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Affine Kac-Moody algebras and semi-infinite flag manifolds. (English) Zbl 0722.17019
With the technique of semi-infinite cohomology over (affine) Kac-Moody algebras the authors realize so-called Wakimoto modules as twisted Verma modules. Among other wonders this brings about a two-sided resolution of these twisted Verma modules leading to a semi-infinite analog of the Borel-Weil-Bott theorem. The exposition is highly illuminating and equally breezy (lots of details are to be published).

MSC:
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
14M15 Grassmannians, Schubert varieties, flag manifolds
17B56 Cohomology of Lie (super)algebras
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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[1] Beilinson, A.: Localization of representations of reductive Lie algebras. In: Proceedings of ICM, Warsawa 1984, Vol. 2, pp. 669–710 · Zbl 0571.20032
[2] Beillinson, A., Bernstein, J.: Localization de g-modules. C.R. Acad. Sci. Paris192, 15–18 (1981) · Zbl 0476.14019
[3] Beilinson, A., Bernstein, J.: Generalization of a theorem of Casselman. Report on Utah Conference on epresentation theory, April 1982. In: Representation theory of reductive groups. Trombi, P. (ed.) DM 40. Boston: Birkhäuser 1983
[4] Bernsteion, J.: Lectures onD-modules (unpublished)
[5] Bernstein, J., Gelfand, I., Gelfand, S.: On the category ofg-modules. Funkt. Anal. Prilozhen.10, 1–8 (1976) (in Russian) · Zbl 0375.35048
[6] Bernstein, J., Gelfand, I., Gelfand, S.: Schubert cells and cohomology of spacesG/P. Usp Mat. Nauk (=Russ. Math. Surv.)28, 3–26 (1973) (in Russian)
[7] Bernstein, J., Gelfand, I., Gelfand, S.: Differential operators on the base affine space and a study ofg-modules. In: Lie groups and their representations, pp. 21–64. New York: Wiley 1975 · Zbl 0338.58019
[8] Borek, A.: Kählerian coset spaces of semi-simple Lie groups Proc. Natl. Acad. Sci. USA40 1147–1151 (1954) · Zbl 0058.16002
[9] Bott, R.: Homogeneous vector bundles. Ann. Math.66, 203–248 (1957) · Zbl 0094.35701
[10] Brylinsky, J.L., Kashiwiara, M.: Kazhdan-Lusztig conjecture and holonomic systems. Invent. Math.64, 487–510 (1981)
[11] Conley, C.C., Zehnder, E.: Morse type index theorem for flows and periodic solutions for hamiltonian equation. Commun. Pure and Appl. Math.34, 207–253 (1984) · Zbl 0559.58019
[12] Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Transformation groups for soliton equations. In: Proceedings of RIMS symposium. Jimbo, M., Miwa, T. (eds.), pp. 39–120 Singapore: World Scientific 1983 · Zbl 0571.35098
[13] Deodhar, V., Gabber, O., Kac, V.: Structure of some categories of representations of infinite-dimensional Lie algebras. Adv. Math.45, 92–116 (1982) · Zbl 0491.17008
[14] Dotsenko, Vl.S., Fateev, V.A.: Conformal algebra and multipoint correlation functions in2D statistical models. Nucl. Phys. B240, F 312, 312–348 (1984)
[15] Dotsenko, Vl.S., Fateev, V.A.: Private communications
[16] Feigin, B.L.: Semi-infinite homology of Kac-Moody and Virasoro Lie algebras. Usp. Mat. Nauk (=Russ. Math. Surv.)39, 195–196 (1984) (in Russian) · Zbl 0544.17009
[17] Feigin, B.L.: On the cohomology of the Lie algebra of vector fields and of the current algebra. Sel. Math. Sov.7, 49–62 (1988) · Zbl 0657.17009
[18] Feigin, B.L., Frenkel, E.V.: The family of representations of affine Lie algebras. Usp. Math. Nauk (=Russ. Math. Surv.)43, 227–228 (1988) (in Russian) · Zbl 0657.17013
[19] Feigin, B.L., Frenkel, E.V.: Representations of affine Kac-Moody algebras and bosonization. To be published in: V. G. Knizhnik Memorial volume. Singapore: World Scientific 1989
[20] Feigin, B.L., Frenkel, E.V.: Representations of affine Kac-Moody algebras, bosonization and resolutions. To be published in Lett. Math. Phys. 19, N3 (1990) · Zbl 0711.17012
[21] Feigin, B.L., Fuchs, D.B.: Representations of the Virasoro algebra. To be publishied in: Representations of inifinite-dimensional Lie groups and Lie algebras. New York: Gordon and Brech 1989
[22] Feigin, B.L., Tsygan, B.L.: Cohomology of generalized Jakobean matrices. Funct. Anal. Prilozhien.17, 86–87 (1983)
[23] Felder, G.: BRST approach to minimal models. Nucl. Phys.B 317, 215–236 (1989)
[24] Floer, A.: Proof of Arnold conjecture for surfaces and generalization to cetain Kähler manifolds. Duke Math. J.53, 1–32 (1986) · Zbl 0607.58016
[25] Fuchs, D.B.: Cohomology of infinite-dimensional Lie algebras. New York: Plenum Press 1986
[26] Gerland, H., Lepowsky, J.: Lie algebra homology and the Macdonald-Kac formulas. Invent. Math.34, 37–76 (1976) · Zbl 0358.17015
[27] Gerasimov, A., Marshakov, A., Morosov, A., Olshanetsky, M., Shatashvili, S.: Moscow preprint, 1989
[28] Hayashi, T.: Sugawara construction and Kac-Kazhdan conjecture. Invent. Math.94, 13–52 (1988) · Zbl 0674.17005
[29] Kac, V.G.: Infinite-dimensional Lie algebras. Boston, MA: Birkhäuser 1983 · Zbl 0537.17001
[30] Kac, V.G., Kazhdan, D.: Structure of representations with highest weight of infinite-dimensional Lie algebras. Adv. Math.34, 97–108 (1979) · Zbl 0427.17011
[31] Kac, V., Peterson, D.: Unpublished, Kac, V.G.: Constructing groups associated to infinite-dimensional Lie algebras. In MSRI Publ., 4 Berlin, Heidelberg, New York: Springer 1985 · Zbl 0625.22014
[32] Kac, V., Peterson, D.: Spin and wedge representation of infinite-dimensional Lie algebras and groups. Proc. Natl. Acad. Sci. USA78, 3308–3312 (1981) · Zbl 0469.22016
[33] Kac, V., Todorov, I.: Superconformal current algebras and their unitary representations. Commun. Math. Phys.102, 337–347 (1985) · Zbl 0599.17011
[34] Kazdan, D., Lusztig, G.: Representation of Coxeter groups and Hecke algebras. Invent. Math.53, 165–184 (1979) · Zbl 0499.20035
[35] Kazhdan, D., Lusztig, G.: Schubert varieties and Poincaré duality. Proc. Symp. Pure Math.36, 185–203 (1980) · Zbl 0461.14015
[36] Kempf, G.: Cousin-Grothendieck complex of induced representations. Adv. Math.29, 310–396 (1978) · Zbl 0393.20027
[37] Kostant, B., Kumar, S.: The nil Hecke ring and cohomology of G/P for a Kac-Moody group. Adv. math.62, 187–237 (1986) · Zbl 0641.17008
[38] Malikov, F.: Singular vectors in Verma modules over affine algebras. Funct. Anal. Prilozhen.23, 76–77 (1989) · Zbl 0675.46011
[39] Malikov, F.: Verma modules over affine rank 2 algebras. To appear in Algebra and Analysis (1989) (in Russian) · Zbl 0676.17012
[40] Peterson, D., Kac, V.: Infinite flag varieties and conjugacy theorem. Proc. Natl. Acad. Sci. USA80, 1778–1782 (1983) · Zbl 0512.17008
[41] Pressley, A., Segal, G.: Loop groups. Oxford: Clarendon Press 1987 · Zbl 0618.22011
[42] Rocha-Caridi, A., Wallach, N.: Projective modules over gradded Lie algebras. Math. Z.180, 151–177 (1982) · Zbl 0475.17002
[43] Tate, J.: Residues of differentials on curves. Ann. Sci. E.N.S.4 Ser., t.1, 149–159 (1968) · Zbl 0159.22702
[44] Tsuchiya, A., Kanie, Y.: Fock space representations of the Virasoro algebria-Intertwining operators. Publ. RIMS, Kyoto Univ.22, 259–327 (1986) · Zbl 0604.17008
[45] Wakimoto, M.: Fock representations of affine Lie algebraA 1 (1) Commun. Math. Phys.104, 604–609 (1986) · Zbl 0587.17009
[46] Wallach, N.: A Class of non-standard modules for affine Lie algebras. Math. Z.196, 303–313 (1987) · Zbl 0637.17011
[47] Zamolodchikov, A.: Unpublished
[48] Bershadsky, M., Ooguri, H.: Hiddensl(n) symmetry in conformal field theory, IASSNS HEP-89/09 · Zbl 0689.17015
[49] Berniard, D., Felder, G.: Fock representations and BRST cohomology ofsl(2) current algebra, ETH-TH/89-26
[50] Bouwknegt, P., McCarthy, J., Pilch, K.: MIT preprint CTP # 1797
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