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Keywords:
deformation; Maurer-Cartan equation; strongly homotopy Lie algebra; deformation quantization
deformation; Maurer-Cartan equation; strongly homotopy Lie algebra; deformation quantization
Summary:
First three sections of this overview paper cover classical topics of deformation theory of associative algebras and necessary background material. We then analyze algebraic structures of the Hochschild cohomology and describe the relation between deformations and solutions of the corresponding Maurer-Cartan equation. In Section we generalize the Maurer-Cartan equation to strongly homotopy Lie algebras and prove the homotopy invariance of the moduli space of solutions of this equation. In the last section we indicate the main ideas of Kontsevich’s proof of the existence of deformation quantization of Poisson manifolds.
References:
[1] André M.: Method simpliciale en algèbre homologique et algébre commutative. Lecture Notes in Mathematics 32, Springer, Berlin, 1967. MR 0214644
[2] Atiyah M. F., MacDonald I. G.: Introduction to Commutative Algebra. Addison-Wesley, 1969. Fifth printing. MR 0242802 | Zbl 0175.03601
[3] Balavoine D.: Deformation of algebras over a quadratic operad. In J. D. Stasheff J.-L. Loday and A. A. Voronov, editors, Operads: Proceedings of Renaissance Conferences, volume 202 of Contemporary Mathematics (1997), 167–205. MR 1436922
[4] Balavoine D.: Homology and cohomology with coefficients, of an algebra over a quadratic operad. J. Pure Appl. Algebra 132 91998), 221–258. MR 1642086 | Zbl 0967.18004
[5] Bayen F., Flato M., Fronsdal C., Lichnerowiscz A., Sternheimer D.: Deformation and quantization I,II. Ann. Physics 111 (1978), 61–151. MR 0496157
[6] Ciocan-Fontanine I., Kapranov M. M.: Derived Quot schemes. Ann. Sci. Ecole Norm. Sup. 34(3), (2001), 403–440. MR 1839580 | Zbl 1050.14042
[7] Ciocan-Fontanine I., Kapranov M. M.: Derived Hilbert schemes. J. Amer. Math. Soc. 15 (4) (2002), 787–815. MR 1915819 | Zbl 1074.14003
[8] Félix Y.: Dénombrement des types de $\mml@font@bold k$-homotopie. Théorie de la déformation. Bulletin Soc. Math. France 108 (3), 1980.
[9] Fox T. F.: The construction of cofree coalgebras. J. Pure Appl. Algebra 84 (2) (1993), 191–198. MR 1201051 | Zbl 0810.16038
[10] Fox T. F.: An introduction to algebraic deformation theory. J. Pure Appl. Algebra 84 (1993), 17–41. MR 1195416 | Zbl 0772.18006
[11] Fox T. F., Markl M.: Distributive laws, bialgebras, and cohomology. In: J.-L. Loday, J. D. Stasheff, A. A. Voronov, editors, Operads: Proceedings of Renaissance Conferences, volume 202 of Contemporary Math., Amer. Math. Soc. (1997), 167–205. MR 1436921 | Zbl 0866.18008
[12] Gerstenhaber M.: The cohomology structure of an associative ring. Ann. of Math. 78 (2) (1963), 267–288. MR 0161898 | Zbl 0131.27302
[13] Gerstenhaber M.: On the deformation of rings and algebras. Ann. of Math. 79 (1), (1964), 59–104. MR 0171807 | Zbl 0123.03101
[14] Gerstenhaber M.: On the deformation of rings and algebras II. Ann. of Math. 88 (1966), 1–19. MR 0207793 | Zbl 0147.28903
[15] Getzler E.: Lie theory for nilpotent $L_\infty $ algebras. preprint math.AT/0404003, April 2004. MR 2521116
[16] Hartshorne R.,. : Algebraic Geometry. volume 52 of Graduate Texts in Mathematics. Springer-Verlag, 1977. MR 0463157 | Zbl 0367.14001
[17] Hazewinkel M.: Cofree coalgebras and multivariable recursiveness. J. Pure Appl. Algebra 183 (1-3), (2003), 61–103. MR 1992043 | Zbl 1048.16022
[18] Hinich V.: Tamarkin’s proof of Kontsevich formality theorem. Forum Math. 15 (2003), 591–614. MR 1978336 | Zbl 1081.16014
[19] Hinich V., Schechtman V. V.: Homotopy Lie algebras. Adv. Soviet Math. 16 (2) (1993), 1–28. MR 1237833 | Zbl 0823.18004
[20] Kadeishvili T. V.: O kategorii differentialnych koalgebr i kategorii $A(\infty )$-algebr. Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR, 77 (1985), 50–70. In Russian. MR 0862919
[21] Kajiura H., Stasheff J.: Homotopy algebras inspired by clasical open-closed field string theory. Comm. Math. Phys. 263 (3) (2006), 553–581. MR 2211816
[22] Kobayashi S., Nomizu K.: Foundations of Differential Geometry. volume I, Interscience Publishers, 1963. MR 0152974 | Zbl 0119.37502
[23] Kontsevich M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66 (3) (2003), 157–216. MR 2062626 | Zbl 1058.53065
[24] Kontsevich M., Soibelman Y.: Deformations of algebras over operads and the Deligne conjecture. In: Dito, G. et al., editor, Conférence Moshé Flato 1999: Quantization, deformation, and symmetries, number 21 in Math. Phys. Stud., pages 255–307. Kluwer Academic Publishers, 2000. MR 1805894
[25] Lada T., Markl M.: Strongly homotopy Lie algebras. Comm. Algebra 23 (6) (1995), 2147–2161. MR 1327129 | Zbl 0999.17019
[26] Lada T., Stasheff J. D.: Introduction to sh Lie algebras for physicists. Internat. J. Theoret. Phys. 32 (7) (1993), 1087–1103. MR 1235010 | Zbl 0824.17024
[27] Mac Lane S.: Homology. Springer-Verlag, 1963.
[28] Mac Lane S.: Natural associativity and commutativity. Rice Univ. Stud. 49 (1) (1963), 28–46. MR 0170925
[29] Mac Lane S.: Categories for the Working Mathematician. Springer-Verlag, 1971. MR 0354798 | Zbl 0232.18001
[30] Markl M.: A cohomology theory for $A(m)$-algebras and applications. J. Pure Appl. Algebra 83 (1992), 141–175. MR 1191090 | Zbl 0801.55004
[31] Markl M.: Cotangent cohomology of a category and deformations. J. Pure Appl. Algebra 113 (2) (1996), 195–218. MR 1415558
[32] Markl M.: Homotopy algebras are homotopy algebras. Forum Math. 16 (1) (2004), 129–160. MR 2034546 | Zbl 1067.55011
[33] Markl M.: Intrinsic brackets and the ${L_\infty }$-deformation theory of bialgebras. preprint math.AT/0411456, November 2004. MR 2812919
[34] Markl M., Remm E.: Algebras with one operation including Poisson and other Lie-admissible algebras. J. Algebra 299 (2006), 171–189. MR 2225770 | Zbl 1101.18004
[35] Markl M., Shnider S., Stasheff J. D.: Operads in Algebra, Topology and Physics. volume 96 of Mathematical Surveys and Monographs, Amer. Math. Soc. Providence, Rhode Island, 2002. MR 1898414 | Zbl 1017.18001
[36] Nijenhuis A., Richardson J.: Cohomology and deformations in graded Lie algebras. Bull. Amer. Math. Soc. 72 (1966), 1–29. MR 0195995 | Zbl 0136.30502
[37] Quillen D.: Homotopical Algebra. Lecture Notes in Math. 43, Springer-Verlag, 1967. MR 0223432 | Zbl 0168.20903
[38] Quillen D.: On the (co-)homology of commutative rings. Proc. Symp. Pure Math. 17 (1970), 65–87. MR 0257068 | Zbl 0234.18010
[39] Serre J.-P.: Lie Algebras and Lie Groups. Benjamin, 1965. Lectures given at Harward University. MR 0218496 | Zbl 0132.27803
[40] Smith J. R.: Cofree coalgebras over operads. Topology Appl. 133 (2) (2003), 105–138. MR 1997960 | Zbl 1032.18004
[41] Stasheff J. D.: Homotopy associativity of H-spaces I,II. Trans. Amer. Math. Soc. 108 (1963), 275–312. MR 0158400 | Zbl 0114.39402
[42] Sullivan D.: Infinitesimal computations in topology. Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269–331. MR 0646078 | Zbl 0374.57002
[43] Tamarkin D. E.: Another proof of M. Kontsevich formality theorem. preprint math.QA/ 9803025, March 1998.
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