Dendriform-Nijenhuis bialgebras and DN-associative Yang-Baxter equations. (English) Zbl 1469.16085
A dendriform-Nijenhuis (briefly, DN) bialgebra, as introduced by P. Leroux [Int. J. Math. Math. Sci. 2004, No. 49–52, 2595–2615 (2004; Zbl 1116.17002)], is a triple \((A,\mu,\Delta)\), where \(\mu\) is an associative coproduct on \(A\), \(\Delta\) is a coassociative coproduct on \(A\), with the compatibility
\[
\Delta(ab)=a\Delta(b)+\Delta(a)- \mu\circ \Delta(a)\otimes b.
\]
Using a 1-cocycle condition, a structure of DN bialgebra on the algebra of planar rooted trees, in the same way as for the Connes-Kreimer Hopf algebra, leading to the notion of free operated DN bialgebra. It is also proved that any DN bialgebra naturally carries a Lie algebra structure.
Other constructions of DN bialgebras on a given algebra are associated to any element \(r\) of \(A\otimes A\): this element should satisfy an invariance condition and a variant of the classical Yang-Baxter equation, called the DN-AYBE. A DN bialgebra obtained in this way is called quasitriangular and it is shown that such an object carries a structure of post-Lie algebras, therefore two extra Lie brackets. A classification of the solutions of the DN-AYBE in dimension 2 and 3 is done in the last section of the paper.
Other constructions of DN bialgebras on a given algebra are associated to any element \(r\) of \(A\otimes A\): this element should satisfy an invariance condition and a variant of the classical Yang-Baxter equation, called the DN-AYBE. A DN bialgebra obtained in this way is called quasitriangular and it is shown that such an object carries a structure of post-Lie algebras, therefore two extra Lie brackets. A classification of the solutions of the DN-AYBE in dimension 2 and 3 is done in the last section of the paper.
Reviewer: Loïc Foissy (Calais)
MSC:
| 16W99 | Associative rings and algebras with additional structure |
| 16S10 | Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) |
| 16T10 | Bialgebras |
| 16T30 | Connections of Hopf algebras with combinatorics |
| 17B38 | Yang-Baxter equations and Rota-Baxter operators |
| 17B60 | Lie (super)algebras associated with other structures (associative, Jordan, etc.) |
Citations:
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