×

Distributive laws between the Three Graces. (English) Zbl 1448.18029

Authors’ abstract: By the Three Graces we refer, following J.-L. Loday, to the algebraic operads \(\mathcal{A}ss\), \(\mathcal{C}om\), and \(\mathcal{L}ie\), each generated by a single binary operation; algebras over these operads are respectively associative, commutative associative, and Lie. We classify all distributive laws (in the categorical sense of Beck) between these three operads. Some of our results depend on the computer algebra system Maple, especially its packages LinearAlgebra and Groebner.

MSC:

18M70 Algebraic operads, cooperads, and Koszul duality
16S37 Quadratic and Koszul algebras
16-XX Associative rings and algebras
17Bxx Lie algebras and Lie superalgebras
17B63 Poisson algebras
PDFBibTeX XMLCite
Full Text: arXiv Link

References:

[1] A. L. Agore, G. Militaru. The global extension problem, crossed products and co-flag noncommutative Poisson algebras. Journal of Algebra 426 (2015) 1-31. · Zbl 1393.17036
[2] F. Akman. On some generalizations of Batalin-Vilkovisky algebras. Journal of Pure and Applied Algebra 120 (1997), no. 2, 105-141. · Zbl 0885.17020
[3] F. Akman. A master identity for homotopy Gerstenhaber algebras. Communications in Mathematical Physics 209 (2000), no. 1, 51-76. · Zbl 0951.55019
[4] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer. Deformation theory and quantization. I. Deformations of symplectic structures. Annals of Physics 111 (1978), no. 1, 61-110. · Zbl 0377.53024
[5] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer. Deformation theory and quantization. II. Physical applications. Annals of Physics 111 (1978), no. 1, 111-151. · Zbl 0377.53025
[6] J. Beck. Distributive laws. In: B. Eckmann, Seminar on Triples and Categorical Homology Theory, pages 119-140. Lecture Notes in Mathematics, 80. Springer, Berlin-Heidelberg 1969. Available online:http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html · Zbl 0186.02902
[7] J. M. Boardman, R. M. Vogt. Homotopy Invariant Algebraic Structures on Topological Spaces. Lecture Notes in Mathematics, 347. Springer-Verlag, Berlin-New York, 1973. · Zbl 0285.55012
[8] M. R. Bremner, V. Dotsenko. Algebraic Operads: An Algorithmic Companion. CRC Press, Boca Raton, FL, 2016. · Zbl 1350.18001
[9] M. R. Bremner, V. Dotsenko. Distributive laws for the Lie and Com operads: Extended abstract. To appear in Proceedings of Maple Conference 2019, to be published in the Springer series Communications in Computer and Information Science. · Zbl 1458.18010
[10] M. R. Bremner, S. Madariaga. Lie and Jordan products in interchange algebras. Communications in Algebra 44 (2016), no. 8, 3485-3508. · Zbl 1402.17001
[11] X. Chen, A. Eshmatov, F. Eshmatov, S. Yang. The derived non-commutative Poisson bracket on Koszul Calabi-Yau algebras. Journal of Noncommutative Geometry 11 (2017), no. 1, 111-160. · Zbl 1453.16010
[12] K. Costello, O. Gwilliam. Factorization Algebras in Quantum Field Theory, Volume 1. New Mathematical Monographs, 31. Cambridge University Press, Cambridge, 2017. · Zbl 1377.81004
[13] V. Dolgushev, D. Tamarkin, B. Tsygan. The homotopy Gerstenhaber algebra of Hochschild cochains of a regular algebra is formal. Journal of Noncommutative Geometry 1 (2007), no. 1, 1-25. · Zbl 1144.18007
[14] D. R. Farkas, G. Letzter. Ring theory from symplectic geometry. Journal of Pure and Applied Algebra 125 (1998), 155-190. · Zbl 0974.17025
[15] V. A. Fok. Naqala kvantovo\(i mehaniki, Nauka, Moskva, 1976 (2nd edition)\)
[16] T. F. Fox, M. Markl. Distributive laws, bialgebras, and cohomology. In: Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), pages 167-205. Contemporary Mathematics, 202. American Mathematical Society, Providence, RI, 1997. · Zbl 0866.18008
[17] J. Francis. The tangent complex and Hochschild cohomology of En-rings. Compositio Mathematica 149 (2013), no. 3, 430-480. · Zbl 1276.18008
[18] I. G´alvez-Carrillo, A. Tonks, B. Vallette. Homotopy Batalin-Vilkovisky algebras. Journal of Noncommutative Geometry 6 (2012), no. 3, 539-602. · Zbl 1258.18005
[19] X. Garc´ıa-Mart´ınez, T. Van der Linden. A characterisation of Lie algebras via algebraic exponentiation. Advances in Mathematics 341 (2019), 92-117. · Zbl 1439.17008
[20] X. Garc´ıa-Mart´ınez, T. Van der Linden. A characterisation of Lie algebras amongst anticommutative algebras. Journal of Pure and Applied Algebra 223 (2019), no. 11, 4857-4870. · Zbl 1480.17002
[21] M. Gerstenhaber. The cohomology structure of an associative ring. Annals of Mathematics (2) 78 (1963) 267-288. · Zbl 0131.27302
[22] E. Getzler. Batalin-Vilkovisky algebras and two-dimensional topological field theories. Communications in Mathematical Physics 159 (1994), no. 2, 265-285. · Zbl 0807.17026
[23] V. Ginzburg and M.M. Kapranov. Koszul duality for operads. Duke Mathematical Journal 76(1) (1994) 203-272. · Zbl 0855.18006
[24] J. Huebschmann. Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras. Annales de l’Institut Fourier (Grenoble) 48 (1998), no. 2, 425-440. · Zbl 0973.17027
[25] Y. Kosmann-Schwarzbach. From Poisson algebras to Gerstenhaber algebras. Annales de l’Institut Fourier (Grenoble) 46 (1996), no. 5, 1243-1274. · Zbl 0858.17027
[26] Y. Kosmann-Schwarzbach. La g´eom´etrie de Poisson, cr´eation du XXe si‘ecle. [Poisson geometry, a twentieth-century creation]. Sim´eon-Denis Poisson, pages 129-172. Hist. Math. Sci. Phys., Ed. Ec. Polytech., Palaiseau, 2013.´ · Zbl 1294.01017
[27] Y. Kosmann-Schwarzbach. Les crochets de Poisson, de la m´ecanique c´eleste ‘a la m´ecanique quantique. [Poisson brackets, from celestial to quantum mechanics]. Sim´eon-Denis Poisson, pages 369- 401, Hist. Math. Sci. Phys., Ed. ´Ec. Polytech., Palaiseau, 2013. · Zbl 1294.01018
[28] Y. Kosmann-Schwarzbach, F. Magri. Poisson-Nijenhuis structures. Annales de l’Institut Henri Poincar´e: Physique Th´eorique 53 (1990), no. 1, 35-81. · Zbl 0707.58048
[29] F. Kubo. Finite-dimensional non-commutative Poisson algebras. Journal of Pure and Applied Algebra 113 (1996), no. 3, 307-314. · Zbl 0872.16020
[30] F. Kubo. Finite-dimensional non-commutative Poisson algebras. II. Communications in Algebra 29 (2001), no. 10, 4655-4669. · Zbl 1037.17024
[31] S. Lack. Composing PROPS. Theory and Applications of Categories 13 (2004), no. 9, 147-163. DISTRIBUTIVE LAWS BETWEEN THE THREE GRACES1341 · Zbl 1062.18007
[32] B. H. Lian, G. J. Zuckerman. New perspectives on the BRST-algebraic structure of string theory. Communications in Mathematical Physics 154 (1993), no. 3, 613-646. · Zbl 0780.17029
[33] M. Livernet, J.-L. Loday. The Poisson operad as a limit of associative operads. Unpublished preprint, March 1998.
[34] J.-L. Loday and B. Vallette. Algebraic Operads. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 346. Springer, Heidelberg, 2012. · Zbl 1260.18001
[35] M. Markl. Operads and PROPs. In: Handbook of Algebra. Vol. 5, pages 87-140. Elsevier/NorthHolland, Amsterdam, 2008. · Zbl 1211.18007
[36] M. Markl. Distributive laws and Koszulness. Annales de l’Institut Fourier (Grenoble) 46 (1996), no. 2, 307-323. · Zbl 0853.18005
[37] M. Markl, E. Remm. Algebras with one operation including Poisson and other Lie-admissible algebras. Journal of Algebra 299 (2006), no. 1, 171-189. · Zbl 1101.18004
[38] M. Markl, E. Remm. Operads for n-ary algebras – calculations and conjectures. Archivum Mathematicum (Brno) 47 (2011), no. 5, 377-387. · Zbl 1265.18015
[39] M. Markl, S. Shnider, and J.D. Stasheff. Operads in Algebra, Topology and Physics. Mathematical Surveys and Monographs, 96. American Mathematical Society, Providence, RI, 2002. · Zbl 1017.18001
[40] J. P. May. The Geometry of Iterated Loop Spaces. Lecture Notes in Mathematics, Vol. 271. SpringerVerlag, Berlin-New York, 1972. · Zbl 0244.55009
[41] S. A. Merkulov. Operads, deformation theory and F-manifolds. In: Hertling K., Marcolli M. (eds): Frobenius Manifolds, pages 213-251. Aspects of Mathematics, vol 36. Vieweg+Teubner Verlag, Wiesbaden, 2004. · Zbl 1075.53094
[42] S. A. Merkulov. Frobenius∞invariants of homotopy Gerstenhaber algebras, Duke Mathematical Journal 105 (2000), no. 3, 411-461. · Zbl 1021.53062
[43] C. Roger. Gerstenhaber and Batalin-Vilkovisky algebras: algebraic, geometric, and physical aspects. Archivum Mathematicum (Brno) 45 (2009), no. 4, 301-324. · Zbl 1212.58004
[44] A. E. Ruuge, F. Van Oystaeyen. Distortion of the Poisson bracket by the noncommutative Planck constants. Communications in Mathematical Physics 304 (2011), no. 2, 369-393. · Zbl 1230.81035
[45] I. P. Shestakov. Quantization of Poisson superalgebras and the specialty of Jordan superalgebras of Poisson type. Algebra i Logika 32 (1993), no. 5, 571-584, 587 (1994); translation in Algebra and Logic 32 (1993), no. 5, 309-317 (1994). · Zbl 0826.17013
[46] D. Sinha. Operads and knot spaces. Journal of the American Mathematical Society 19 (2006), no. 2, 461-486. · Zbl 1112.57004
[47] D. Sinha. The (non-equivariant) homology of the little disks operad. OPERADS 2009, pages 253-279. S´eminaires et Congr‘es, 26. Soci´et´e Math´ematique de France, Paris, 2013. · Zbl 1277.18012
[48] R. Street. The formal theory of monads. Journal of Pure and Applied Algebra 2 (1972), no. 2, 149-168. · Zbl 0241.18003
[49] V. Turaev. Poisson-Gerstenhaber brackets in representation algebras. Journal of Algebra 402 (2014) 435-478. · Zbl 1308.17027
[50] M. Van den Bergh. Double Poisson algebras. Transactions of the American Mathematical Society 360 (2008), no. 11, 5711-5769. · Zbl 1157.53046
[51] T. Voronov. Graded manifolds and Drinfel’d doubles for Lie bialgebroids. Contemporary Mathematics 315 (2002) 131-168. · Zbl 1042.53056
[52] T. Voronov. On the Poisson hull of a Lie algebra: a “noncommutative” moment space. Funktsional’ny˘ı Analiz i ego Prilozheniya 29 (1995) no. 3, 61-64. · Zbl 0860.17036
[53] P. Xu. Gerstenhaber algebras and BV-algebras in Poisson geometry. Communications in Mathematical Physics 200 (1999), no. 3, 545-560. · Zbl 0941.17016
[54] G. W. Zinbiel. Encyclopedia of types of algebras 2010. Operads and Universal Algebra, pages 21- 297. Nankai Series in Pure, Applied Mathematics and Theoretical Physics, 9. World Scientific, Hackensack, 2012. · Zbl 1351.17001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.