Nijenhuis algebras, NS algebras and N-dendriform algebras

TL;DR

This paper explores the structure of Nijenhuis algebras and their connections to NS and N-dendriform algebras, providing explicit constructions and revealing new algebraic relations.

Contribution

It introduces explicit constructions of free Nijenhuis algebras and the universal enveloping Nijenhuis algebra of NS algebras, and defines the new N-dendriform algebra.

Findings

Constructed free Nijenhuis algebras explicitly

Established the universal enveloping Nijenhuis algebra for NS algebras

Defined the N-dendriform algebra with more relations than NS algebra

Abstract

In this paper we study (associative) Nijenhuis algebras, with emphasis on the relationship between the category of Nijenhuis algebras and the categories of NS algebras. This is in analogy to the well-known theory of the adjoint functor from the category of Lie algebras to that of associative algebras, and the more recent results on the adjoint functor from the categories of dendriform and tridendriform algebras to that of Rota-Baxter algebras. We first give an explicit construction of free Nijenhuis algebras and then apply it to obtain the universal enveloping Nijenhuis algebra of an NS algebra. We further apply the construction to determine the binary quadratic nonsymmetric algebra, called the N-dendriform algebra, that is compatible with the Nijenhuis algebra. As it turns out, the N-dendriform algebra has more relations than the NS algebra.