The Batalin-Vilkovisky Algebra on Hochschild Cohomology Induced by Infinity Inner Products

[L’algèbre de Batalin-Vilkovisky sur la cohomologie de Hochschild]

Tradler, Thomas

Annales de l'Institut Fourier, Tome 58 (2008), p. 2351-2379 / Harvested from Numdam

Résumé

On définit une structure de BV sur la cohomologie de Hochschild d’une algèbre associative unitaire munie d’une forme bilinéaire symétrique non dégénérée. La structure d’algèbre de Gerstenhaber induite est celle introduite dans l’article originel de Gerstenhaber sur la cohomologie de Hochschild. On étend ce résultat au cas d’une algèbre $A$-infinie unitaire munie d’une forme bilinéaire symétrique $A$-infinie non dégénérée.

Abstract

We define a BV-structure on the Hochschild cohomology of a unital, associative algebra $A$ with a symmetric, invariant and non-degenerate inner product. The induced Gerstenhaber algebra is the one described in Gerstenhaber’s original paper on Hochschild-cohomology. We also prove the corresponding theorem in the homotopy case, namely we define the BV-structure on the Hochschild-cohomology of a unital $A_\infty $-algebra with a symmetric and non-degenerate $\infty $-inner product.

Détail

Publié le : 2008-01-01

MR 2498354 | Zbl pre05505486 | 2 citations dans Numdam

DOI : https://doi.org/10.5802/aif.2417
Classification:  16E40
Mots clés: cohomologie de Hochschild, algèbre de Batalin Vilkovisky

Bibliographie

[1] Chas, M.; Sullivan, D. String Topology (1999) (preprint GT/9911159)

[2] Cohen, R. L.; Jones, J. D. S. A Homotopy Theoretic Realization Of String Topology, Math. Ann., Tome 324 (2002), pp. 773-798 | Article | MR 1942249 | Zbl 1025.55005

[3] Cohen, R. L.; Jones, J. D. S.; Yan, J. The loop homology algebra of spheres and projective spaces, Birkhäuser, Basel, Progr. Math., Tome 215 (2004) | MR 2039760 | Zbl 1054.55006

[4] Connes, A. Non-commutative differential geometry, Publ. Math. IHÉS, Tome 62 (1985), pp. 257-360 | Numdam | MR 823176 | Zbl 0592.46056

[5] Costello, K. Topological conformal field theories and Calabi-Yau categories, Adv. Math., Tome 210 (2007), pp. 165-214 | Article | MR 2298823 | Zbl pre05132553

[6] Felix, Y.; Thomas, J.-C. Rational BV-algebra in String Topology (2007) (arXiv:0705.4194) | Numdam | MR 2415345

[7] Felix, Y.; Thomas, J.-C.; Vigue-Poirrier, M. Loop homology algebra of a closed manifold (arXiv:math/0203137v2)

[8] Gerstenhaber, M. The Cohomology Structure Of An Associative Ring, Ann. of Math., Tome 78 (1963), pp. 267-288 | Article | MR 161898 | Zbl 0131.27302

[9] Getzler, E.; Jones, J. D. S. Operads, homotopy algebra and iterated integrals for double loop spaces (1994) (Preprint hep-th/9403055)

[10] Jones, J. D. S. Cyclic homology and equivariant homology, Invent. Math., Tome 87 (1987), pp. 403-423 | Article | MR 870737 | Zbl 0644.55005

[11] Kaufmann, R. M. A proof of a cyclic version of Deligne’s conjecture via cacti (2004) (arXiv:QA/0403340)

[12] Lawrence, R.; Sullivan, D. A free differential Lie algebra for the interval (2006) (arXiv:math/0610949v2)

[13] Loday, J.-L. Cyclic Homology, Springer-Verlag Tome 301 (1992) | MR 1217970 | Zbl 0780.18009

[14] Markl, M.; Shnider, S.; Stasheff, J. Operads in Algebra, Topology and Physics, Amer. Math. Soc., Providence, RI Tome 96 (2002) | MR 1898414 | Zbl 1017.18001

[15] Menichi, L. String topology for spheres (arXiv:math/0609304)

[16] Menichi, L. Batalin-Vilkovisky algebras and cyclic cohomology of Hopf algebras, K-Theory, Tome 32 (2004), pp. 231-251 | Article | MR 2114167 | Zbl 1101.19003

[17] Merkulov, S. A. De Rham model for string topology, Int. Math. Res. Not., Tome 55 (2004), pp. 2955-2981 | Article | MR 2099178 | Zbl 1066.55008

[18] Stasheff, J. Homotopy associativity of $H$-spaces I, Trans. AMS, Tome 108 (1963), pp. 275-292 | Article | MR 158400 | Zbl 0114.39402

[19] Stasheff, J. The intrinsic bracket on the deformation complex of an associative algebra, J. Pure Applied Algebra, Tome 89 (1993), pp. 231-235 | Article | MR 1239562 | Zbl 0786.57017

[20] Tradler, T. Infinity-inner-products on $A$-infinity algebras (to be published in J. Homotopy and Related Structures)

[21] Tradler, T.; Zeinalian, M. On the cyclic Deligne conjecture, J. Pure Appl. Algebra, Tome 204 (2006) no. 2, pp. 280-299 | Article | MR 2184812 | Zbl pre02242211

[22] Tradler, T.; Zeinalian, M. Algebraic string operations, K-Theory, Tome 38 (2007) no. 1, pp. 59-82 | Article | MR 2353864 | Zbl 1144.55012

[23] Tradler, T.; Zeinalian, M.; Sullivan, D. Infinity structure of Poincaré duality spaces, Algebr. Geom. Topol., Tome 7 (2007), pp. 233-260 | Article | MR 2308943 | Zbl 1137.57025

[24] Yang, T. A Batalin-Vilkovisky Algebra structure on the Hochschild Cohomology of Truncated Polynomials (arXiv:0707.4213)