Prop of ribbon hypergraphs and strongly homotopy involutive Lie bialgebras

2023 • In *International Mathematics Research Notices, 7*, p. 5685-5727

Peer reviewed

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Keywords :

Lie bialgebra; string topology; graph complexes

Abstract :

[en] For any integer d we introduce a prop RHrad of d-oriented ribbon hypergraphs (in which "edges" can connect more than two vertices) and prove that there exists a canonical morphism Holieb⋄d⟶RHrad from the minimal resolution Holieb⋄d of the (degree shifted) prop of involutive Lie bialgebras into the prop of ribbon hypergraphs which is non-trivial on each generator of Holieb⋄d. As an application we show that for any graded vector space W equipped with a family of cyclically (skew)symmetric higher products the associated vector space of cyclic words in elements of W has a combinatorial Holieb⋄d-structure. As an illustration we construct for each natural number N≥1 an explicit combinatorial strongly homotopy involutive Lie bialgebra structure on the vector space of cyclic words in N graded letters which extends the well-known Schedler's necklace Lie bialgebra structure from the formality theory of the Goldman-Turaev Lie bialgebra in genus zero.

Disciplines :

Mathematics

Merkulov, Sergei ^{}; University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit

External co-authors :

no

Language :

English

Title :

Prop of ribbon hypergraphs and strongly homotopy involutive Lie bialgebras

Publication date :

2023

Journal title :

International Mathematics Research Notices

ISSN :

1687-0247

Publisher :

Oxford University Press, Oxford, United Kingdom

Volume :

7

Pages :

5685-5727

Peer reviewed :

Peer reviewed

Available on ORBilu :

since 07 January 2019

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WoS citations^{™}

1

- Alekseev, A., N. Kawazumi, Y. Kuno, and F. Naef. “The Goldman–Turaev Lie bialgebra in genus zero and the Kashiwara–Vergne problem.” Adv. Math. 326 (2018): 1–53.
- Alekseev, A., N. Kawazumi, Y. Kuno, and F. Naef. “The Goldman–Turaev Lie bialgebra and the Kashiwara–Vergne problem in higher genera.” (Forthcoming) arXiv:1703.0581.
- Alekseev, A. and F. Naef. “Goldman–Turaev formality from the Knizhnik–Zamolodchikov connection.” C. R. Math. Acad. Sci. Paris 355, no. 11 (2017): 1138–47.
- Barannikov, S. “Noncommutative Batalin–Vilkovisky geometry and matrix integrals.” C. R. Math. Acad. Sci. Paris 348, no. 7–8 (2010): 359–62.
- Barannikov, S. “Matrix De Rham complex and quantum A-infinity algebras.” Lett. Math. Phys. 104 (2014): 373–95.
- Chas, M. and D. Sullivan. “String topology.” (Forthcoming) arXiv:math/9911159.
- Chas, M. and D. Sullivan. “Closed String Operators in Topology Leading to Lie Bialgebras and Higher String Algebra.” In The Legacy of Niels Henrik Abel. 771–84. Berlin: Springer, 2004.
- Campos, R., S. Merkulov, and T. Willwacher. “The Frobenius properad is Koszul.” Duke Math. J. 165, no. 1 (2016): 2921–89.
- Chas, M. “Combinatorial Lie bialgebras of curves on surfaces.” Topology 43, no. 3 (2004): 543–68.
- Chen, X., F. Eshmatov, and W. L. Gan. “Quantization of the Lie bialgebra of string topology.” Comm. Math. Phys. 301, no. 1 (2011): 37–53.
- Cieliebak, K., K. Fukaya, and J. Latschev. “Homological algebra related to surfaces with boundary.” (2015): preprint arxiv:1508.02741.
- Drinfeld, V. “Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations.” Soviet Math. Dokl. 27 (1983): 68–71.
- Drinfeld, V. “On quasitriangular quasi-Hopf algebras and a group closely connected with Gal Q/Q .” Leningrad Math. J 2, no. 4 (1991): 829–60.
- Drummond-Cole, G. C., J. Terilla, and T. Tradler. “Algebras over Cobar(coFrob).” J. Homotopy Relat. Struct. 5, no. 1 (2010): 15–36.
- Etingof, P. and O. Schiffmann. Lectures on Quantum Groups. Somerville, MA: International Press, 2002.
- Goldman, W. “Invariant functions on Lie groups and Hamiltonian flows of surface group representations.” Invent. Math. 85, no. 2 (1986): 263–302.
- Kravchenko, O. “Deformations of Batalin–Vilkovisky Algebras.” In Poisson Geometry (Warsaw, 1998), vol. 51. Banach Center Publ. 131–9. Warsaw: Polish Acad. Sci., 2000.
- Kuno, Y. “Private communication.” (2016).
- Lambrechts, P. and D. Stanley. “Poincaré duality and commutative differential graded algebras.” Ann. Sci. Éc. Norm. Supér. (4) 41, no. 4 (2008): 495–509.
- Lando, S. K. and A. K. Zvonkin. Graphs on Surfaces and their Applications. Berlin: Springer, 2004.
- Markl, M. and A. A. Voronov. “PROPped up Graph Cohomology.” In Algebra, Arithmetic, and Geometry: In Honor of Yu I. Manin, Vol II. Progr. Math., 270. 249–81. Boston, MA: Birkhäuser Boston, Inc., 2009.
- Massuyeau, G. “Formal descriptions of Turaev’s loop operations.” Quantum Topol. 9 (2018): 39–117.
- Merkulov, S. and B. Vallette. “Deformation theory of representations of prop(erad)s II.” J. Reine Angew. Math. 634 (2009): 51–106. 636, 123–174.
- Merkulov, S. and T. Willwacher. “Props of ribbon graphs, involutive Lie bialgebras and moduli spaces of curves.” (2015): 51. Preprint arXiv:1511.07808.
- Merkulov, S. A. and T. Willwacher. “Classification of universal formality maps for quantizations of Lie bialgebras.” Compos. Math. 156 (2020): 2111–48.
- Naef, F. and T. Willwacher. “String topology and configuration spaces of two points.” (2019): preprint arXiv:1911.06202.
- Schedler, T. “A Hopf algebra quantizing a necklace Lie algebra canonically associated to a quiver.” Int. Math. Res. Not. IMRN 12 (2005): 725–60.
- Turaev, V. G. “Skein quantization of Poisson algebras of loops on surfaces.” Ann. Sci. Éc. Norm. Supér. (4) 24, no. 6 (1991): 635–704.
- Vallette, B. “A Koszul duality for props.” Trans. Amer. Math. Soc. 359 (2007): 4865–943.