On embedding of dendriform algebras into Rota---Baxter algebras

TL;DR

This paper generalizes the embedding of dendriform algebras into Rota-Baxter algebras across various algebraic varieties, extending previous work to broader contexts and establishing new embedding results.

Contribution

It introduces the concept of dendriform di- and trialgebras in arbitrary algebraic varieties and proves their embeddings into Rota-Baxter algebras of specific weights.

Findings

Dendriform dialgebras embed into Rota-Baxter algebras of weight zero.

Dendriform trialgebras embed into Rota-Baxter algebras of nonzero weight.

The results apply to various algebraic structures like associative, commutative, and Poisson algebras.

Abstract

Following a recent work by C. Bai, O. Bellier, L. Guo, and X. Ni (arXiv:1106.6080) we define what is a dendriform di- or trialgebra in an arbitrary variety Var of algebras (associative, commutative, Poisson, etc.). We prove that every dendriform dialgebra in Var can be embedded into a Rota---Baxter algebra of weight zero in the same variety, and every dendriform trialgebra can be embedded into a Rota---Baxter algebra of nonzero weight.