Lie algebras of differential operators, their central extensions, and $W$-algebras

A. O. Radul

Funkcionalʹnyj analiz i ego priloženiâ, Tome 25 (1991) no. 1, pp. 33-49.

Bibliographie

[1] Belavin A. A., Polyakov A. M., Zamolodchikov A. V., “Infinite conformal symmetry in two-dimensional quantum field theory”, Nucl. Phys., B 241 (1984), 333–380 | DOI | MR | Zbl

[2] Moore G., Seiberg N., Classical and Quantum Conformal Field Theory, Preprint / IASSNS / HEP - 88/39, 1988 | MR

[3] Zamolodchikov A.B., “Beskonechnye dopolnitelnye simmetrii v dvumernoi kvantovoi konformnoi teorii polya”, TMF, 65 (1985), 347–359 | MR

[4] Lukyanov S.L., “Kvantovanie skobki Gelfanda–Dikogo”, Funktsion. analiz i ego pril., 22:4 (1988), 1–10 | MR

[5] Lukyanov S.L., Fateev V.A., “Konformno-invariantnye modeli dvumernoi kvantovoi teorii polya s simmetriei $Z_n$”, ZhETF, 94:3 (1988), 23–37 | MR

[6] Drinfeld V.G., Sokolov V.V., “Lie algebras and Korteweg–deVries type equations”, J. Sov. Math., 30 (1985), 1975–1998 | DOI

[7] Belavin A.A., “Doclad in Proc. of the second Yukava Memorial Symp.”, Ser. Proc. in Phys., 31, Nishinomiya. Japan, 1987, 132

[8] Bershadsky M., Ooguri H., Quantum Hamiltonian reduction and $W$-algebras, Preprint / IASSNS / HEP 1989/09, 1989 | MR

[9] Bakas I., “The large-N limit of extended conformal symmetries”, Phys. Lett., B 228 (1989), 57–63 | DOI | MR

[10] Morozov A., On the concept of universal $W$-algebra, ITEP preprint / ITEP / 148-89, 1989 | MR

[11] Gelfand I.M., Dikii L.A., “A family ofsHamiltonian structures for nonlinear integrable equations”, Gelfand I.M. Collected papers, 1, eds. S.G. Gindikin, V.W. Guillemin, A.A. Kirillov, B. Kostant, S. Sternberg, Springer, Berlin, Heidelberg, New York, 1989, 625–641 | MR

[12] Khovanova T.G., “Algebry Li Gelfanda–Dikogo i algebra Virasoro”, Funktsion. analiz i ego pril., 20:4 (1986), 89–90 | MR | Zbl

[13] Radul A.O., “Tsentralnoe rasshirenie algebry Li differentsialnykh operatorov na okruzhnosti i $W$-algebry”, Pisma v ZhETF, 50:8 (1989), 341–343 | MR

[14] Radul A.O., “Non-trivial central extensions of Lie algebras of differential operators in two and higher dimensions”, Phys. Lett., 1990 (to appear) | MR

[15] Kravchenko O.S, Khesin B.A., “Netrivialnoe tsentralnoe rasshirenie algebry Li (psevdo-) differentsialnykh simvolov na okruzhnosti”, Funktsion. analiz i ego pril., 23:3 (1989), 78–79 | MR

[16] Brylinski J.L., Getzler E., “The Homology of Algebras of Pseudo-Differential Symbols and the Noncommutative Residue”, $K$-Theory, 1 (1987), 385–403 | DOI | MR | Zbl

[17] Wodzicki M., “Non-commutative residue. Fundamentals”, LN in Math., 1289, 1985, 320–399 | MR

[18] Manin Yu.I., “Neveu-Schwarz sheaves and differential equations for Mamford superforms”, JGP, 5:2 (1988), 161–181 | DOI | MR | Zbl

[19] Adler M., “On a trace functional for formal pseudodifferential operators and symblectic structure of the Korteweg de Vries type equations”, Invent. math., 50 (1979), 219–248 | DOI | MR | Zbl

[20] Manin Yu.I., Algebraicheskie aspekty nelineinykh differentsialnykh uravnenii, Itogi nauki i tekhniki. Sovr. probl. matem., 11, VINITI, M., 1978 | MR

[21] Shubin M.A., Psevdodifferentsialnye operatory i spektralnaya teoriya, Nauka, M., 1990 | MR

[22] Drinfeld V.G., “Doclad in Quantum Groups”, Proc. Int. Congr. Math. (Berkeley, 1986), Amer. Math. Soc., 1988 | MR

[23] Semenov-Tian-Shansky M.A., “What is classical $R$-matrix?”, Funct. Anal. Appl., 17 (1983), 259–273 | DOI | MR

[24] Semenov-Tian-Shansky M.A., “Pressing transformations and Poissonian group actions”, Publ. RIMS, 21, no. 6, Kyoto Univ., 1985, 1203–1221 | DOI | MR

[25] Arnold V.I., Givental A.B., “Simplekticheskaya geometriya”, Itogi nauki i tekhniki. Sovr. probl. matem. fundam. napravl., 4, VINITI, M., 1985, 5–139 | MR

[26] Feigin B.L., “Algebry Li $\mathfrak{gl}(\lambda)$ i kogomologii algebry Li differentsialnykh operatorov”, UMN, 43:2 (1988), 157–158 | MR | Zbl

[27] Presli A., Sigal G., Gruppy petel, Mir, M., 1990 | MR

[28] Leites D.A., “Vvedenie v teoriyu supermnogoobrazii”, UMN, 35:1(211) (1980), 3–57 | MR | Zbl

[29] Manin Yu.I., Kalibrovochnye polya i golomorfnaya geometriya, Nauka, M., 1984 | MR

[30] Manin Yu.I., Radul A.O., “A supersymmetric Extension of the Kadomtsev Petviashvili Hierarchy”, Gomm. Math. Phys., 98 (1985), 65–77 | DOI | MR | Zbl

[31] Leites D.A., Feigin B.L., “Novye superalgebry Li strunnykh teorii”, Teoretiko-gruppovye metody v fizike, t. 1, Nauka, M., 1983, 269–278

[32] Bilal A., “A remark on $N\to\infty$ limit of $W_N$-algebras”, Phys. Lett., B 227 (1989), 406–411 | DOI | MR

[33] Pope C., Romans L., Shen X., “The complete structure of $W$”, Phys. Lett., B 236 (1990), 173–178 | DOI | MR

[34] Bakas I., Kiritis E., On the structure of $W$-algebras, Maryland preprint / UMD-PP90-160, 1990 | MR

[35] Gervais J.-Z., “$W$-algebras and two-dimensional Toda equations”, Phys. Lett., B160 (1985), 277–284 | DOI | MR

[36] Marshakov A., Morozov A., Note on $W_3$-algebra, ITEP preprint / ITEP / 146-89, 1989 | MR

[37] Belinson A.A., Manin Yu.I., Schechtman V.V., “Sheaves of the Virasoro and Neveu-Schwarz algebras”, LN in Math., 1289, 1987, 52–71 | MR

[38] Kac V.G, Peterson P., Representations of infinite dimensional Lie algebras, Preprint/MIT. 27/87, 1987 | MR

[39] Radul A.O., “Superanalogues of Schwarz derivative and Bott cocycle”, Seminar on supermanifolds, 21, ed. D. Leites, Univ. of Stockholm, 1986, 40–57