Lett.B236, 173 (1990); Nucl. Phys.B339, 191 (1990) · doi:10.1016/0370-2693(90)90822-N [5] Bakas, I., Kiritsis, E.: Nucl. Phys.B343, 185 (1990); Int. J. Mod. Phys.A6, 2871 (1991) · doi:10.1016/0550-3213(90)90600-I [6] Wodzicki, M.: Duke Math. J.54, 641 (1987); Getzler, E.: Proc. Am. Math. Soc.104, 729 (1988); Feigin, B.L.: Russ. Math. Surv.43(2), 157 (1988); Beilinson, A. Manin, Yu., Schechtman, V.: In:K-Theory, Arithmetic and Geometry. Lect. Notes in Math. vol.1289, Manin, Yu. (ed.) Berlin, Heidelberg, New York: Springer 1987 · Zbl 0635.18010 · doi:10.1215/S0012-7094-87-05426-3 [7] Gelfand, I.M., Fuchs, D.B.: Funct. Anal. Appl.2, 342 (1968) · Zbl 0176.11501 · doi:10.1007/BF01075687 [8] Kravchenko, O.S., Khesin, B.A.: Funct. Anal. Appl.25, 152 (1991) · Zbl 0729.35154 · doi:10.1007/BF01079603 [9] Kac, V.G., Peterson, D.H.: Proc. Nat. Acad. Sci. USA78, 3308 (1981) · Zbl 0469.22016 · doi:10.1073/pnas.78.6.3308 [10] Radul, A.O.: JETP Lett.50, 341 (1989) [11] Pope, C.N., Romans, L.J., Shen, X.: Phys. Lett.B 242, 401 (1990) · doi:10.1016/0370-2693(90)91782-7 [12] Fairlie, D.B., Nuyts, J.: Commun. Math. Phys.134, 413 (1990) · Zbl 0719.17019 · doi:10.1007/BF02097709 [13] Radul, A.O., Vaysburd, I.: Differential Operators andW-Algebra. Racah Institute preprint 9/91 (1991) [14] Yu, F., Wu, Y.-S.: Phys. Lett.B263, 220 (1991) · doi:10.1016/0370-2693(91)90589-I [15] Bergshoeff, E., Pope, C.N., Romans, L.J., Sezgin, E., Shen, X.: Phys. Lett.B 245, 447 (1990); Depireux, D.A.: Phys. Lett.B252, 586 (1990) · doi:10.1016/0370-2693(90)90672-S [16] Bakas, I., Kiritsis, E.: Mod. Phys. Lett.A5, 2039 (1990); Progr. Theor. Phys. [Supp.]102, 15 (1990) · Zbl 1020.81809 · doi:10.1142/S0217732390002328 [17] Odake, S., Sano, T.: Phys. Lett.B258, 369 (1991) · doi:10.1016/0370-2693(91)91101-Z [18] Khesin, B.: Int. Math. Res. Not. (in Duke Math. J.)1, 1 (1992) · Zbl 0761.17022 · doi:10.1155/S1073792892000011 [19] Radul, A.O.: Funct. Anal. Appl.25, 25 (1991); Phys. Lett.B265, 86 (1991) · Zbl 0809.47044 · doi:10.1007/BF01090674 [20] Wodzicki, M.: In: K-Theory, Arithmetic and Geometry. Lect. Notes in Math. vol.1289, Manin, Yu. (ed.) Berlin, Heidelberg, New York: Springer 1987 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.