Embedding of dendriform algebras into Rota-Baxter algebras

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 RotaBaxter algebra of nonzero weight in Var. MSC:18D50, 17B69

Rota–Baxter coalgebras and Rota–Baxter bialgebras

“…Rota-Baxter algebras were introduced in [21] in the context of differential operators on commutative Banach algebras and since [4] intensively studied in probability and combinatorics, and more recently in mathematical physics, such as dendriform algebras * Corresponding author 1 (see [3,6,7,11]), free Rota-Baxter algebras (see [5,8,9,13,14]), Lie algebras (see [2,18]), multiple zeta values (see [8,15]) and Connes-Kreimer renormalization theory in quantum field theory (see [10]), etc. One can refer to the book [12] for the detailed theory of Rota-Baxter algebras.…”

Universal Enveloping Commutative Rota–Baxter Algebras of Pre- and Post-Commutative Algebras

“…In 2000, M. Aguiar showed [7] that a solution of the associative Yang-Baxter equation (AYBE) [8] gives rise to a structure of the associative RB-algebra. There are many applications of RB-operators in mathematical physics, combinatorics, number theory, and operads [9][10][11][12].…”

“…In 2013 [11], it was proved that any pre-Var-algebra (post-Var-algebra) injectively embeds into its universal enveloping RB-algebra of variety Var and weight equal (not equal) to zero.…”

“…Finally, a variety can be considered as an operad [37], a particular case of the notion of multicategory. From this point of view, identities and varieties of preand post-algebras were investigated in [11,30].…”

“…The history of investigations into defining identities for varieties of pre-and post-algebras can be roughly divided into two periods: the first came before the principal work of C. Bai et al [30] appeared on arXiv in 2011 (see also [11]; for dialgebras and Koszul duals for pre-algebras, the breakthrough was made in 2008 [38]); the second period has been going on since this work. During the first period, the names of the varieties of pre-and post-algebras were in some sense chaotic (pre-Lie algebras, dendriform algebras, Zinbiel algebras, post-Lie algebras, tridendriform algebras, and commutative tridendriform algebras).…”