The deformation and construction of Nijenhuis paired modules

Year 2024, Early Access, 1 - 18

Yunfei Fang Ximu Wang Liangyun Zhang

https://doi.org/10.24330/ieja.1580177

Abstract

In this paper, we introduce the notion of Nijenhuis paired module, and give characterizations of Nijenhuis paired modules. Finally, we construct Nijenhuis paired modules from Hopf algebras, Hopf modules, dimodules and weak Hopf modules.

Keywords

Nijenhuis algebra, Nijenhuis paired module, Hopf algebra, Hopf module

References

• G. Böhm, F. Nill and K. Szlachanyi, Weak Hopf algebras: I. Integral theory and $C^{*}$-structure, J. Algebra, 221(2) (1999), 385-438.
• S. Caenepeel, F. Van Oystaeyen and Y. H. Zhang, Quantum Yang-Baxter module algebras, K-Theory, 8(3) (1994), 231-255.
• J. F. Carinena, J. Grabowski and G. Marmo, Quantum bi-Hamiltonian systems, Internat. J. Modern Phys. A, 15(30) (2000), 4797-4810.
• Y. Fang, H. H. Zheng and L. Y. Zhang, Modified Rota-Baxter paired modules and modified Rota-Baxter paired comodules, J. Shangdong Univ. Nat. Sci., 58(2) (2023), 93-104.
• I. Z. Golubchik and V. V. Sokolov, Generalized operator Yang-Baxter equations, integrable ODEs and nonassociative algebras, J. Nonlinear Math. Phys., 7(2) (2000), 184-197.
• I. Z. Golubchik and V. V. Sokolov, One more kind of the classical Yang-Baxter equation, Funct. Anal. Appl., 34(4) (2000), 296-298.
• L. Guo, An Introduction to Rota-Baxter Algebra, Surveys of Modern Mathematics, 4, International Press, Somerville, MA; Higher Education Press, Beijing, 2012.
• L. Guo, H. L. Lang and Y. H. Sheng, Integration and geometrization of Rota-Baxter Lie algebras, Adv. Math., 387 (2021), 107834 (34 pp).
• L. Guo and Z. Z. Lin, Representations and modules of Rota-Baxter algebras, Asian J. Math., 25(6) (2021), 841-870.
• R. Q. Jian, Construction of Rota-Baxter algebras via Hopf module algebras, Sci. China Math., 57 (2014), 2321-2328.
• P. Lei and L. Guo, Nijenhuis algebras, NS algebras, and N-dendriform algebras, Front. Math. China, 7 (2012), 827-846.
• P. Leroux, Construction of Nijenhuis operators and dendriform trialgebras, Int. J. Math. Math. Sci., 49 (2004), 2595-2615.
• Z. H. Li and S. K. Wang, Rota-Baxter systems and skew trusses, J. Algebra, 623 (2023), 447-480.
• F. Magri, A simple model of the integrable Hamiltonian equatio}, J. Math. Phys., 19(5) (1978), 1156-1162.
• B. Mondal and R. Saha, Nijenhuis operators on Leibniz algebras, J. Geom. Phys., 196 (2024), 105057 (14 pp).
• A. Nijenhuis, $\mathrm{X_{n-1}}$-Forming sets of eigenvectors, Indag. Math., 13 (1951), 200-212.
• M. E. Sweedler, Hopf Algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969.
• Q. Wang, Y. H. Sheng, C. M. Bai and J. F. Liu, Nijenhuis operators on pre-Lie algebras, Commun. Contemp. Math., 21(7) (2019), 1850050 (37 pp).
• Y. Wang and L. Y. Zhang, The structure theorem and duality theorem for endomorphism algebras of weak Hopf algebras, J. Pure Appl. Algebra, 215(6) (2011), 1133-1145.
• X. M. Wang, C. X. Zhang and L. Y. Zhang, Rota-Baxter operators on skew braces, Mathematics, 12(11) (2024), 1671 (14 pp).
• L. Y. Zhang, Long bialgebras, dimodule algebras and quantum Yang-Baxter modules over Long bialgebras, Acta Math. Sin. (Engl. Ser.), 22(4) (2006), 1261-1270.
• L. Y. Zhang, The structure theorem of weak comodule algebras}, Comm. Algebra, 38(1) (2010), 254-260.
• L. Y. Zhang and W. T. Tong, Quantum Yang-Baxter $H$-module algebras and their braided products, Comm. Algebra, 31(5) (2003), 2471-2495.
• H. H. Zheng, L. Guo and L. Y. Zhang, Rota-Baxter paired modules and their constructions from Hopf algebras, J. Algebra, 559 (2020), 601-624.