# Questions tagged [combinatorial-hopf-algebras]

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### The shuffle algebra over the rationals is isomorphic to the polynomial algebra in the Lyndon words

On this wikipedia page is stated that over the rational numbers, the shuffle algebra (over a set $X$) is isomorphic to the polynomial algebra in the Lyndon words (on $X$). I was wondering if you can ...
0answers
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### Hopf algebra interpretation of hypergraph duality?

The work of Aguiar and Ardila (https://arxiv.org/abs/1709.07504) on Hopf monoids for generalized permutohedra gives a Hopf monoid structure on the collection of hypergraphs; see sections 19 and 20 of ...
1answer
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### Is Leray's theorem on commutative Hopf algebras proven in Milnor-Moore?

Question 1. Is a correct proof of Leray's theorem (the one that says that a connected graded Hopf algebra $H$ over a field of characteristic $0$ is isomorphic as an algebra to the symmetric ...
1answer
223 views

### Hopf structures on “pictorial” descriptions of permutations

There is a well-known Hopf structure on permutations, due to Malvenuto and Reutenauer. Here, the F basis elements are permutations in one-line notation, the product of two permutations is their ...
1answer
244 views

### Problem with Eisenbud's Lemma “Symmetry of Diagonalization”?

This question was first asked on MathSE but nobody answered. In his proof of Lemma A2.5 in his book Commutative Algebra with a View towards Algebraic Geometry, Prof. Eisenbud writes something like ...
3answers
1k views

### Classification of Hopf algebras (state of the art)

I assume that the classification of (certain families of) Hopf algebras is still an open problem, am I right? My question is the following: What is the current state of the art? What is known about ...
6answers
295 views

### Do stunted exponential series give projections of a cocommutative bialgebra on its coradical filtration?

Let $k$ be a field of characteristic $0$. Let $H$ be a cocommutative connected filtered bialgebra over $k$. ("Connected" means that the counit, restricted to the $0$-th part of the filtration, is an ...
7answers
4k views

### Shuffle Hopf algebra: how to prove its properties in a slick way?

Let $k$ be a commutative ring with $1$, and let $V$ be a $k$-module. Let $TV$ be the $k$-module $\bigoplus\limits_{n\in\mathbb N}V^{\otimes n}$, where all tensor products are over $k$. We define a $k$...