Nijenhuis operators and associative D-bialgebras

Abstract

Firstly, we prove that a quasitriangular associative D-bialgebra together with a coquasitriangular structure can induce a Nijenhuis algebra. This shows that Nijenhuis operators and associative D-bialgebras are closely related. Secondly, we investigate some properties of a representation of a Nijenhuis algebra and introduce the notion of a corepresentation of a Nijenhuis coalgebra, which also can be seen as an infinitesimal deformation of compatible bicomodule. Thirdly, a Nijenhuis associative D-bialgebra can be established by a Nijenhuis algebra and a Nijenhuis coalgebra satisfying some compatible conditions. Furthermore, we find that a Nijenhuis associative D-bialgebra is equivalent to a matched pair of Nijenhuis algebras or a double construction of a Nijenhuis Frobenius algebra. Lastly, we provide some methods to construct Nijenhuis associative D-bialgebras through admissible associative Yang-Baxter equations (aYBes for short) and O-operators on Nijenhuis algebras.