Noncommutative Bell polynomials, quasideterminants and incidence Hopf algebras

Bell polynomials appear in several combinatorial constructions throughout mathematics. Perhaps most naturally in the combinatorics of set partitions, but also when studying compositions of diffeomorphisms on vector spaces and manifolds, and in the study of cumulants and moments in probability theory. We construct commutative and noncommutative Bell polynomials and explain how they give rise to Fa\`a di Bruno Hopf algebras. We use the language of incidence Hopf algebras, and along the way provide a new description of antipodes in noncommutative incidence Hopf algebras, involving quasideterminants. We also discuss M\"obius inversion in certain Hopf algebras built from Bell polynomials.Comment: 37 pages, final version, to appear in IJA

Bell polynomials in combinatorial Hopf algebras

Partial multivariate Bell polynomials have been defined by E.T. Bell in 1934. These polynomials have numerous applications in Combinatorics, Analysis, Algebra, Probabilities, etc. Many of the formulae on Bell polynomials involve combinatorial objects (set partitions, set partitions in lists, permutations, etc.). So it seems natural to investigate analogous formulae in some combinatorial Hopf algebras with bases indexed by these objects. The algebra of symmetric functions is the most famous example of a combinatorial Hopf algebra. In a first time, we show that most of the results on Bell polynomials can be written in terms of symmetric functions and transformations of alphabets. Then, we show that these results are clearer when stated in other Hopf algebras (this means that the combinatorial objects appear explicitly in the formulae). We investigate also the connexion with the Fa{\`a} di Bruno Hopf algebra and the Lagrange-B{\"u}rmann formula

Free cumulants, Schr\"oder trees, and operads

The functional equation defining the free cumulants in free probability is lifted successively to the noncommutative Fa\`a di Bruno algebra, and then to the group of a free operad over Schr\"oder trees. This leads to new combinatorial expressions, which remain valid for operator-valued free probability. Specializations of these expressions give back Speicher's formula in terms of noncrossing partitions, and its interpretation in terms of characters due to Ebrahimi-Fard and Patras.Comment: 23 page

On the Lie enveloping algebra of a post-Lie algebra

We consider pairs of Lie algebras $g$ and $\bar{g}$, defined over a common vector space, where the Lie brackets of $g$ and $\bar{g}$ are related via a post-Lie algebra structure. The latter can be extended to the Lie enveloping algebra $U(g)$. This permits us to define another associative product on $U(g)$, which gives rise to a Hopf algebra isomorphism between $U(\bar{g})$ and a new Hopf algebra assembled from $U(g)$ with the new product. For the free post-Lie algebra these constructions provide a refined understanding of a fundamental Hopf algebra appearing in the theory of numerical integration methods for differential equations on manifolds. In the pre-Lie setting, the algebraic point of view developed here also provides a concise way to develop Butcher's order theory for Runge--Kutta methods.Comment: 25 page

Quotient Hopf algebras of the free bialgebra with PBW bases and GK-dimensions

Let $\mathbb{K}$ be a field. We study the free bialgebra $\mathcal{T}$ generated by the coalgebra $C=\mathbb{K} g \oplus \mathbb{K} h$ and its quotient bialgebras (or Hopf algebras) over $\mathbb{K}$. We show that the free noncommutative Fa\`a di Bruno bialgebra is a sub-bialgebra of $\mathcal{T}$, and the quotient bialgebra $\overline{\mathcal{T}}:=\mathcal{T}/(E_{\alpha}|~\alpha(g)\ge 2)$ is an Ore extension of the well-known Fa\`a di Bruno bialgebra. The image of the free noncommutative Fa\`a di Bruno bialgebra in the quotient $\overline{\mathcal{T}}$ gives a more reasonable non-commutative version of the commutative Fa\`a di Bruno bialgebra from the PBW basis point view. If char $\mathbb{K}=p>0$, we obtain a chain of quotient Hopf algebras of $\overline{\mathcal{T}}$: $\overline{\mathcal{T}} \twoheadrightarrow \overline{\mathcal{T}}_{n}\twoheadrightarrow \overline{\mathcal{T}}_{n}'(p)\twoheadrightarrow \overline{\mathcal{T}}_{n}(p)\twoheadrightarrow \overline{\mathcal{T}}_{n}(p;d_{1}) \twoheadrightarrow \ldots \twoheadrightarrow \overline{\mathcal{T}}_{n}(p;d_{j},d_{j-1},\ldots,d_{1}) \twoheadrightarrow \ldots \twoheadrightarrow \overline{\mathcal{T}}_{n}(p;d_{p-2},d_{p-3},\ldots,d_{1})$ with finite GK-dimensions. Furthermore, we study the homological properties and the coradical filtrations of those quotient Hopf algebras.Comment: 27 pages, typos correcte