Drinfeld Double of Braided Lie Bialgebras and Braided Infinitesimal Hopf Algebras

Abstract

The present paper is a continuation of [Comm. Algebra, 2014, 42(5): 2195–2212], where we constructed the Drinfeld double of braided infinitesimal Hopf algebras (i.e., infinitesimal Hopf algebras in the Yetter–Drinfeld category \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${}_{H}^{H}{\mathcal{Y}}{\mathcal{D}}$$\end{document}). In this paper we show the Drinfeld double of braided Lie bialgebras (i.e., Lie bialgebras in \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${}_{H}^{H}{\mathcal{Y}}{\mathcal{D}}$$\end{document}) and present the Drinfeld double of Sl₂ in Yetter–Drinfeld categories over the group algebra H = kG. Finally, we consider the relation between the Drinfeld double of braided infinitesimal Hopf algebras and braided Lie bialgebras.

Keywords

Braided infinitesimal bialgebra / braided Lie bialgebra / Yetter–Drinfeld category / Drinfeld double / balanceator / 16T05 / 16T15

References

[1] Aguiar M. Infinitesimal Hopf algebras. New Trends in Hopf Algebra Theory (La Falda, 1999), 2000, Providence, RI, Amer. Math. Soc.: 1-29 267

[2] Aguiar M. On the associative analog of Lie bialgebras. J. Algebra, 2001, 244(2): 492-532

[3] Aguiar M. Infinitesimal Hopf algebras and the cd-index of polytopes. Discrete Comput. Geom., 2002, 27(1): 3-28

[4] Artenstein D, González A, Ronco M. Primitive elements in infinitesimal bialgebras. J. Algebra, 2025, 681: 218-269

[5] Chen D, Luo Y, Peng X, Zhang Y. Weighted infinitesimal unitary bialgebras on free monoid algebras. Colloq. Math., 2021, 166(1): 1-22

[6] Drinfel’d VG. Quantum groups. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 1987, Providence, RI, Amer. Math. Soc.: 798-820

[7] Ehrenborg R, Readdy M. Coproducts and the cd-index. J. Algebraic Combin., 1998, 8(3): 273-299

[8] Etingof P, Schiffmann O. Lectures on Quantum Groups, 1998, Boston, MA, International Press

[9] Foissy L. The infinitesimal Hopf algebra and the poset of planar forests. J. Algebraic Combin., 2009, 30(3): 277-309

[10] Hirschhorn PS, Raphael L. Coalgebraic foundations of the method of divided differences. Adv. Math., 1992, 91(1): 75-135

[11] Hong Y, Bai C. On antisymmetric infinitesimal conformal bialgebras. J. Algebra, 2021, 586: 325-356

[12] Joni SA, Rota G-C. Coalgebras and bialgebras in combinatorics. Stud. Appl. Math., 1979, 61(2): 93-139

[13] Joyal A, Street R. Braided tensor categories. Adv. Math., 1993, 102(1): 20-78

[14] Kassel C. Quantum Groups, 1995, New York, Springer-Verlag155

[15] Liu L, Makhlouf A, Menini C, Panaite F. BiHom-Novikov algebras and infinitesimal BiHom-bialgebras. J. Algebra, 2020, 560: 1146-1172

[16] Ma T, Li B, Li J, Chen M. A new approach to antisymmetric infinitesimal bialgebras. Czechoslovak Math. J., 2023, 73(3): 755-764

[17] Ma T, Makhlouf A. Infinitesimal (BiHom-)bialgebras of any weight (I): Basic definitions and properties. Front. Math., 2026, 21(1): 167-202

[18] Ma T., Yang H., Drinfeld double for infinitesimal BiHom-bialgebras. Adv. Appl. Clifford Algebr., 2020, 30(3): Paper No. 42, 22 pp.

[19] Radford DE. The structure of Hopf algebras with a projection. J. Algebra, 1985, 92(2): 322-347

[20] Radford DE, Towber J. Yetter–Drinfel’d categories associated to an arbitrary bialgebra. J. Pure Appl. Algebra, 1993, 87(3): 259-279

[21] Sheng Y, Wang Y. Quasi-triangular and factorizable antisymmetric infinitesimal bialgebras. J. Algebra, 2023, 628: 415-433

[22] Sweedler ME. Hopf Algebras, 1969, New York, W. A. Benjamin, Inc.

[23] Wang S. On the generalized H-Lie structure of associative algebras in Yetter–Drinfeld categories. Comm. Algebra, 2002, 30(1): 307-325

[24] Wang S. From braided infinitesimal bialgebras to braided Lie bialgebras. Adv. Pure Math., 2017, 7: 366-374

[25] Wang S, Chen H. Hopf–Galois coextensions and braided Lie coalgebras. Progr. Natur. Sci. (English Ed.), 2002, 12(4): 264-270

[26] Wang SX, Wang SH. Drinfeld double for braided infinitesimal Hopf algebras. Comm. Algebra, 2014, 42(5): 2195-2212

[27] Yau D., Infinitesimal Hom-bialgebras and Hom-Lie bialgebras, 2010, arXiv:1001.5000v1

[28] Zhang S.C., Zhang T., Double bicrosssum of braided Lie algebras. 2007, arXiv:0701217v2

[29] Zhang Y, Gao X, Luo Y. Weighted infinitesimal unitary bialgebras of rooted forests, symmetric cocycles and pre-Lie algebras. J. Algebraic Combin., 2021, 53(3): 771-803

[30] Zhang Y, Zheng J, Luo Y. Weighted infinitesimal unitary bialgebras, pre-Lie, matrix algebras, and polynomial algebras. Comm. Algebra, 2019, 47(12): 5164-5181

RIGHTS & PERMISSIONS

Peking University

Just Accepted

This article has successfully passed peer review and final editorial review, and will soon enter typesetting, proofreading and other publishing processes. The currently displayed version is the accepted final manuscript. The officially published version will be updated with format, DOI and citation information upon launch. We recommend that you pay attention to subsequent journal notifications and preferentially cite the officially published version. Thank you for your support and cooperation.