Hochschild cohomology of the Weyl algebra and traces in deformation quantization. (English) Zbl 1106.53055
The authors obtain an explicit expression for the canonical trace in deformation quantization of symplectic manifolds.
Let \(M\) be a symplectic manifold. Fedosov’s and Nest and Tsygan’s works give explicit formulas for the trace in the case of the constant function 1 and prove the index theorem for \(M\). The purpose of this paper is to generalize these results by using a noncommutative version of Fedosov’s construction.
The authors first give a formula for a cocycle generating the Hochschild cohomology of the Weyl algebra. The main result of the paper is then proved. Interesting formulas are finally obtained for the trace on general functions on \(M\).
Let \(M\) be a symplectic manifold. Fedosov’s and Nest and Tsygan’s works give explicit formulas for the trace in the case of the constant function 1 and prove the index theorem for \(M\). The purpose of this paper is to generalize these results by using a noncommutative version of Fedosov’s construction.
The authors first give a formula for a cocycle generating the Hochschild cohomology of the Weyl algebra. The main result of the paper is then proved. Interesting formulas are finally obtained for the trace on general functions on \(M\).
Reviewer: Angela Gammella (Creil)
MSC:
| 53D17 | Poisson manifolds; Poisson groupoids and algebroids |
| 53D55 | Deformation quantization, star products |
References:
| [1] | F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer, Deformation theory and quantization, I, II , Ann. Physics 111 (1977), 61–110.; 111–151. ; Mathematical Reviews (MathSciNet): · Zbl 0377.53024 · doi:10.1016/0003-4916(78)90224-5 |
| [2] | H. Cartan, “La transgression dans un groupe de Lie et dans un espace fibré principal” in Colloque de topologie: Espaces fibrés (Brussels, 1950) , Masson, Paris, 1951, 57–71.; reprinted in Oeuvres, Vol. III , Springer, Berlin, 1979, 1268–1283. ; Mathematical Reviews (MathSciNet): |
| [3] | –. –. –. –., “Notions d’algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Lie” in Colloque de topologie: Espaces fibrés (Brussels, 1950) , Masson, Paris, 1951, 15–27.; reprinted in Ouevres, Vol. III , Springer, Berlin, 1979, 1255–1267. ; Mathematical Reviews (MathSciNet): · Zbl 0045.30601 |
| [4] | A. Connes, M. Flato, and D. Sternheimer, Closed star products and cyclic cohomology , Lett. Math. Phys. 24 (1992), 1–12. · Zbl 0767.55005 · doi:10.1007/BF00429997 |
| [5] | B. Fedosov, Deformation Quantization and Index Theory , Math. Top. 9 , Akademie, Berlin, 1996. · Zbl 0867.58061 |
| [6] | B. L. Feĭgin and B. L. Tsygan, Cohomology of the Lie algebras of generalized Jacobi matrices (in Russian), Funktsional. Anal. i Prilozhen. 17 , no. 2 (1983), 86–87.; English translation in Funct. Anal. Appl. 17 , no. 2 (1983), 153–155. |
| [7] | –. –. –. –., “Riemann-Roch theorem and Lie algebra cohomology, I” in Proceedings of the Winter School on Geometry and Physics (Srní, Czech Rep., 1988) , Rend. Circ. Mat. Palermo (2) Suppl. 21 , Circ. Mat. Palermo, Palermo, Italy, 1989, 15–52. |
| [8] | D. B. Fuks [Fuchs], Cohomology of Infinite-Dimensional Lie Algebras , Contemp. Soviet Math., Consultants Bureau, New York, 1986. · Zbl 0667.17005 |
| [9] | I. M. Gel’fand and D. A. Kazhdan, Some problems of differential geometry and the calculation of the cohomology of Lie algebras of vector fields , Sov. Math. Dokl. 12 (1971), 1367–1370. · Zbl 0238.58001 |
| [10] | G. Hochschild and J.-P. Serre, Cohomology of Lie algebras , Ann. of Math. (2) 57 (1953), 591–603. · Zbl 0053.01402 · doi:10.2307/1969740 |
| [11] | M. Kontsevich, Quantization of Poisson manifolds , Lett. Math. Phys. 66 (2003), 157–216. · Zbl 1058.53065 · doi:10.1023/B:MATH.0000027508.00421.bf |
| [12] | S. Mac Lane, Homology , Classics Math., Springer, Berlin, 1975. |
| [13] | R. Nest and B. Tsygan, Algebraic index theorem , Comm. Math. Phys. 172 (1995), 223–262. · Zbl 0887.58050 · doi:10.1007/BF02099427 |
| [14] | –. –. –. –., “On the cohomology ring of an algebra” in Advances in Geometry , Progr. Math. 172 , Birkhäuser, Boston, 1999, 337–370. · Zbl 0919.57019 |
| [15] | B. Shoikhet, A proof of the Tsygan formality conjecture for chains , Adv. Math. 179 (2003), 7–37. · Zbl 1163.53350 · doi:10.1016/S0001-8708(02)00023-3 |
| [16] | B. Tsygan, “Formality conjectures for chains” in Differential Topology, Infinite-Dimensional Lie Algebras, and Applications , Amer. Math. Soc. Transl. Ser. 2 194 , Amer. Math. Soc., Providence, 1999, 261–274. · Zbl 0962.18008 |
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