Infinitesimal bialgebras, pre-Lie and dendriform algebras

By Marcelo Aguiar. 2002

93 citations recorded by the source corpus.

We introduce the categories of infinitesimal Hopf modules and bimodules over an infinitesimal bialgebra. We show that they correspond to modules and bimodules over the infinitesimal version of the double. We show that there is a natural, but non-obvious way to construct a pre-Lie algebra from an arbitrary infinitesimal bialgebra and a dendriform algebra from a quasitriangular infinitesimal bialgebra. As consequences, we obtain a pre-Lie structure on the space of paths on an arbitrary quiver, and a striking dendriform structure on the space of endomorphisms of an arbitrary infinitesimal bialgebra, which combines the convolution and composition products. We extend the previous constructions to the categories of Hopf, pre-Lie and dendriform bimodules. We construct a brace algebra structure from an arbitrary infinitesimal bialgebra; this refines the pre-Lie algebra construction. In two appendices, we show that infinitesimal bialgebras are comonoid objects in a certain monoidal category and discuss a related construction for counital infinitesimal bialgebras.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/54ed5900af18ed2e9610c58008a4a23f7d0ff1f7

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Citations in the corpus

Algebras with one operation including Poisson and other Lie-admissible algebras (2004, background): “To interpret X in a similar way, wee need to recall that an infinitesimal bialgebra [1] (also called a mock bialgebra in [5]) is a triple (V, μ, δ), where μ is an associative multiplication, δ is a coassociative comultiplication and δ(μ(u, v)) = δ(1)(u) ⊗ μ(δ(2)(u), v) + μ(u, δ(1)(v)) ⊗ δ(2)(v) for ”
On quadri-algebras (2014, background): “It is also suspected that the sub-quadri-algebra of FQSym generated by the permutation (12) is free.”
Nonabelian Generalized Lax Pairs, the Classical Yang-Baxter Equation and PostLie Algebras (2009, methodology): “When the operation · of the dendriform trialgebra and the bracket [, ] of the PostLie algebra are trivial, we obtain the following commutative diagram introduced in [12] (see also [2, 3 ,42]):”
Bidendriform bialgebras, trees, and free quasi-symmetric functions (2005, background): “1 Dendriform algebras and coalgebras Definition 1 (See [1, 15, 16, 17, 19]).”