Dendriform-Nijenhuis bialgebras and DN-associative Yang-Baxter equations

Abstract Involving a symmetric Hochschild 1-cocycle condition, we equip the space of decorated planar rooted forests with a coproduct which turns the space into a dendriform-Nijenhuis bialgebra. We combine dendriform-Nijenhuis bialgebras with operated algebras and introduce the notation of an Ω-operated DN-bialgebra. Applying the universal property of the underlying operated algebras, we construct free objects in the category of Ω-cocycle DN-bialgebras. We introduce the notation of a DN-associative Yang-Baxter equation (AYBE) and show that the dendriform-Nijenhuis bialgebra offers an algebraic framework of the DN-associative Yang-Baxter equation. We construct a Leroux's TD operator from a solution of the DN-AYBE. We also give two different ways to derive Lie algebras from quasitriangular DN-bialgebras. Finally, we classify the solutions of the DN-AYBE in the unitary algebras of dimensions two and three over the field of complex numbers.

[1]  Li Guo,et al.  Matching Rota-Baxter algebras, matching dendriform algebras and matching pre-Lie algebras , 2019, 1909.10577.

[2]  Philippe Leroux,et al.  Construction of Nijenhuis operators and dendriform trialgebras , 2004, Int. J. Math. Math. Sci..

[3]  Kurusch Ebrahimi-Fard.  On the Associative Nijenhuis Relation , 2004, Electron. J. Comb..

[4]  Mircea Cras¸M Areanu.  First Integrals Generated by Pseudosymmetries in Nambu-Poisson Mechanics , 2000 .

[5]  L. Foissy,et al.  Les algèbres de Hopf des arbres enracinés décorés, I , 2001, math/0105212.

[6]  Li Guo,et al.  Nonabelian Generalized Lax Pairs, the Classical Yang-Baxter Equation and PostLie Algebras , 2009, 0910.3262.

[7]  Jean-Louis Loday,et al.  On the structure of cofree Hopf algebras , 2004, math/0405330.

[8]  K. Uchino.  Twisting on associative algebras and Rota-Baxter type operators , 2007, 0710.4309.

[9]  Martin Hairer,et al.  Algebraic renormalisation of regularity structures , 2016, Inventiones mathematicae.

[10]  Jean-Louis Loday,et al.  Hopf Algebra of the Planar Binary Trees , 1998 .

[11]  Benjamin Peirce,et al.  Linear Associative Algebra , 2007 .

[12]  M. AGUIAR,et al.  Infinitesimal Hopf algebras , 2007 .

[13]  Li Guo,et al.  Hopf algebras of rooted forests, cocyles, and free Rota-Baxter algebras , 2016, 1605.09531.

[14]  Yanfeng Luo,et al.  Left counital Hopf algebras on bi-decorated planar rooted forests and Rota-Baxter systems , 2020 .

[15]  Yanfeng Luo,et al.  Weighted infinitesimal unitary bialgebras on rooted forests and weighted cocycles , 2018, Pacific Journal of Mathematics.

[16]  Marcelo Aguiar,et al.  Pre-Poisson Algebras , 2000 .

[17]  E. Study.  Über Systeme complexer Zahlen und ihre Anwendung in der Theorie der Transformationsgruppen , 1890 .

[18]  A. Nijenhuis.  Xn-1-Forming Sets of Eigenvectors , 1951 .

[19]  O. Ogievetsky,et al.  $R$-matrices in Rime , 2007, 0704.1947.

[20]  Richard G. Larson,et al.  Hopf-algebraic structure of families of trees , 1989 .

[21]  Li Guo,et al.  Left counital Hopf algebra structures on free commutative Nijenhuis algebras , 2017, SCIENTIA SINICA Mathematica.

[22]  Bruno Vallette,et al.  Homology of generalized partition posets , 2007 .

[23]  Li Guo,et al.  Operated semigroups, Motzkin paths and rooted trees , 2007, 0710.0429.

[24]  Quantum Bi-Hamiltonian Systems , 2000, math-ph/0610011.

[25]  Franco Magri,et al.  Poisson-Nijenhuis structures , 1990 .

[26]  Li Guo,et al.  Braided dendriform and tridendriform algebras, and braided Hopf algebras of rooted trees , 2019, 1906.06454.

[27]  Li Guo,et al.  Operator forms of nonhomogeneous associative classical Yang-Baxter equation , 2020, 2007.10939.

[28]  Gian-Carlo Rota,et al.  Coalgebras and Bialgebras in Combinatorics , 1979 .

[29]  Li Guo,et al.  O-OPERATORS ON ASSOCIATIVE ALGEBRAS AND ASSOCIATIVE YANG-BAXTER EQUATIONS , 2009, 0910.3261.

[30]  Yanfeng Luo,et al.  Weighted infinitesimal unitary bialgebras of rooted forests, symmetric cocycles and pre-Lie algebras , 2020, Journal of Algebraic Combinatorics.

[31]  Ralf Holtkamp,et al.  Comparison of Hopf algebras on trees , 2003 .

[32]  Xing Gao,et al.  Infinitesimal unitary Hopf algebras and planar rooted forests , 2018, Journal of Algebraic Combinatorics.

[33]  K. Ebrahimi-Fard,et al.  Generalized Shuffles Related to Nijenhuis and TD-Algebras , 2006, math/0606164.

[34]  L. Foissy.  The infinitesimal Hopf algebra and the poset of planar forests , 2008, 0802.0442.

[35]  I. Moerdijk.  On the Connes-Kreimer construction of Hopf algebras , 1999, math-ph/9907010.

[36]  Li Guo,et al.  Hopf algebra of multi-decorated rooted forests, free matching Rota-Baxter algebras and Gr\"obner-Shirshov bases. , 2020, 2002.02864.

[37]  Xing Gao,et al.  Left counital Hopf algebras on free Nijenhuis algebras , 2018, 1805.07693.

[38]  Yanfeng Luo,et al.  Weighted infinitesimal unitary bialgebras, pre-Lie, matrix algebras, and polynomial algebras , 2018, Communications in Algebra.

[39]  Li Guo,et al.  An introduction to Rota-Baxter algebra , 2012 .

[40]  D. Simson,et al.  Elements of the Representation Theory of Associative Algebras , 2007 .

[41]  Alain Connes,et al.  Hopf Algebras, Renormalization and Noncommutative Geometry , 1998 .

[42]  Frederic Chapoton,et al.  Pre-Lie algebras and the rooted trees operad , 2000 .

[43]  Li Guo,et al.  Nijenhuis algebras, NS algebras, and N-dendriform algebras , 2012, 1210.1821.

[44]  Xing Gao,et al.  Hopf algebras of planar binary trees: an operated algebra approach , 2019, Journal of Algebraic Combinatorics.