We define a BV-structure on the Hochschild cohomology of a unital, associative algebra 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 -algebra with a symmetric and non-degenerate -inner product.
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 -infinie unitaire munie d’une forme bilinéaire symétrique -infinie non dégénérée.
Keywords: Hochschild cohomology, Batalin Vilkovisky algebra
Mot clés : cohomologie de Hochschild, algèbre de Batalin Vilkovisky
Tradler, Thomas 1
@article{AIF_2008__58_7_2351_0, author = {Tradler, Thomas}, title = {The {Batalin-Vilkovisky} {Algebra} on {Hochschild} {Cohomology} {Induced} by {Infinity} {Inner} {Products}}, journal = {Annales de l'Institut Fourier}, pages = {2351--2379}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {58}, number = {7}, year = {2008}, doi = {10.5802/aif.2417}, mrnumber = {2498354}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2417/} }
TY - JOUR AU - Tradler, Thomas TI - The Batalin-Vilkovisky Algebra on Hochschild Cohomology Induced by Infinity Inner Products JO - Annales de l'Institut Fourier PY - 2008 SP - 2351 EP - 2379 VL - 58 IS - 7 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2417/ DO - 10.5802/aif.2417 LA - en ID - AIF_2008__58_7_2351_0 ER -
%0 Journal Article %A Tradler, Thomas %T The Batalin-Vilkovisky Algebra on Hochschild Cohomology Induced by Infinity Inner Products %J Annales de l'Institut Fourier %D 2008 %P 2351-2379 %V 58 %N 7 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2417/ %R 10.5802/aif.2417 %G en %F AIF_2008__58_7_2351_0
Tradler, Thomas. The Batalin-Vilkovisky Algebra on Hochschild Cohomology Induced by Infinity Inner Products. Annales de l'Institut Fourier, Volume 58 (2008) no. 7, pp. 2351-2379. doi : 10.5802/aif.2417. https://aif.centre-mersenne.org/articles/10.5802/aif.2417/
[1] String Topology (1999) (preprint GT/9911159)
[2] A Homotopy Theoretic Realization Of String Topology, Math. Ann., Volume 324 (2002), pp. 773-798 | DOI | MR | Zbl
[3] The loop homology algebra of spheres and projective spaces, Progr. Math., 215, Birkhäuser, Basel, 2004 | MR | Zbl
[4] Non-commutative differential geometry, Publ. Math. IHÉS, Volume 62 (1985), pp. 257-360 | Numdam | MR | Zbl
[5] Topological conformal field theories and Calabi-Yau categories, Adv. Math., Volume 210 (2007), pp. 165-214 | DOI | MR
[6] Rational BV-algebra in String Topology (2007) (arXiv:0705.4194) | Numdam | MR
[7] Loop homology algebra of a closed manifold (arXiv:math/0203137v2)
[8] The Cohomology Structure Of An Associative Ring, Ann. of Math., Volume 78 (1963), pp. 267-288 | DOI | MR | Zbl
[9] Operads, homotopy algebra and iterated integrals for double loop spaces (1994) (Preprint hep-th/9403055)
[10] Cyclic homology and equivariant homology, Invent. Math., Volume 87 (1987), pp. 403-423 | DOI | MR | Zbl
[11] A proof of a cyclic version of Deligne’s conjecture via cacti (2004) (arXiv:QA/0403340)
[12] A free differential Lie algebra for the interval (2006) (arXiv:math/0610949v2)
[13] Cyclic Homology, 301, Springer-Verlag, 1992 | MR | Zbl
[14] Operads in Algebra, Topology and Physics, 96, Amer. Math. Soc., Providence, RI, 2002 | MR | Zbl
[15] String topology for spheres (arXiv:math/0609304)
[16] Batalin-Vilkovisky algebras and cyclic cohomology of Hopf algebras, K-Theory, Volume 32 (2004), pp. 231-251 | DOI | MR | Zbl
[17] De Rham model for string topology, Int. Math. Res. Not., Volume 55 (2004), pp. 2955-2981 | DOI | MR | Zbl
[18] Homotopy associativity of -spaces I, Trans. AMS, Volume 108 (1963), pp. 275-292 | DOI | MR | Zbl
[19] The intrinsic bracket on the deformation complex of an associative algebra, J. Pure Applied Algebra, Volume 89 (1993), pp. 231-235 | DOI | MR | Zbl
[20] Infinity-inner-products on -infinity algebras (to be published in J. Homotopy and Related Structures)
[21] On the cyclic Deligne conjecture, J. Pure Appl. Algebra, Volume 204 (2006) no. 2, pp. 280-299 | DOI | MR
[22] Algebraic string operations, K-Theory, Volume 38 (2007) no. 1, pp. 59-82 | DOI | MR | Zbl
[23] Infinity structure of Poincaré duality spaces, Algebr. Geom. Topol., Volume 7 (2007), pp. 233-260 | DOI | MR | Zbl
[24] A Batalin-Vilkovisky Algebra structure on the Hochschild Cohomology of Truncated Polynomials (arXiv:0707.4213)
Cited by Sources: