From left modules to algebras over an operad: application to combinatorial Hopf algebras

Livernet, Muriel

doi:10.5802/ambp.278

Cited by 8 publications

(8 citation statements)

References 23 publications

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“…The functor K preserves a number of properties, including freeness: if h is free as a monoid, then K(h) is free as an algebra[4, Proposition 18.7].Combining these remarks with Corollary 3.3 we deduce that for any connected Hopf monoid h, the algebra K(h) is free. This result is due to Livernet[7, Theorem 4.2.2].…”

mentioning

confidence: 88%

On the Hadamard Product of Hopf Monoids

Aguiar¹,

Mahajan²

2014

Can. j. math.

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Abstract. Combinatorial structures that compose and decompose give rise to Hopf monoids in Joyal's category of species. The Hadamard product of two Hopf monoids is another Hopf monoid. We prove two main results regarding freeness of Hadamard products. The first one states that if one factor is connected and the other is free as a monoid, their Hadamard product is free (and connected). The second provides an explicit basis for the Hadamard product when both factors are free.The first main result is obtained by showing the existence of a one-parameter deformation of the comonoid structure and appealing to a rigidity result of Loday and Ronco that applies when the parameter is set to zero. To obtain the second result, we introduce an operation on species that is intertwined by the free monoid functor with the Hadamard product. As an application of the first result, we deduce that the Boolean transform of the dimension sequence of a connected Hopf monoid is nonnegative.

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confidence: 88%

On the Hadamard Product of Hopf Monoids

Aguiar¹,

Mahajan²

2014

Can. j. math.

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“…Operads. Operads originated in algebraic topology [16,63] and, while they continue to be very important in that area (see, e.g., [11,28,29,65]), they have found applications in several other branches of mathematics, including geometry [35,43], algebra [57,64], combinatorics [3,56] and category theory [4,7,54]. Recently, operads have started to be considered also in theoretical computer science [23].…”

Section: Contents Introductionmentioning

confidence: 99%

On operads, bimodules and analytic functors

Gambino

Joyal²

2017

Memoirs of the AMS

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Background 1.1. Review of bicategory theory 1.2. V-categories and presentable V-categories 1.3. Distributors Chapter 2. Monoidal distributors 2.1. Monoidal V-categories and V-rigs 2.2. Monoidal distributors 2.3. Symmetric monoidal V-categories and symmetric V-rigs 2.4. Symmetric monoidal distributors Chapter 3. Symmetric sequences 3.1. Free symmetric monoidal V-categories 3.2. S-distributors 3.3. Symmetric sequences and analytic functors 3.4. Cartesian closure of categorical symmetric sequences Chapter 4. The bicategory of operad bimodules 4.1. Monads, modules and bimodules 4.2. Tame bicategories and bicategories of bimodules 4.3. Monad morphisms and bimodules 4.4. Tameness of bicategories of symmetric sequences 4.5. Analytic functors Chapter 5. Cartesian closure of operad bimodules 5.1. Cartesian closed bicategories of bimodules 5.2. Monad theory in tame bicategories 5.3. Monad theory in bicategories of bimodules 5.4. Bicategories of bimodules as Eilenberg-Moore completions Appendix A. A compendium of bicategorical definitions Appendix B. A technical proof B.1. Preliminaries B.2. The proof Bibliography

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“…In [19] and [10], an associative monoid in (S-Mod, ⊗ S ) is called a twisted associative algebra or an As-algebra in the category S-Mod, respectively.…”

Section: Shuffle Algebrasmentioning

confidence: 99%

“…Shuffle algebras are a particular case of monoids in the category of Smodules, as described in [19] and [10], where the operations do not preserve the action of the symmetric group. Applying this notion we are able to introduce the notion of shuffle bialgebra, in such a way that the Hopf algebra structures defined on the space spanned by maps between finite sets are induced by shuffle bialgebra structures on these spaces.…”

Section: Introductionmentioning

confidence: 99%

Shuffle bialgebras

Ronco¹

2011

Ann. inst. Fourier

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The goal of our work is to study the spaces of primitive elements of the Hopf algebras associated to the permutahedra and the associahedra. We introduce the notion of shuffle bialgebras, and compute the subpaces of primitive elements associated to these algebras. These spaces of primitive elements have natural structure of some type of algebras which we describe in terms of generators and relations. Applying these results we are able to compute primitive elements of other combinatorial Hopf algebras, and describe the algebraic theories associated to them.

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From left modules to algebras over an operad: application to combinatorial Hopf algebras

Cited by 8 publications

References 23 publications

On the Hadamard Product of Hopf Monoids

On the Hadamard Product of Hopf Monoids

On operads, bimodules and analytic functors

Shuffle bialgebras

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