### Session S16 - Quantum symmetries

Wednesday, July 14, 15:30 ~ 15:55 UTC-3

## Dendriform operads and brace algebras

### Maria Ronco

#### Universidad de Talca, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.

Dendriform algebras were introduced by J.-L. Loday in [8], even if the first example of them was yet described by S. Eilenberg and S. MacLane in [1]. On the other hand, a brace algebra $A$ is equivalent to a formal deformation $*$ of the shuffle product on $T(A)$, which makes $(A, *, \Delta^c)$ a Hopf algebra, for the deconcatenation coproduct, and is given by a family of $n+1$-ary products $M_{1,n}$, for $n\geq 1$.

The operad of dendriform algebras is regular and Hopf, which means that the tensor product of two dendriform algebras is dendriform and the notion of dendriform bialgebra is well-defined. In [10], we defined a functor from dendriform algebras to brace algebras, and proved that the subspace of primitive elements of a dendriform bialgebra are closed under the brace structure, more precisely any conilpotent dendriform bialgebra is isomorphic to the {\it dendriform} enveloping algebra of the brace algebra of its primitive elements.

The construction of the functor from dendriform algebras to brace algebras has been applied to several combinatorial Hopf algebras recently, see for instance [2], [4], [5], [6], [7], [11], [3] for the last two years. There exist new families of examples, concerning constructions on the faces of associahedra and $m$-versions of planar binary trees, partitions and permutations (see [11]and [9]) which involve more operations than the dendriform structure. In these cases we still have a notion of bialgebra, and therefore we are able to compute the operad of primitive elements associated to these algebras, but most results published on triples of operads or rigidity structure theorems failed to provide Cartier-Milnor-Moore structure theorems in these cases.

In a joint work with M. Livernet, we give a description of dendriform in terms of generators and relations, envolving braces. This result allows us to give an adjoint {\it free} functor to ${\mbox{Dend-alg}}\longrightarrow {\mbox{Brace-alg}}$, and to get structure theorems for dendriform algebras equipped with some additional structure, like tridendriform and $m$-Dyck algebras.

{\bf References}

[1] S. Eilenberg, S. Maclane, On the groups $H(\Pi, n)$, I}, Annals of Maths., Vol. 58}, (1) (1953), 55--106

[2] K Ebrahimi-Fard, F Patras, Shuffle group laws: applications in free probability, Proceedings of the London Math. Soc. 119 (3) (2019)

[3] K. Ebrahimi-Fard, L. Foissy, J. Kock, F. Patras, Operads of (noncrossing) partitions, interacting bialgebras, and moment-cumulant relations, Adv. in Math. 369 (2020).

[4] L Foissy, Algebraic structures associated to operads, preprint arXiv:1702.05344 (2017).

[5] M. N. Hounkonnou, G. D. Houndedji, Solutions of associative Yang-Baxter equation and equation in low dimensions and associated Frobenius algebras and Connes cocycles, J. of Algebra and Its Applications, Vol. 17, No. 1 (2018).

[6] Y. Li, Q. Mo, X. Zhao, The Freiheitssatz and automorphisms for free brace algebras, Comm. in Algebra 47(10) (2019) 4125--4136

[7] Y. Li, Q. Mo, L.A. Bokut, Gr\"obner-Shirshov bases for symmetric brace algebras, Comm. in Algebra 49 (3) (2021) 1368--1369.

[8] J.-L. Loday, Dialgebras in Dialgebras and related operads, Lecture Notes in Math., 1763, Springer, Berlin, 2001.7--66

[9] D. L\'opez, L.-F. Pr\'eville-Ratelle, M. Ronco, A simplicial complex splitting associativity, J. of Pure and Applied Algebra, Vol 224, (5) (2020).

[10] M. Ronco, Eulerian idempotents and Milnor-Moore theorem for certain non-cocommutative Hopf algebras, J. of Algebra. 254(1) (2002)152-172.

[11] Y. Zhang, X. Gao, D. Manchon, {\it Free (tri) dendriform family algebras}, J. of Algebra 547 (2020) 456--493

Joint work with Muriel Livernet, Univ. Paris-Diderot..