Cohomology theories for homotopy algebras and noncommutative geometry

Hamilton, Alastair¹ ; Lazarev, Andrey²

¹Mathematics Department, University of Connecticut, 196 Auditorium Road, Storrs CT 06269-3009, USA
²Department of Mathematics, University of Leicester, Leicester LE1 7RH, England

Algebraic and Geometric Topology, Tome 9 (2009) no. 3, pp. 1503-1583

Résumé

This paper builds a general framework in which to study cohomology theories of strongly homotopy algebras, namely A∞–, C∞– and L∞–algebras. This framework is based on noncommutative geometry as expounded by Connes and Kontsevich. The developed machinery is then used to establish a general form of Hodge decomposition of Hochschild and cyclic cohomology of C∞–algebras. This generalises and puts in a conceptual framework previous work by Loday and Gerstenhaber–Schack.

Détail

DOI : 10.2140/agt.2009.9.1503

Keywords: infinity-algebra, cyclic cohomology, Harrison cohomology, symplectic structure, Hodge decomposition

Bibliographie

[1] M Alexandrov, A Schwarz, O Zaboronsky, M Kontsevich, The geometry of the master equation and topological quantum field theory, Internat. J. Modern Phys. A 12 (1997) 1405

[2] M Barr, Harrison homology, Hochschild homology and triples, J. Algebra 8 (1968) 314

[3] X Z Cheng, E Getzler, Transferring homotopy commutative algebraic structures, J. Pure Appl. Algebra 212 (2008) 2535

[4] J Conant, K Vogtmann, On a theorem of Kontsevich, Algebr. Geom. Topol. 3 (2003) 1167

[5] A Connes, Noncommutative geometry, Academic Press (1994)

[6] A Fialowski, M Penkava, Deformation theory of infinity algebras, J. Algebra 255 (2002) 59

[7] M Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. $(2)$ 78 (1963) 267

[8] M Gerstenhaber, S D Schack, A Hodge-type decomposition for commutative algebra cohomology, J. Pure Appl. Algebra 48 (1987) 229

[9] E Getzler, Lie theory for nilpotent $L_\infty$–algebras

[10] E Getzler, A Darboux theorem for Hamiltonian operators in the formal calculus of variations, Duke Math. J. 111 (2002) 535

[11] E Getzler, J D S Jones, Operads, homotopy algebra, and iterated integrals for double loop spaces

[12] E Getzler, J D S Jones, $A_\infty$–algebras and the cyclic bar complex, Illinois J. Math. 34 (1990) 256

[13] E Getzler, M M Kapranov, Cyclic operads and cyclic homology, from: "Geometry, topology, physics" (editor S T Yau), Conf. Proc. Lecture Notes Geom. Topology, IV, Int. Press (1995) 167

[14] V Ginzburg, Non-commutative symplectic geometry, quiver varieties, and operads, Math. Res. Lett. 8 (2001) 377

[15] W M Goldman, J J Millson, The deformation theory of representations of fundamental groups of compact Kähler manifolds, Inst. Hautes Études Sci. Publ. Math. (1988) 43

[16] A Hamilton, On the classification of Moore algebras and their deformations, Homology Homotopy Appl. 6 (2004) 87

[17] A Hamilton, A super-analogue of Kontsevich's theorem on graph homology, Lett. Math. Phys. 76 (2006) 37

[18] A Hamilton, A Lazarev, Characteristic classes of $A_\infty$–algebras, J. Homotopy Relat. Struct. 3 (2008) 65

[19] D K Harrison, Commutative algebras and cohomology, Trans. Amer. Math. Soc. 104 (1962) 191

[20] V Hinich, Descent of Deligne groupoids, Internat. Math. Res. Notices (1997) 223

[21] T V Kadeishvili, The algebraic structure in the homology of an $A(\infty )$–algebra, Soobshch. Akad. Nauk Gruzin. SSR 108 (1982)

[22] M Kapranov, Noncommutative geometry based on commutator expansions, J. Reine Angew. Math. 505 (1998) 73

[23] C Kassel, Quantum groups, Graduate Texts in Math. 155, Springer (1995)

[24] B Keller, Introduction to $A$–infinity algebras and modules, Homology Homotopy Appl. 3 (2001) 1

[25] M Kontsevich, Formal (non)commutative symplectic geometry, from: "The Gel$'$fand Mathematical Seminars, 1990–1992" (editors L Corwin, I Gel′fand, J Lepowsky), Birkhäuser (1993) 173

[26] M Kontsevich, Feynman diagrams and low-dimensional topology, from: "First European Congress of Mathematics, Vol. II (Paris, 1992)", Progr. Math. 120, Birkhäuser (1994) 97

[27] M Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003) 157

[28] M Kontsevich, A L Rosenberg, Noncommutative smooth spaces, from: "The Gelfand Mathematical Seminars, 1996–1999" (editors I M Gelfand, V S Retakh), Birkhäuser (2000) 85

[29] M Kontsevich, A L Rosenberg, Noncommutative spaces and flat descent, MPIM Preprint 2004-36, Max-Planck Insitut für Mathematik (2004)

[30] I Kříž, J P May, Operads, algebras, modules and motives, Astérisque (1995)

[31] A Lazarev, Hochschild cohomology and moduli spaces of strongly homotopy associative algebras, Homology Homotopy Appl. 5 (2003) 73

[32] J L Loday, Cyclic homology, Grund. der Math. Wissenschaften 301, Springer (1998)

[33] M Markl, A cohomology theory for $A(m)$–algebras and applications, J. Pure Appl. Algebra 83 (1992) 141

[34] M Markl, Homotopy algebras are homotopy algebras, Forum Math. 16 (2004) 129

[35] M Markl, S Shnider, J Stasheff, Operads in algebra, topology and physics, Math. Surveys and Monogr. 96, Amer. Math. Soc. (2002)

[36] J Milles, Andre–Quillen cohomology of algebras over an operad

[37] T Natsume, S D Schack, A decomposition for the cyclic cohomology of a commutative algebra, J. Pure Appl. Algebra 61 (1989) 273

[38] M Penkava, A Schwarz, $A_\infty$ algebras and the cohomology of moduli spaces, from: "Lie groups and Lie algebras: E B Dynkin's Seminar" (editors S G Gindikin, E B Vinberg), Amer. Math. Soc. Transl. Ser. 2 169, Amer. Math. Soc. (1995) 91

[39] A Polishchuk, Extensions of homogeneous coordinate rings to $A_\infty$–algebras, Homology Homotopy Appl. 5 (2003) 407

[40] D Quillen, Rational homotopy theory, Ann. of Math. $(2)$ 90 (1969) 205

[41] M Schlessinger, J Stasheff, Deformation theory and rational homotopy type, Preprint

[42] J Stasheff, Homotopy associativity of $H$–spaces. I, Trans. Amer. Math. Soc. 108 (1963) 275

[43] J Stasheff, Homotopy associativity of $H$–spaces. II, Trans. Amer. Math. Soc. 108 (1963) 293

[44] J Stasheff, The intrinsic bracket on the deformation complex of an associative algebra, J. Pure Appl. Algebra 89 (1993) 231

[45] D Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. (1977)

[46] A A Voronov, Notes on universal algebra, from: "Graphs and patterns in mathematics and theoretical physics" (editors M Lyubich, L Takhtajan), Proc. Sympos. Pure Math. 73, Amer. Math. Soc. (2005) 81

[47] B Zwiebach, Closed string field theory: quantum action and the Batalin–Vilkovisky master equation, Nuclear Phys. B 390 (1993) 33