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Extensions of vertex algebras. Constructions and applications
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 72 (2017) no. 4, pp. 707-763 Cet article a éte moissonné depuis la source Math-Net.Ru

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This paper discusses the main known constructions of vertex operator algebras. The starting point is the lattice algebra. Screenings distinguish subalgebras of lattice algebras. Moreover, one can construct extensions of vertex algebras. Combining these constructions gives most of the known examples. A large class of algebras with big centres is constructed. Such algebras have applications to the geometric Langlands programme. Bibliography: 46 titles.
Keywords: vertex operator algebras, screenings, opers, quantum groups. 
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B. L. Feigin. Extensions of vertex algebras. Constructions and applications. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 72 (2017) no. 4, pp. 707-763. http://geodesic.mathdoc.fr/item/RM_2017_72_4_a3/

[1] A. A. Belavin, A. M. Polyakov, A. B. Zamolodchikov, “Infinite conformal symmetry in two-dimensional quantum field theory”, Nuclear Phys. B, 241:2 (1984), 333–380 | DOI | MR | Zbl

[2] A. Beilinson, V. Drinfeld, Chiral algebras, Amer. Math. Soc. Colloq. Publ., 51, Amer. Math. Soc., Providence, RI, 2004, vi+375 pp. | DOI | MR | Zbl

[3] E. Frenkel, D. Ben-Zvi, Vertex algebras and algebraic curves, Math. Surveys Monogr., 88, 2nd ed., Amer. Math. Soc., Providence, RI, 2004, xiv+400 pp. | DOI | MR | Zbl

[4] M. Wakimoto, “Fock representations of the affine Lie algebra $A_1^{(1)}$”, Comm. Math. Phys., 104:4 (1986), 605–609 | DOI | MR | Zbl

[5] B. Feigin, E. Frenkel, “Free field resolutions in affine Toda field theories”, Phys. Lett. B, 276:1-2 (1992), 79–86 | DOI | MR

[6] C. Dong, J. Lepowsky, Generalized vertex algebras and relative vertex operators, Progr. Math., 112, Birkhäuser Boston, Boston, MA, 1993, x+202 pp. | DOI | MR | Zbl

[7] D. Bakalov, V. G. Kac, “Generalized vertex algebras”, Lie theory and its applications in physics VI, Heron Press, Sofia, 2006, 3–25

[8] D. Kazhdan, G. Lusztig, “Affine Lie algebras and quantum groups”, Internat. Math. Res. Notices, 1991:2 (1991), 21–29 | DOI | MR | Zbl

[9] V. G. Drinfel'd, “Quasi-Hopf algebras and Knizhnik–Zamolodchikov equations”, Problems of modern quantum field theory (Alushta, 1989), Res. Rep. Phys., Springer, Berlin, 1989, 1–13 | DOI | MR

[10] V. Ostrik, “Module categories over the {Drinfeld} double of a finite group”, Int. Math. Res. Not., 2003:27 (2003), 1507–1520 | DOI | MR | Zbl

[11] B. Feigin, S. Parkhomenko, “Regular representation of affine Kac–Moody algebras”, Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996, 415–424 | MR | Zbl

[12] B. L. Feigin, E. V. Frenkel, “Representations of affine Kac–Moody algebras, bosonization and resolutions”, Lett. Math. Phys., 19:4 (1990), 307–317 | DOI | MR | Zbl

[13] S. Arkhipov, D. Gaitsgory, “Differential operators on the loop group via chiral algebras”, Int. Math. Res. Not., 2002:4 (2002), 165–210 | DOI | MR | Zbl

[14] A. Beilinson, V. Drinfeld, Quantization of Hitchin's integrable system and Hecke eigensheaves, preliminary version, 384 \par pp. http://www.math.uchicago.edu/~mitya/langlands.html

[15] A. A. Beilinson, V. G. Drinfeld, “Quantization of Hitchin's fibration and Langland's program”, Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996, 3–7 | DOI | MR | Zbl

[16] E. Frenkel, “Lectures on the Langlands program and conformal field theory”, Frontiers in number theory, physics, and geometry. II, Springer, Berlin, 2007, 387–533 | DOI | MR | Zbl

[17] V. V. Batyrev, L. A. Borisov, “On Calabi–Yau complete intersections in toric varieties”, Higher-dimensional complex varieties (Trento, 1994), de Gruyter, Berlin, 1996, 39–65 | MR | Zbl

[18] V. Gorbounov, F. Malikov, V. Schechtman, “Gerbes of chiral differential operators”, Math. Res. Lett., 7:1 (2000), 55–66 ; (1999), 12 pp., arXiv: math/9906117 | DOI | MR | Zbl

[19] W. L. Gan, V. Ginzburg, “Quantization of Slodowy slices”, Int. Math. Res. Not., 2002:5 (2002), 243–255 | DOI | MR | Zbl

[20] A. Premet, “Special transverse slices and their enveloping algebras”, Adv. Math., 170:1 (2002), 1–55 | DOI | MR | Zbl

[21] B. Feigin, E. Frenkel, “Quantization of the Drinfel'd–Sokolov reduction”, Phys. Lett. B, 246:1-2 (1990), 75–81 | DOI | MR | Zbl

[22] D. Kazhdan, G. Lusztig, “Tensor structures arising from affine Lie algebras. I”, J. Amer. Math. Soc., 6:4 (1993), 905–947 | DOI | MR | Zbl

[23] D. Kazhdan, G. Lusztig, “Tensor structures arising from affine {Lie} algebras. II”, J. Amer. Math. Soc., 6:4 (1993), 949–1011 | DOI | MR | Zbl

[24] D. Kazhdan, G. Lusztig, “Tensor structures arising from affine Lie algebras. III”, J. Amer. Math. Soc., 7:2 (1994), 335–381 | DOI | MR | Zbl

[25] D. Kazhdan, G. Lusztig, “Tensor structures arising from affine Lie algebras. IV”, J. Amer. Math. Soc., 7:2 (1994), 383–453 | DOI | MR | Zbl

[26] G. Lusztig, “Canonical bases arising from quantized enveloping algebras”, J. Amer. Math. Soc., 3:2 (1990), 447–498 | DOI | MR | Zbl

[27] P. Deligne, Une description de catégorie tressée (inspiré par {Drinfeld}), Letter to V. Drinfeld, 1990, unpublished

[28] B. L. Feigin, “The semi-infinite homology of Kac–Moody and Virasoro Lie algebras”, Russian Math. Surveys, 39:2 (1984), 155–156 | DOI | MR | Zbl

[29] B. Bakalov, A. Kirillov, Jr., Lectures on tensor categories and modular functors, Univ. Lecture Ser., 21, Amer. Math. Soc., Providence, RI, 2001, x+221 pp. | MR | Zbl

[30] B. L. Feigin, A. M. Gainutdinov, A. M. Semikhatov, I. Yu. Tipunin, “Logarithmic extensions of minimal models: characters and modular transformations”, Nuclear Phys. B, 757:3 (2006), 303–343 | DOI | MR | Zbl

[31] B. L. Feigin, I. Yu. Tipunin, Logarithmic CFTs connected with simple Lie algebras, 2010, 14 pp., arXiv: 1002.5047

[32] A. Tsuchiya, S. Wood, “The tensor structure on the representation category of the $\mathscr W_p$ triplet algebra”, J. Phys. A, 46:44 (2013), 445203, 40 pp. | DOI | MR | Zbl

[33] A. Beilinson, V. Drinfeld, Opers, 2005, 29 pp., arXiv: math.AG/0501398

[34] O. Gamayun, N. Iorgov, O. Lisovyy, “How instanton combinatorics solves Painlevé VI, V and IIIs”, J. Phys. A, 46:33 (2013), 335203, 29 pp. | DOI | MR | Zbl

[35] M. A. Bershtein, A. I. Shchechkin, “$q$-deformed Painlevé $\tau$ function and $q$-deformed conformal blocks”, J. Phys. A, 50:8 (2017), 085202, 22 pp. | DOI | MR | Zbl

[36] B. L. Feigin, A. M. Semikhatov, “$\mathscr W^{(2)}_n$ algebras”, Nuclear Phys. B, 698:3 (2004), 409–449 | DOI | MR | Zbl

[37] M. Bershadsky, “Conformal field theories via Hamiltonian reduction”, Comm. Math. Phys., 139:1 (1991), 71–82 | DOI | MR | Zbl

[38] A. M. Polyakov, “Gauge transformations and diffeomorphisms”, Internat. J. Modern Phys. A, 5:05 (1990), 833–842 | DOI | MR | Zbl

[39] J. E. Marsden, G. Misiołek, J.-P. Ortega, M. Perlmutter, T. S. Ratiu, Hamiltonian reduction by stages, Lecture Notes in Math., 1913, Springer, Berlin, 2007, xvi+519 pp. | DOI | MR | Zbl

[40] J. Brundan, A. Kleshchev, “Shifted Yangians and finite $W$-algebras”, Adv. Math., 200:1 (2006), 136–195 | DOI | MR | Zbl

[41] J. Brundan, A. Kleshchev, Representations of shifted Yangians and finite $W$-algebras, Mem. Amer. Math. Soc., 196, No 918, Amer. Math. Soc., Providence, RI, 2008, viii+107 pp. ; 2006 (v1 – 2005), 109 pp., arXiv: math/0508003 | DOI | MR | Zbl

[42] G. I. Olshanskii, “Twisted Yangians and infinite-dimensional classical Lie algebras”, Quantum groups (Leningrad, 1990), Lecture Notes in Math., 1510, Springer, Berlin, 1992, 104–119 | DOI | MR | Zbl

[43] A. I. Molev, “Yangians and their applications”, Handbook of algebra, v. 3, Elsevier/North-Holland, Amsterdam, 2003, 907–959 | DOI | MR | Zbl

[44] P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik, Tensor categories, Math. Surveys Monogr., 205, Amer. Math. Soc., Providence, RI, 2015, xvi+343 pp. | DOI | MR | Zbl

[45] V. G. Kac, Infinite dimensional Lie algebras, 3rd ed., Cambridge Univ. Press, Cambridge, 1990, xxii+400 pp. | DOI | MR | MR | Zbl | Zbl

[46] M. Atiyah, “Topological quantum field theories”, Inst. Hautes Études Sci. Publ. Math., 68:1 (1988), 175–186 | DOI | MR | Zbl