Seminar of Algebra

Dendriform algebras are associative algebras whose product splits as the sum of two binary operations, satisfying certain relations. The first example of this structure was introduced by S. Eilenberg and S. MacLane, in their work {\it On the groups $H(\Pi, n), I$} (Annals of Mathematics, Second Series, Vol. 58, No. 1 (Jul., 1953), pp. 55-106), under the name of half-product. The notion of dendriform algebra is due to J.-L. Loday, who found these type of algebras as the Koszul duals of enveloping algebras for a certain non-symmetric version of Lie algebras: the Leibniz algebras. Any associative product given by shuffles, or by a Rota-Baxter structure, comes from a dendriform structure.

Free dendriform algebras are described easily on spaces spanned by plane rooted binary trees, with colored internal nodes. A non-commutative version of A. Connes and D. Kreimer Hopf algebra, shows that any free dendriform algebra admits a structure of bialgebra, which is not cocommutative. From this fact we easily obtained a definition of dendriform bialgebra. Moreover, there exist a homomorphism from the operad of brace algebras ( as introduced by M. Gerstenhaber and A. Voronov) and the operad of dendriform algebras, which satisfies that the subspace of primitive elements of any dendriform bialgebra ${\mathcal D}$ is a brace subalgebra of ${\mathcal D}$.

The idea of splitting associativity has been studied in recent works by different authors. We shall introduce two generalizations of dendriform algebras:

1. The ${\mbox {Dyck}^m}$ algebras, whose operad is described by $m$-Dyck paths (this is a joint work with D. Lop\'ez Neumann and L.-F. Pr\'eville-Rattelle),
2. The dipterous algebras, introduced with J.-L. Loday, which provide enveloping structures for multibrace algebras.