Rota-Baxter bialgebra structures arising from (co-)quasi-idempotent elements

Year 2021, Volume: 50 Issue: 1, 216 - 223, 04.02.2021

Tianshui Ma , Jie Li Haiyan Yang

https://doi.org/10.15672/hujms.685742

Cited By: 2

https://izlik.org/JA38HP96BM

Abstract

In this note, we construct Rota-Baxter (coalgebras) bialgebras by (co-)quasi-idempotent elements and prove that every finite dimensional Hopf algebra admits nontrivial Rota-Baxter bialgebra structures and tridendriform bialgebra structures. We give all the forms of (co)-quasi-idempotent elements and related structures of tridendriform (co, bi)algebras and Rota-Baxter (co, bi)algebras on the well-known Sweedler's four-dimensional Hopf algebra.

Keywords

Rota-Baxter bialgebras , (co-)quasi-idempotent element , tridendriform bialgebra

References

• [1] G. Baxter, An analytic problem whose solution follows from a simple algebraic iden- tity, Pacific J. Math. 10, 731–742, 1960.
• [2] L. Guo, An Introduction to Rota-Baxter Algebra, Surveys of Modern Mathematics, 4. International Press, Somerville, MA; Higher Education Press, Beijing, 2012.
• [3] L. Guo, Properties of free Baxter algebras, Adv. Math. 151, 346–374, 2000.
• [4] L. Guo and W. Keigher, Baxter algebras and shuffle products, Adv. Math. 150, 117– 149, 2000.
• [5] L. Guo and B. Zhang, Polylogarithms and multiple zeta values from free Rota-Baxter algebras, Sci. China Math. 53 (9), 2239–2258, 2010.
• [6] L. Guo, J.-Y. Thibon and H. Yu, Weak composition quasi-symmetric functions, Rota- Baxter algebras and Hopf algebras, Adv. Math. 344, 1–34, 2019.
• [7] R.Q. Jian, Quasi-idempotent Rota-Baxter operators arising from quasi-idempotent elements, Lett. Math. Phys. 107, 367–374, 2017.
• [8] R.Q. Jian and J. Zhang, Rota-Baxter coalgebras, arXiv:1409.3052.
• [9] T.S. Ma and L.L. Liu, Rota-Baxter coalgebras and Rota-Baxter bialgebras, Linear Multilinear Algebra, 64 (5), 968–979, 2016.
• [10] D.E. Radford, Hopf Algebras, KE Series on Knots and Everything, World Scientific, Vol. 49, New Jersey, 2012.
• [11] G.C. Rota, Baxter algebras and combinatorial identities I, II, Bull. Amer. Math. Soc. 75 (2), 325–329, 330–334, 1969.
• [12] E.J. Taft, The order of the antipode of finite dimensional Hopf algebra, Proc. Nat. Acad. Sci. USA. 68, 2631–2633, 1971.