The Expansion ⋆ mod ō (ℏ⁴) and Computer-Assisted Proof Schemes in the Kontsevich Deformation Quantization

References

• Ammar, M., Chloup, V., Gutt, S. (2008). Universal star products. Lett. Math. Phys. 84: 199–215.
   Web of Science ®Google Scholar
• Banks, P., Panzer, E., Pym, B. (2018). Multiple zeta values in deformation quantization. Preprint. arXiv:1812.11649 [math.QA].
   Google Scholar
• Bauer, C., Frink, A., Kreckel, R. (2002). Introduction to the GiNaC Framework for Symbolic Computation within the C++ Programming Language. J. Symb. Comp. 33: 1–12. http://www.ginac.de.
   Web of Science ®Google Scholar
• Bayen, F., Flato, M., Frønsdal, C., Lichnerowicz, A., Sternheimer, D. (1978). Deformation theory and quantization. I. Deformations of symplectic structures. Ann. Phys. (N. Y.). 111: 61–110.
   Web of Science ®Google Scholar
• Bayen, F., Flato, M., Frønsdal, C., Lichnerowicz, A., Sternheimer, D. (1978). Deformation theory and quantization. II. Physical applications. Ann. Phys. (N. Y.). 111: 111–151.
   Web of Science ®Google Scholar
• Ben Amar, N. (2007). A comparison between Rieffel’s and Kontsevich’s deformation quantizations for linear Poisson tensors. Pacific J. Math. 229: 1–24. (1)
   Web of Science ®Google Scholar
• Bouisaghouane, A., Buring, R., Kiselev, A. V. (2017). The Kontsevich tetrahedral flow revisited. J. Geom. Phys. 119: 272–285. Preprint arXiv:1608.01710 [q-alg]).
   Web of Science ®Google Scholar
• Bouisaghouane, A., Kiselev, A. V. (2017). Do the Kontsevich tetrahedral flows preserve or destroy the space of Poisson bi-vectors?. J. Phys: Conf. Ser. 804: 1–10. (Preprint arXiv:1609.06677 [q-alg])
   Google Scholar
• Buring, R. Software package Kontsevich-graph-series-cpp. https://github.com/rburing/kontsevich_graph_series-cpp
   Google Scholar
• Buring, R. List of integrands for the 149 basic Kontsevich graphs at ℏ4. http://rburing.nl/kontsevich_graph_weight_integrands4.txt
   Google Scholar
• Buring, R., Kiselev, A. V. (2019). Formality morphism as the mechanism of ⋆-product associativity: how it works. Collected Works Inst. Math. Kiev. 16: 22–43.
   Google Scholar
• Buring, R., Kiselev, A. V. (2017). On the Kontsevich ⋆-product associativity mechanism. Phys. Part. Nuclei Lett. 14: 403–407. Preprint arXiv:1602.09036 [q-alg]).
   Web of Science ®Google Scholar
• Buring, R., Kiselev, A. V. (2019). The orientation morphism: from graph cocycles to deformations of Poisson structures. J. Phys. Conf. Ser. 1194. Proc. 32nd Int. colloquium on Group-theoretical methods in Physics: Group32 (9–13 July 2018, CVUT Prague, Czech Republic), Paper 012017, 10 p. (Preprint arXiv:1811.07878 [math.CO]).
   Google Scholar
• Buring, R., Kiselev, A. V. (2015). The Table of Weights for Graphs with ⩽3 Internal Vertices in Kontsevich’s Deformation Quantization Formula. (3rd International Workshop on Symmetries of Discrete Systems & Processes, 3–7 August 2015, CVUT Děčín, Czech Republic), see Appendix A.1 in this paper.
   Google Scholar
• Cattaneo, A. S., Felder, G. (2000). A path integral approach to the Kontsevich quantization formula. Comm. Math. Phys. 212: 591–611. (3)
   Web of Science ®Google Scholar
• Dito, G. (1999). Kontsevich star product on the dual of a Lie algebra. Lett. Math. Phys. 4: 291–306.
   Google Scholar
• Felder, G., Willwacher, T. (2010). On the (ir)rationality of Kontsevich weights. Int. Math. Res. Not. 2010: 701–716. (4)
   Web of Science ®Google Scholar
• Felder, G., Shoikhet, B. (2000). Deformation quantization with traces. Lett. Math. Phys. 53: 75–86.
   Web of Science ®Google Scholar
• Gerstenhaber, M. (1964). On the deformation of rings and algebras. Ann. Math. 79: 59–103.
   Web of Science ®Google Scholar
• Grabowski, J., Marmo, G., Perelomov, A. M. (1993). Poisson structures: towards a classification. Mod. Phys. Lett. A. A8: 1719–1733.
   Google Scholar
• Kathotia, V. (1998). Kontsevich’s universal formula for deformation quantization and the Campbell – Baker – Hausdorff formula, I. Preprint arXiv:9811174 (v2) [math.QA].
   Google Scholar
• Kiselev, A. V. (2012). The twelve lectures in the (non) commutative geometry of differential equations. Preprint IHÉS/M/12/13 (Bures-sur-Yvette, France), 140 p.
   Google Scholar
• Kiselev, A. V. (2013). The geometry of variations in Batalin – Vilkovisky formalism. J. Phys: Conf. Ser. 474: 1–51. (Preprint arXiv:1312.1262 [math-ph])
   Google Scholar
• Kiselev, A. V. (2014). The Jacobi identity for graded-commutative variational Schouten bracket revisited. Phys. Part. Nuclei Lett. 11: 950–953. Preprint arXiv:1312.4140 [math-ph])
   Google Scholar
• Kiselev, A. V. (2017). The deformation quantization mapping of Poisson to associative structures in field theory. Banach Center Publ. 113: 219–242. Preprint arXiv:1705.01777 [q-alg])
   Google Scholar
• Kiselev, A. V. (2016). The right-hand side of the Jacobi identity: to be naught or not to be? J. Phys.: Conf. Ser. 670 Proc. XXIII Int. conf. ‘Integrable Systems and Quantum Symmetries’. (23–27 June 2015, CVUT Prague, Czech Republic), Paper 012030, 1–17. Preprint arXiv:1410.0173 [math-ph]).
   Google Scholar
• Kiselev, A. V. (2018). The calculus of multivectors on noncommutative jet spaces. J. Geom. Phys. 130: 130–167. (Preprint arXiv:1210.0726 [math.DG]).
   Google Scholar
• Kontsevich, M. (1993). Formal (non) commutative symplectic geometry. In: Corwin, L., Gelfand, I., Lepowsky, J., eds. The Gel’fand Mathematical Seminars, 1990-1992. Boston, MA: Birkhäuser, pp. 173–187.
   Google Scholar
• Kontsevich, M. (1994). Feynman diagrams and low-dimensional topology. First Europ. Congr. of Math. 2 (Paris, 1992), Progr. Math. 120, Birkhäuser, Basel, pp. 97–121.
   Google Scholar
• Kontsevich, M. (1995). Homological algebra of mirror symmetry. Proc. Intern. Congr. Math. 1 (Zürich, 1994), Birkhäuser, Basel, pp. 120–139.
   Google Scholar
• Kontsevich, M. (1997). Formality Conjecture. Deformation Theory and Symplectic Geometry (Ascona 1996, D. Sternheimer, J. Rawnsley and S. Gutt, eds), Math. Phys. Stud. 20, Kluwer Acad. Publ., Dordrecht, 139–156.
   Google Scholar
• Kontsevich, M. (2003). Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66: 157–216. Preprint q-alg/9709040)
   Web of Science ®Google Scholar
• Laurent – Gengoux, C., Picherau, A., Vanhaecke, P. (2013). Poisson Structures. Gründlehren Der Mathematischen Wissenschaften 347. Berlin: Springer – Verlag.
   Google Scholar
• Panzer, E., Pym, B. (2017). Private communication. https://www.mathematics.uni-bonn.de/veranstaltungskalender/17251
   Google Scholar
• Penkava, M., Vanhaecke, P. (2000). Deformation quantization of polynomial poisson algebras. J. Algebra. 227: 365–393. Preprint arXiv:math/9804022)
   Web of Science ®Google Scholar
• Polyak, M. (2003). Quantization of linear Poisson structures and degrees of maps. Lett. Math. Phys. 66: 15–35.
   Web of Science ®Google Scholar
• Vanhaecke, P. (1996). Integrable systems in the realm of algebraic geometry, Lect. Notes Math. 1638, Springer – Verlag, Berlin.
   Google Scholar
• Willwacher, T. (2014). The obstruction to the existence of a loopless star product. C. R. Math. Acad. Sci. Paris. 352: 881–883.
   Google Scholar