On the structure of cofree Hopf algebras

By Jean-Louis Loday, Maria O. Ronco. 2004

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Abstract We prove an analogue of the Poincaré-Birkhoff-Witt theorem and of the Cartier-Milnor-Moore theorem for non-cocommutative Hopf algebras. The primitive part of a cofree Hopf algebra is a nondifferential B ∞-algebra. We construct a universal enveloping functor U2 from nondifferential B ∞-algebras to 2-associative algebras, i.e. algebras equipped with two associative operations. We show that any cofree Hopf algebra ℋ is of the form U2(Prim ℋ). We take advantage of the description of the free 2as-algebra in terms of planar trees to unravel the operad associated to nondifferential B ∞-algebras.

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

View Paper: https://www.semanticscholar.org/paper/6d48ccebed4765a9b2cb032a88d15d2a86415407

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Citations in the corpus

Generalized bialgebras and triples of operads (2006, background): “It was shown in [54] that the universal idempotent is given by the geometric series: e = Id− Id ⋆ Id + · · ·+ (−1)Id + · · · .”
On the algebra of quasi-shuffles (2005, background): “In order to get the equivalence of the three assertions, it is sufficient to verify three hypotheses, which, in our case, read as follows: (H0) the compatibility relation is distributive, (H1) the free CTD-algebra is naturally a Asc-CTD-bialgebra, (H2epi) the natural coalgebra map ϕ(V ) : CTD(V ) →…”
FREE ROTA–BAXTER ALGEBRAS AND ROOTED TREES (2005, background): “So k F with the operations ⊔ and ⋄ forms a 2-associative algebra in the sense of [ 50 , 53].”
Cocommutative Hopf Algebras of Permutations and Trees (2004, background): “In the classical Hopf algebra literature usually only one Hopf algebra structure on Q(V ) is considered: the shuffle Hopf algebra.”
HOPF MONOIDS IN THE CATEGORY OF SPECIES (2012, background): “ algebra, or Lie algebra, then UV is a comonoid, bimonoid, Hopf monoid, or Lie monoid. When V is a Hopf algebra with antipode s, the antipode of UV is simply sI := (−1) |I s. This follows from either (42) or (43). 3. The Hadamard product Multiplying species term by term yields the Hadamard product. ”