{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T15:25:13Z","timestamp":1772119513157,"version":"3.50.1"},"reference-count":18,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2024,1,20]],"date-time":"2024-01-20T00:00:00Z","timestamp":1705708800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,1,20]],"date-time":"2024-01-20T00:00:00Z","timestamp":1705708800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UID\/MAT\/04459\/2020"],"award-info":[{"award-number":["UID\/MAT\/04459\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100005765","name":"Universidade de Lisboa","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100005765","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Algebr Represent Theor"],"published-print":{"date-parts":[[2024,4]]},"abstract":"Abstract<\/jats:title>\n \n We propose a quantization algebra of the Loday-Ronco Hopf algebra\n \n \n $$k[Y^\\infty ]$$<\/jats:tex-math>\n \n \n k<\/mml:mi>\n [<\/mml:mo>\n \n Y<\/mml:mi>\n \u221e<\/mml:mi>\n <\/mml:msup>\n ]<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , based on the Topological Recursion formula of Eynard and Orantin. We have shown in previous works that the Loday-Ronco Hopf algebra of planar binary trees is a space of solutions for the genus 0 version of Topological Recursion, and that an extension of the Loday Ronco Hopf algebra as to include some new graphs with loops is the correct setting to find a solution space for arbitrary genus. Here we show that this new algebra\n \n \n $$k[Y^\\infty ]_h$$<\/jats:tex-math>\n \n \n k<\/mml:mi>\n \n \n [<\/mml:mo>\n \n Y<\/mml:mi>\n \u221e<\/mml:mi>\n <\/mml:msup>\n ]<\/mml:mo>\n <\/mml:mrow>\n h<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is still a Hopf algebra that can be seen in some sense to be made precise in the text as a quantization of the Hopf algebra of planar binary trees, and that the solution space of Topological Recursion\n \n \n $$\\mathcal {A}^h_{\\text {TopRec}}$$<\/jats:tex-math>\n \n \n \n A<\/mml:mi>\n <\/mml:mrow>\n TopRec<\/mml:mtext>\n h<\/mml:mi>\n <\/mml:msubsup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is a subalgebra of a quotient algebra\n \n \n $$\\mathcal {A}_{\\text {Reg}}^h$$<\/jats:tex-math>\n \n \n A<\/mml:mi>\n \n Reg<\/mml:mtext>\n <\/mml:mrow>\n h<\/mml:mi>\n <\/mml:msubsup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n obtained from\n \n \n $$k[Y^\\infty ]_h$$<\/jats:tex-math>\n \n \n k<\/mml:mi>\n \n \n [<\/mml:mo>\n \n Y<\/mml:mi>\n \u221e<\/mml:mi>\n <\/mml:msup>\n ]<\/mml:mo>\n <\/mml:mrow>\n h<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n that nevertheless doesn\u2019t inherit the Hopf algebra structure. We end the paper with a discussion on the cohomology of\n \n \n $$\\mathcal {A}^h_{\\text {TopRec}}$$<\/jats:tex-math>\n \n \n \n A<\/mml:mi>\n <\/mml:mrow>\n TopRec<\/mml:mtext>\n h<\/mml:mi>\n <\/mml:msubsup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n in low degree.\n <\/jats:p>","DOI":"10.1007\/s10468-024-10253-1","type":"journal-article","created":{"date-parts":[[2024,1,20]],"date-time":"2024-01-20T03:02:12Z","timestamp":1705719732000},"page":"1177-1201","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Quantization of the Loday-Ronco Hopf Algebra"],"prefix":"10.1007","volume":"27","author":[{"given":"Jo\u00e3o N.","family":"Esteves","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,1,20]]},"reference":[{"key":"10253_CR1","unstructured":"Abe, E.: Hopf algebras, Cambridge Tracts in Mathematics, Cambridge University Press, 9780521604895, (2004) https:\/\/books.google.pt\/books?id=D0AIcewz5-8C"},{"issue":"2","key":"10253_CR2","doi-asserted-by":"publisher","first-page":"473","DOI":"10.1016\/j.jalgebra.2005.06.021","volume":"295","author":"M Aguiar","year":"2006","unstructured":"Aguiar, M., Sottile, F.: Structure of the Loday-Ronco Hopf algebra of trees. J. Algebra 295(2), 473\u2013511 (2006). https:\/\/doi.org\/10.1016\/j.jalgebra.2005.06.021","journal-title":"J. Algebra"},{"key":"10253_CR3","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1007\/BF02698547","volume":"68","author":"MF Atiyah","year":"1988","unstructured":"Atiyah, M.F.: Topological quantum field theory. Publications Math\u00e9matiques de l\u2019IH\u00c9S 68, 175\u2013186 (1988)","journal-title":"Publications Math\u00e9matiques de l\u2019IH\u00c9S"},{"key":"10253_CR4","doi-asserted-by":"crossref","unstructured":"Di\u00a0Francesco, P., Ginsparg, P.H., Zinn-Justin, J.: 2-D Gravity and random matrices, Phys. Rept., 254, 1\u2013133, (1995). https:\/\/www.sciencedirect.com\/science\/article\/abs\/pii\/037015739400084G?via%3Dihub","DOI":"10.1016\/0370-1573(94)00084-G"},{"key":"10253_CR5","doi-asserted-by":"crossref","unstructured":"incollection, Dumitrescu, O., Mulase, M.: Lectures on the topological recursion for higgs bundles and quantum curves: the geometry, topology and physics of moduli spaces of higgs bundles. World Scientific, 103\u2013198 (2018). https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/9789813229099_0003","DOI":"10.1142\/9789813229099_0003"},{"key":"10253_CR6","doi-asserted-by":"publisher","unstructured":"incollection, Dumitrescu, O, Mulase, M, Safnuk, B, Sorkin, A, The spectral curve of the Eynard-Orantin recursion via the Laplace transform: algebraic and geometric aspects of integrable systems and random matrices. Contemp. Math. Amer. Math. Soc., Providence, RI, 593, 263\u2013315 (2013). https:\/\/doi.org\/10.1090\/conm\/593\/11867","DOI":"10.1090\/conm\/593\/11867"},{"key":"10253_CR7","doi-asserted-by":"crossref","unstructured":"incollection, Esteves, JN.: Hopf algebras and topological recursion: Topological recursion and its influence in analysis, geometry, and topology. In: Liu, C.-C.M., Mulase, M. (eds.) Am. Math. Soc. Proc. Sympos. Pure Math, pp. 333\u2013357. (2018)","DOI":"10.1090\/pspum\/100\/10"},{"key":"10253_CR8","doi-asserted-by":"crossref","unstructured":"Esteves, JN.: Hopf algebras and topological recursion, J. Phys. A Math. Theor., 48(44), 445205 (2015) http:\/\/stacks.iop.org\/1751-8121\/48\/i=44\/a=445205","DOI":"10.1088\/1751-8113\/48\/44\/445205"},{"issue":"2","key":"10253_CR9","doi-asserted-by":"publisher","first-page":"347","DOI":"10.4310\/CNTP.2007.v1.n2.a4","volume":"1","author":"B Eynard","year":"2007","unstructured":"Eynard, B., Orantin, N.: Invariants of algebraic curves and topological expansion. Commun. Number Theory Phys. 1(2), 347\u2013452 (2007). https:\/\/doi.org\/10.4310\/CNTP.2007.v1.n2.a4","journal-title":"Commun. Number Theory Phys."},{"issue":"2","key":"10253_CR10","doi-asserted-by":"publisher","first-page":"483","DOI":"10.1007\/s00220-015-2361-5","volume":"337","author":"B Eynard","year":"2015","unstructured":"Eynard, B., Orantin, N.: Computation of open gromov-witten invariants for toric calabi-yau 3-folds by topological recursion, a proof of the bkmp conjecture. Commun. Math. Phys. 337(2), 483\u2013567 (2015). https:\/\/doi.org\/10.1007\/s00220-015-2361-5","journal-title":"Commun. Math. Phys."},{"key":"10253_CR11","unstructured":"Eynard, B.: The geometry of integrable systems. tau functions and homology of spectral curves. Perturbative definition (2019), arXiv:1706.04938"},{"issue":"1","key":"10253_CR12","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1023\/A:1008723801201","volume":"13","author":"A Frabetti","year":"2001","unstructured":"Frabetti, A.: Simplicial properties of the set of planar binary trees. J. Algebraic Combin. 13(1), 41\u201365 (2001). https:\/\/doi.org\/10.1023\/A:1008723801201","journal-title":"J. Algebraic Combin."},{"key":"10253_CR13","doi-asserted-by":"crossref","unstructured":"Iliopoulos, J., Itzykson, C., Martin, A.: Functional methods and perturbation theory. Rev. Mod. Phys. 47, 165\u2013192 (1975). https:\/\/link.aps.org\/doi\/10.1103\/RevModPhys.47.165","DOI":"10.1103\/RevModPhys.47.165"},{"key":"10253_CR14","doi-asserted-by":"crossref","unstructured":"Liu, C.C.M., Mulase, M.: Topological recursion and its influence in analysis, geometry, and topology. Proceedings of Symposia in Pure Mathematics, American Mathematical Society, 9781470435417 (2018) https:\/\/books.google.pt\/books?id=39V7DwAAQBAJ","DOI":"10.1090\/pspum\/100"},{"issue":"2","key":"10253_CR15","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1006\/aima.1998.1759","volume":"139","author":"J-L Loday","year":"1998","unstructured":"Loday, J.-L., Ronco, M.O.: Hopf algebra of the planar binary trees. Adv. Math. 139(2), 293\u2013309 (1998). https:\/\/doi.org\/10.1006\/aima.1998.1759","journal-title":"Adv. Math."},{"key":"10253_CR16","doi-asserted-by":"crossref","unstructured":"Milnor, J.W., Moore, J.C.: On the structure of hopf algebras. Annals Math. 81(2), 211\u2013264, 0003486X (1965). http:\/\/www.jstor.org\/stable\/1970615","DOI":"10.2307\/1970615"},{"key":"10253_CR17","doi-asserted-by":"crossref","unstructured":"Mulase, M., Sulkowski, P.: Spectral curves and the schr\u00f6dinger equations for the eynard-orantin recursion. Adv. Theor. Math. Phys. 19(5), 9551015 (2015). https:\/\/www.intlpress.com\/site\/pub\/pages\/journals\/items\/atmp\/content\/vols\/0019\/0005\/a002\/","DOI":"10.4310\/ATMP.2015.v19.n5.a2"},{"key":"10253_CR18","unstructured":"Wang, Z., Zhou, J.: A formalism of abstract quantum field theory of summation of fat graphs (2021), arXiv:2108.10498"}],"container-title":["Algebras and Representation Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10468-024-10253-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10468-024-10253-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10468-024-10253-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,5,3]],"date-time":"2024-05-03T05:27:23Z","timestamp":1714714043000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10468-024-10253-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,20]]},"references-count":18,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2024,4]]}},"alternative-id":["10253"],"URL":"https:\/\/doi.org\/10.1007\/s10468-024-10253-1","relation":{"has-preprint":[{"id-type":"doi","id":"10.21203\/rs.3.rs-3006766\/v1","asserted-by":"object"}]},"ISSN":["1386-923X","1572-9079"],"issn-type":[{"value":"1386-923X","type":"print"},{"value":"1572-9079","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,1,20]]},"assertion":[{"value":"31 May 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 January 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 January 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"Not applicable.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing Interests"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical Approval"}}]}}