Embedding of dendriform algebras into Rota-Baxter algebras

Vsevolod Gubarev; Pavel Kolesnikov

Open Mathematics (2013)

  • Volume: 11, Issue: 2, page 226-245
  • ISSN: 2391-5455

Abstract

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Following a recent work [Bai C., Bellier O., Guo L., Ni X., Splitting of operations, Manin products, and Rota-Baxter operators, Int. Math. Res. Not. IMRN (in press), DOI: 10.1093/imrn/rnr266] we define what is a dendriform dior trialgebra corresponding to an arbitrary variety Var of binary algebras (associative, commutative, Poisson, etc.). We call such algebras di- or tri-Var-dendriform algebras, respectively. We prove in general that the operad governing the variety of di- or tri-Var-dendriform algebras is Koszul dual to the operad governing di- or trialgebras corresponding to Var!. We also prove that every di-Var-dendriform algebra can be embedded into a Rota-Baxter algebra of weight zero in the variety Var, and every tri-Var-dendriform algebra can be embedded into a Rota-Baxter algebra of nonzero weight in Var.

How to cite

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Vsevolod Gubarev, and Pavel Kolesnikov. "Embedding of dendriform algebras into Rota-Baxter algebras." Open Mathematics 11.2 (2013): 226-245. <http://eudml.org/doc/269180>.

@article{VsevolodGubarev2013,
abstract = {Following a recent work [Bai C., Bellier O., Guo L., Ni X., Splitting of operations, Manin products, and Rota-Baxter operators, Int. Math. Res. Not. IMRN (in press), DOI: 10.1093/imrn/rnr266] we define what is a dendriform dior trialgebra corresponding to an arbitrary variety Var of binary algebras (associative, commutative, Poisson, etc.). We call such algebras di- or tri-Var-dendriform algebras, respectively. We prove in general that the operad governing the variety of di- or tri-Var-dendriform algebras is Koszul dual to the operad governing di- or trialgebras corresponding to Var!. We also prove that every di-Var-dendriform algebra can be embedded into a Rota-Baxter algebra of weight zero in the variety Var, and every tri-Var-dendriform algebra can be embedded into a Rota-Baxter algebra of nonzero weight in Var.},
author = {Vsevolod Gubarev, Pavel Kolesnikov},
journal = {Open Mathematics},
keywords = {Dendriform algebra; Dialgebra; Trialgebra; Rota-Baxter operator; Operad; Manin product; dendriform algebra; dialgebra; trialgebra; operad},
language = {eng},
number = {2},
pages = {226-245},
title = {Embedding of dendriform algebras into Rota-Baxter algebras},
url = {http://eudml.org/doc/269180},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Vsevolod Gubarev
AU - Pavel Kolesnikov
TI - Embedding of dendriform algebras into Rota-Baxter algebras
JO - Open Mathematics
PY - 2013
VL - 11
IS - 2
SP - 226
EP - 245
AB - Following a recent work [Bai C., Bellier O., Guo L., Ni X., Splitting of operations, Manin products, and Rota-Baxter operators, Int. Math. Res. Not. IMRN (in press), DOI: 10.1093/imrn/rnr266] we define what is a dendriform dior trialgebra corresponding to an arbitrary variety Var of binary algebras (associative, commutative, Poisson, etc.). We call such algebras di- or tri-Var-dendriform algebras, respectively. We prove in general that the operad governing the variety of di- or tri-Var-dendriform algebras is Koszul dual to the operad governing di- or trialgebras corresponding to Var!. We also prove that every di-Var-dendriform algebra can be embedded into a Rota-Baxter algebra of weight zero in the variety Var, and every tri-Var-dendriform algebra can be embedded into a Rota-Baxter algebra of nonzero weight in Var.
LA - eng
KW - Dendriform algebra; Dialgebra; Trialgebra; Rota-Baxter operator; Operad; Manin product; dendriform algebra; dialgebra; trialgebra; operad
UR - http://eudml.org/doc/269180
ER -

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