On the Maurer-Cartan simplicial set of a complete curved $$A_\infty $$-algebra

In this paper, we develop the $$A_\infty $$ A ∞ -analog of the Maurer-Cartan simplicial set associated to an $$L_\infty $$ L ∞ -algebra and show how we can use this to study the deformation theory of $$\infty $$ ∞ -morphisms of algebras over non-symmetric operads. More precisely, we first recall and prove some of the main properties of $$A_\infty $$ A ∞ -algebras like the Maurer-Cartan equation and twist. One of our main innovations here is the emphasis on the importance of the shuffle product. Then, we define a functor from the category of complete (curved) $$A_\infty $$ A ∞ -algebras to simplicial sets, which sends a complete curved $$A_\infty $$ A ∞ -algebra to the associated simplicial set of Maurer-Cartan elements. This functor has the property that it gives a Kan complex. In all of this, we do not require any assumptions on the field we are working over. We also show that this functor can be used to study deformation problems over a field of characteristic greater than or equal to 0. As a specific example of such a deformation problem, we study the deformation theory of $$\infty $$ ∞ -morphisms of algebras over non-symmetric operads.

A Hopf Algebra on Permutations Arising from Super-Shuffle Product

In this paper, we first prove that any atom of a permutation obtained by the super-shuffle product of two permutations can only consist of some complete atoms of the original two permutations. Then, we prove that the super-shuffle product and the cut-box coproduct on permutations are compatible, which makes it a bialgebra. As this algebra is graded and connected, it is a Hopf algebra.

Double shuffle and Hoffman’s relations for interpolated multiple zeta values

The polynomials interpolating multiple zeta values and multiple zeta-star values using a parameter [Formula: see text] were introduced by Yamamoto. We call these [Formula: see text]-MZVs. In this paper, we establish the shuffle product for [Formula: see text]-MZVs to prove Hoffman type relation and double shuffle relation for [Formula: see text]-MZVs.

Homologie de Hochschild Et Espaces Algébriques

The purpose of this article is to define the Hochschild and cyclic homologies of an algebraic space $X$. We consider two definitions of Hochschild homology of $X$ and show that these definitions coincide in the case of a separated algebraic space. Moreover, we construct a shuffle product on the Hochschild homologies of $X$. Finally, we show that one has a pairing between the Hochschild homologies and cohomologies of $X$. Résumé Le but de cet article est de définir les homologies de Hochschild et les homologies cycliques d'un espace algébrique $X$. Nous considérons deux définitions de l'homologie de Hochschild et nous montrons que ces définitions sont les mêmes en cas d'un espace algébrique séparé. De plus, nous construisons un produit shuffle sur les homologies de Hochschild de $X$. Enfin, nous montrons qu'on a un appariement entre les homologies et les cohomologies de Hochschild de $X$.