FREE MODIFIED ROTA-BAXTER ALGEBRAS AND HOPF ALGEBRAS

Year 2019, Volume: 25 Issue: 25, 12 - 34, 08.01.2019

Xigou Zhang Xing Gao Li Guo

https://doi.org/10.24330/ieja.504101

Cited By: 2

Abstract

The notion of a modied Rota-Baxter algebra comes from the

combination of those of a Rota-Baxter algebra and a modied Yang-Baxter

equation. In this paper, we rst construct free modied Rota-Baxter algebras.

We then equip a free modied Rota-Baxter algebra with a bialgebra structure

by a cocycle construction. Under the assumption that the generating algebra

is a connected bialgebra, we further equip the free modied Rota-Baxter alge-

bra with a Hopf algebra structure.

Keywords

Modied Rota-Baxter algebra, Rota-Baxter algebra, Hopf algebra, bracketed word, cocycle

References

• M. Aguiar, On the associative analog of Lie bialgebras, J. Algebra, 244(2) (2001), 492-532.
• F. V.Atkinson, Some aspects of Baxter's functional equation, J. Math. Anal. Appl., 7 (1963), 1-30.
• C. Bai, O. Bellier, L. Guo and X. Ni, Splitting of operations, Manin products, and Rota-Baxter operators, Int. Math. Res. Not. IMRN, (2013) 3 (2013), 485- 524.
• C. Bai, L. Guo and X. Ni, Nonabelian generalized Lax pairs, the classical Yang- Baxter equation and PostLie algebras, Comm. Math. Phys., 297(2) (2010), 553-596.
• C. Bai, L. Guo and X. Ni, Generalizations of the classical Yang-Baxter equation and O-operators, J. Math. Phys., 52(6) (2011), 063515 (17 pp).
• C. Bai, L. Guo and X. Ni, O-operators on associative algebras and associative Yang-Baxter equations, Pacic J. Math., 256(2) (2012), 257-289.
• C. Bai, L. Guo and X. Ni, O-operators on associative algebras, associative Yang-Baxter equations and dendriform algebras, In \Quantized Algebra and Physics", Nankai Ser. Pure Appl. Math. Theoret. Phys., 8, World Sci. Publ., Hackensack, NJ, (2012), 10-51.
• G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pacic J. Math., 10 (1960), 731-742.
• M. Bordemann, Generalized Lax pairs, the modied classical Yang-Baxter equa- tion, and ane geometry of Lie groups, Comm. Math. Phys., 135(1) (1990), 201-216.
• P. Cartier, On the structure of free Baxter algebras, Advances in Math., 9 (1972), 253-265.
• A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommuta- tive geometry, Comm. Math. Phys., 199(1) (1998), 203-242.
• K. Ebrihimi-Fard, Loday-type algebras and the Rota-Baxter relation, Lett. Math. Phys., 61(2) (2002), 139-147.
• K. Ebrahimi-Fard and L. Guo, Quasi-shues, mixable shues and Hopf alge- bras, J. Algebraic Combin., 24(1) (2006), 83-101.
• K. Ebrahimi-Fard and L. Guo, Rota-Baxter algebras and dendriform algebras, J. Pure Appl. Algebra, 212(2) (2008), 320-339.
• X. Gao, L. Guo and T. Zhang, Bialgebra and Hopf algebra structures on free Rota-Baxter algebra, arXiv:1604.03238.
• L. Guo, An Introduction to Rota-Baxter Algebra, Surveys of Modern Mathematics, 4, International Press, Somerville, MA; Higher Education Press, Beijing, 2012.
• L. Guo and W. Keigher, Baxter algebras and shue products, Adv. Math., 150(1) (2000), 117-149.
• L. Guo and W. Y. Sit Enumeration and generating functions of Rota-Baxter words, Math. Comput. Sci., 4(2-3) (2010), 313-337.
• R. Jian and J. Zhang, Rota-Baxter coalgebras, arXiv:1409.3052. [20] Y. Kosmann-Schwarzbach, Lie bialgebras, Poisson Lie groups and dressing transformations, in \Integrability of nonlinear systems", Lecture Notes in Phys., 495, Springer, Berlin, (1997), 104-170.
• T. Ma and L. Liu, Rota-Baxter coalgebras and Rota-Baxter bialgebras, Linear Multilinear Algebra, 64(5) (2016), 968-979.
• A. Makhlouf and D. Yau, Rota-Baxter Hom-Lie-admissible algebras, Comm. Algebra, 42(3) (2014), 1231-1257.
• D. Manchon, Hoft algebras, from basics to applications to renormalization, Comptes-rendus des Rencontres mathematiques de Glanon, 2001.
• M. Marcolli and X. Ni, Rota-Baxter algebras, singular hypersurfaces, and renormalization on Kausz compactications, J. Singul., 15 (2016), 80-117.
• J. Pei, C. Bai and L. Guo, Splitting of operads and Rota-Baxter operators on operads, Appl. Categ. Structures, 25(4) (2017), 505-538.
• G.-C. Rota, Baxter algebras and combinatorial identities I, II, Bull. Amer. Math. Soc., 75 (1969), 325-329, 330-334.
• M. A. Semenov-Tian-Shansky, What is a classical R-matrix? Functional Anal. Appl. 17(4) (1983), 259-272.
• T. Zhang, X. Gao and L. Guo, Hopf algebras of rooted forests, cocyles, and free Rota-Baxter algebras, J. Math. Phys., 57(10) (2016), 101701 (16 pp).
• X. Zhang, X. Gao and L. Guo, Commutative modied Rota-Baxter algebras, shuffe products and Hopf algebras, Bull. Malays. Math. Sci. Soc., (2018), https://doi.org/10.1007/s40840-018-0648-3.