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Keywords:
infinitesimal bialgebra; quasitriangular infinitesimal bialgebra
Summary:
We present a notion of an anti-covariant bialgebra extending the anti-symmetric infinitesimal bialgebra and also provide some equivalent characterizations of it. We also prove that an anti-associative Yang-Baxter pair can produce a special Rota-Baxter system.
References:
[1] Aguiar, M.: Infinitesimal Hopf algebras. New Trends in Hopf Algebra Theory Contemporary Mathematics 267. AMS, Providence (2000), 1-29. DOI 10.1090/conm/267 | MR 1800704 | Zbl 0982.16028
[2] Aguiar, M.: On the associative analog of Lie bialgebras. J. Algebra 244 (2001), 492-532. DOI 10.1006/jabr.2001.8877 | MR 1859038 | Zbl 0991.16033
[3] Bai, C.: Double constructions of Frobenius algebras, Connes cocycles and their duality. J. Noncommut. Geom. 4 (2010), 475-530. DOI 10.4171/JNCG/64 | MR 2718800 | Zbl 1250.17028
[4] Bai, C., Guo, L., Ma, T.: Bialgebras, Frobenius algebras and associative Yang-Baxter equations for Rota-Baxter algebras. Available at https://arxiv.org/abs/2112.10928 (2021), 27 pages. MR 4534711
[5] Brzeziński, T.: Rota-Baxter systems, dendriform algebras and covariant bialgebras. J. Algebra 460 (2016), 1-25. DOI 10.1016/j.jalgebra.2016.04.018 | MR 3510392 | Zbl 1376.16039
[6] Drinfel'd, V. G.: Hamiltonian structures on Lie groups, Lie bialgebras and geometric meaning of the classical Yang-Baxter equations. Sov. Math., Dokl. 27 (1983), 67-71 translation from Dokl. Akad. Nauk SSSR 268 1983 285-287. MR 0688240 | Zbl 0526.58017
[7] Gao, X., Wang, X.: Infinitesimal unitary Hopf algebras and planar rooted forests. J. Algebr. Comb. 49 (2019), 437-460. DOI 10.1007/s10801-018-0830-6 | MR 3954430 | Zbl 1437.16030
[8] Joni, S. A., Rota, G.-C.: Coalgebras and bialgebras in combinatorics. Stud. Appl. Math. 61 (1979), 93-139. DOI 10.1002/sapm197961293 | MR 0544721 | Zbl 0471.05020
[9] Liu, L., Makhlouf, A., Menini, C., Panaite, F.: BiHom-Novikov algebras and infinitesimal BiHom-bialgebras. J. Algebra 560 (2020), 1146-1172. DOI 10.1016/j.jalgebra.2020.06.012 | MR 4117866 | Zbl 07239031
[10] Loday, J.-L., Ronco, M.: On the structure of cofree Hopf algebras. J. Reine Angew. Math. 592 (2006), 123-155. DOI 10.1515/CRELLE.2006.025 | MR 2222732 | Zbl 1096.16019
[11] Ma, T., Li, J.: Nonhomogeneous associative Yang-Baxter equations. Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 65 (2022), 97-118. MR 4408202
[12] Ma, T., Li, J., Yang, T.: Coquasitriangular infinitesimal BiHom-bialgebras and related structures. Commun. Algebra 49 (2021), 2423-2443. DOI 10.1080/00927872.2021.1871913 | MR 4255016 | Zbl 1476.17017
[13] Ma, T., Makhlouf, A., Silvestrov, S.: Rota-Baxter cosystems and coquasitriangular mixed bialgebras. J. Algebra Appl. 20 (2021), Article ID 2150064, 28 pages. DOI 10.1142/S021949882150064X | MR 4251744 | Zbl 1476.16030
[14] Ma, T., Yang, H.: Drinfeld double for infinitesimal BiHom-bialgebras. Adv. Appl. Clifford Algebr. 30 (2020), Article ID 42, 22 pages. DOI 10.1007/s00006-020-01071-x | MR 4118445 | Zbl 1473.17056
[15] Ma, T., Yang, H., Zhang, L., Zheng, H.: Quasitriangular covariant monoidal BiHom-bialgebras, associative monoidal BiHom-Yang-Baxter equations and Rota-Baxter paired monoidal BiHom-modules. Colloq. Math. 161 (2020), 189-221. DOI 10.4064/cm7993-9-2019 | MR 4097065 | Zbl 1465.16033
[16] Wang, S., Wang, S.: Drinfeld double for braided infinitesimal Hopf algebras. Commun. Algebra 42 (2014), 2195-2212. DOI 10.1080/00927872.2013.766796 | MR 3169699 | Zbl 1301.16035
[17] Yau, D.: Infinitesimal Hom-bialgebras and Hom-Lie bialgebras. Available at https://arxiv.org/abs/1001.5000 (2010), 35 pages. MR 2660540
[18] Zhang, Y., Chen, D., Gao, X., Luo, Y.-F.: Weighted infinitesimal unitary bialgebras on rooted forests and weighted cocycles. Pac. J. Math. 302 (2019), 741-766. DOI 10.2140/pjm.2019.302.741 | MR 4036749 | Zbl 1435.16005
[19] Zhang, Y., Gao, X.: Weighted infinitesimal bialgebras. Available at https://arxiv.org/abs/1810.10790v3 (2022), 44 pages.
[20] Zhang, Y., Gao, X., Luo, Y.: Weighted infinitesimal unitary bialgebras of rooted forests, symmetric cocycles and pre-Lie algebras. J. Algebr. Comb. 53 (2021), 771-803. DOI 10.1007/s10801-020-00942-7 | MR 4258069 | Zbl 1476.16041
[21] Zhelyabin, V. N.: Jordan bialgebras and their connection with Lie bialgebras. Algebra Logic 36 (1997), 1-15. DOI 10.1007/BF02671949 | MR 1454688 | Zbl 0935.17014
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