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International audience; An interpolation problem is defined by a set of linear forms on the (multivariate) polynomial ring and values to be achieved by an interpolant. For Lagrange interpolation the linear forms consist of evaluations at some nodes,while Hermite interpolation also considers the values of successive derivatives. Both are examples of ideal interpolation in that the kernels of the linear forms intersect into an ideal. For an ideal interpolation problem with symmetry, we address the simultaneous computation of a symmetry adapted basis of the least interpolation space and the symmetry adapted H-basis of the ideal. Beside its manifest presence in the output, symmetry is exploited computationally at all stages of the algorithm. For an ideal invariant, under a group action, defined by a Groebner basis, the algorithm allows to obtain a symmetry adapted basis of the quotient and of the generators. We shall also note how it applies surprisingly but straightforwardly to compute fundamental invariants and equivariants of a reflection group.

Teaching abstract algebra, seen as the study of structures and properties of structures, at a university level appears to be a challenge for both students and faculty. The professors in our study describe this passage through abstraction for the student body as a "killing game", an "impassable wall" or a "leap through abstraction". In this paper, based on a case study (Candy, 2020) we will investigate the choices of professors teaching abstract algebra at university. These professors were chosen because they teach abstract algebra at university in France, and they gave us access to their course corpus and agreed to be interviewed. In this article, we will choose to study in particular the teaching of the concept of ideal. An epistemological analysis will allow us to highlight its central role in the construction of abstract algebra. Then, using the anthropological theory of didactics (Chevallard, 1998), we will try to specify the place of examples and exercises in the praxis of the student body., Enseigner l'algèbre abstraite, vue comme l'étude des structures et des propriétés des structures, à un niveau universitaire apparaît être un challenge pour les corps étudiant et professoral. Les professeurs de notre étude décrivent ce passage à travers l'abstraction pour le corps étudiant comme un « jeu de massacre », un « mur infranchissable » ou encore « un saut à travers l'abstraction ». Dans cet article, en appui sur une étude de cas (Candy, 2020) nous allons étudier les choix de professeurs enseignant l'algèbre abstraite à l'université. Ces professeurs ont été choisis car ils enseignent l'algèbre abstraite à l'université en France, de plus ils nous ont donné accès à leur corpus de cours et ont accepté d'être interviewés. Dans cet article, nous choisirons d'étudier en particulier l'enseignement du concept d'idéal. Une analyse épistémologique nous permettra de mettre en évidence son rôle central dans la construction de l'algèbre abstraite. Puis, en utilisant la théorie anthropologique du didactique (Chevallard, 1998), nous chercherons à préciser la place des exemples et exercices dans la praxis du corps étudiant.

International audience

In this paper, we first construct the free Rota-Baxter family algebra generated by some set $X$ in terms of typed angularly $X$-decorated planar rooted trees. As an application, we obtain a new construction of the free Rota-Baxter algebra only in terms of angularly decorated planar rooted trees (not forests), which is quite different from the known construction via angularly decorated planar rooted forests by K. Ebrahimi-Fard and L. Guo. We then embed the free dendriform (resp. tridendriform) family algebra into the free Rota-Baxter family algebra of weight zero (resp. one). Finally, we prove that the free Rota-Baxter family algebra is the universal enveloping algebra of the free (tri)dendriform family algebra., Reference precised in the introduction. This text uses in an essential way the description of the free dendriform and tridendriform family algebras given in arXiv:1909.08946

International audience; The formal integrability of systems of partial differential equations plays a fundamental role in different analysis and synthesis problems for both linear and nonlinear differential control systems. Following Spencer's theory, to test the formal integrability of a system of partial differential equations, we must study when the symbol of the system, namely, the top-order part of the linearization of the system, is 2-acyclic or involutive, i.e., when certain Spencer cohomology groups vanish. Combining the fact that Spencer cohomology is dual to Koszul homology and symbolic computation methods, we show how to effectively compute the homology modules defined by the Koszul complex of a finitely presented module over a commutative polynomial ring. These results are implemented using the OreMorphisms package. We then use these results to effectively characterize 2-acyclicity and involutivity of the symbol of a linear system of partial differential equations. Finally, we show explicit computations on two standard examples.

International audience; In this paper, the algebraic analysis approach to linear state-space systems is further developed using rings of integro-differential operators. The module structure of linear state-space systems is investigated over these rings. The module associated with a linear state-space system is shown to be the direct sum of the stably free module defined by the linear system without inputs and the free module defined by the inputs of the system.

Let $H$ be a Hopf algebra and let $A\subset H$ be a right coideal subalgebra. We show that if $A$ is a direct summand in $H$ as a right $A$-module, then $H$ is faithfully flat as a right $A$-module.

Condition (Fg) was introduced in [6] to ensure that the theory of support varieties of a finite dimensional algebra, established by Snashall and Solberg, has some similar properties to that of a group algebra. In this paper we give some easy to check combinatorial conditions that are equivalent to (Fg) for monomial d-Koszul algebras. We then extend this to monomial (D, A)-stacked algebras. We also extend the description of the Yoneda algebra of a d-Koszul algebra in [11] to (D, A)-stacked monomial algebras.

We prove that there exists a functorial correspondence between MV-algebras and partially cyclically ordered groups which are wound round of lattice-ordered groups. It follows that some results about cyclically ordered groups can be stated in terms of MV-algebras. For example, the study of groups together with a cyclic order allows to get a first-order characterization of groups of unimodular complex numbers and of finite cyclic groups. We deduce a characterization of pseudofinite MV-chains and of pseudo-simple MV-chains (i.e. which share the same first-order properties as some simple ones). We can generalize these results to some non-lineraly ordered MV-algebras, for example hyper-archimedean MV-algebras.

International audience; We compute the simple finite-dimensional modules and the center of the Drinfeld double of the Jordan plane introduced in [AP1] assuming that the characteristic is zero.

We introduce a weak division-like property for noncommutative rings: a nontrivial ring is fadelian if for all nonzero $a,x$ there exist $b,c$ such that $x=ab+ca$. We prove properties of fadelian rings, and construct examples of such rings which are not division rings, as well as non-Noetherian and non-Ore examples. We have also formalized some of these results in the Lean proof assistant., 15 pages. 1 figure, 1 appendix (Lean code). Comments welcome!

Let $k$ be a perfect field of characteristic $p > 0$, and let $K = k((u))$ be the field of Laurent series over $K$. We study the skew polynomial ring $K[T, \Phi]$, where $\Phi$ is an endomorphism of $K$ that extends a Frobenius endomorphism of $k$. We give a description of the irreducible skew polynomials, develop an analogue of the theory of the Newton polygon in this context, and classify the similarity classes of irreducible elements.

International audience; In this paper, we initiate a new algebraic analysis approach to linear differential systems based on rings of integro-differential operators. Within this algebraic analysis approach, we first interpret the method of variations of constants as an operator identity. Using this result, we show that the module associated with a state-space representation of a linear system is the same as the one associated with its standard convolution representation. This finitely presented module over the ring of integro-differential operators is proved to be stably free. Finally, we show how the reachability property can be expressed within this algebraic analysis approach.

Let $A$ be a definite quaternion algebra over $\mathbb Q$, with discriminant $D_A$, and $O$ a maximal order of $A$. We show that the minimum of the positive definite hamiltonian binary forms over $O$ with discrimiminant $−1$ is $\sqrt{D_A}$. When the different of $O$ is principal, we provide an explicit form representing this minimum, and when $O$ is principal, we give the list of the equivalence classes of all such forms. We also give criteria and algorithms to determine when the different of $O$ is principal.; Étant donné un ordre maximal $O$ d'une algèbre de quaternions rationnelle définie $A$ de discriminant $D_A$, nous montrons que le minimum des formes hamiltoniennes binaires sur $O$, définies positives et de discriminant $−1$, est $\sqrt{D_A}$. Lorsque la différente de $O$ est principale, nous explicitons une forme atteignant cette valeur, et lorsque $O$ est principal, nous donnons la liste exacte des formes atteignant cette valeur. Nous donnons des critères et des algorithmes pour déterminer quand la différente de $O$ est principale.

A correspondence functor is a functor from the category of finite sets and correspondences to the category of k k -modules, where k k is a commutative ring. We determine exactly which simple correspondence functors are projective. We also determine which simple modules are projective for the algebra of all relations on a finite set. Moreover, we analyze the occurrence of such simple projective functors inside the correspondence functor F F associated with a finite lattice and we deduce a direct sum decomposition of F F .

Hyperfields and systems are two algebraic frameworks which have been developed to provide a unified approach to classical and tropical structures. All hyperfields, and more generally hyperrings, can be represented by systems. We show that, conversely, we show that the systems arising in this way, called hypersystems, are characterized by certain elimination axioms. Systems are preserved under standard algebraic constructions; for instance matrices and polynomials over hypersystems are systems, but not hypersystems. We illustrate these results by discussing several examples of systems and hyperfields, and constructions like matroids over systems.

The algebra B of bicomplex numbers is viewed as a complexification of the Archimedean f-algebra of hyperbolic numbers D. This lattice-theoretic approach allows us to establish new properties of the so-called D-norms. In particular, we show that D-norms generate the same topology in B. We develop the D-trigonometric form of a bicomplex number which leads us to a geometric interpretation of the nth roots of a bicomplex number in terms of polyhedral tori. We use the concepts developed, in particular that of Riesz subnorm of a D-norm, to study the uniform convergence of the bicomplex zeta and gamma functions. The main result of this paper is the generalization to the bicomplex case of the Riemann functional equation and Euler's reflection formula., 23 pages, 1 figure. Submitted

International audience; In this work, we propose an algebraic method to study linear differential time-varying delay (DTVD) systems. Our goal is to give an effective construction of the ring of DTVD operators as an Ore extension, thanks to the concept of skew polynomial rings developed by Ore in the 30s. Some algebraic properties of the DTVD operators ring are analyzed, such as its Noetherianity, its homological and Krull dimensions, and the existence of Gröbner bases, all given in terms of the time-varying delay function. The algebraic analysis framework for linear systems theory allows us to study linear DTVD systems and essential properties such as the existence of autonomous elements, controllability, parametrizability, flatness, etc., through methods coming from module theory, homological algebra, and constructive algebra.

We study from the point of view of rational equivalence the enveloping algebras of Lie algebras of dimension 3 whose derived Lie subalgebra is of dimension 2, over an algebraically closed base field in arbitrary characteristics.

We introduce a generalization of tridendriform algebras, where each of the three products are replaced by a family of products indexed by a set $\Omega$. We study the needed structure on $\Omega$ for free $\Omega$-tridendriform algebras to be built on Schr{\"o}der trees (as it is the case in the classical case), with convenient decorations on their leaves. We obtain in this way extended triassociative semigroups. We describe commutative $\Omega$-tridendriform algebras in terms of typed words. We also study links with generalizations of Rota-Baxter algebras and describe the Koszul duals of the corresponding operads.

We introduce a generalization of parametrized Rota-Baxter algebras, which includes family and matching Rota-Baxter algebras. We study the structure needed on the set $\Omega$ of parameters in order to obtain that free Rota-Baxter algebras are described in terms of typed and angularly decorated planar rooted trees: we obtain the notion of $\lambda$-extended diassociative semigroup, which includes sets (for matching Rota-Baxter algebras) and semigroups (for family Rota-Baxter algebras), and many other examples. We also describe free commutative $\Omega$-Rota-Baxter algebras generated by a commutative algebra A in terms of typed words.

International audience; The Cauchy problem characterizes the solutions of a linear ordinary differential equation that satisfies initial conditions. In this paper, we investigate the converse problem, namely, given a function that is known to satisfy a linear ordinary differential equation of a fixed order, determine the coefficients of the ordinary differential equation and the initial conditions. The techniques used to investigate the inverse Cauchy problem come from the algebraic estimation problem introduced by Fliess and Sira-Ramírez. From the perfect observation of the solution, i.e., without external perturbation and noise corrupting it, the initial value problem can be explicitly reconstructed using only iterative indefinite integrals of the solution.

We derive a priori bounds on the size of the structure constants of the free Lie algebra over a set of indeterminates, relative to its Hall bases. We investigate their asymptotic growth, especially as a function of the length of the involved Lie brackets. First, using the classical recursive decomposition algorithm, we obtain a rough upper bound valid for all Hall bases. We then introduce new notions (which we call alphabetic subsets and relative foldings) related to structural properties of the Lie brackets created by the algorithm, which allow us to prove a sharp upper bound for the general case. We also prove that the length of the relative folding provides a strictly decreasing indexation of the recursive rewriting algorithm. Moreover, we derive lower bounds on the structure constants proving that they grow at least geometrically in all Hall bases. Second, for the celebrated historical length-compatible Hall bases and the Lyndon basis, we prove tighter sharp upper bounds, which turn out to be geometric in the length of the brackets. Third, we construct two new Hall bases, illustrating two opposite behaviors in the two-indeterminates case. One is designed so that its structure constants have the minimal growth possible, matching exactly the general lower bound, linked with the Fibonacci sequence. The other one is designed so that its structure constants grow super-geometrically. Eventually, we investigate asymmetric growth bounds which isolate the role of one particular indeterminate. Despite the existence of super-geometric Hall bases, we prove that the asymmetric growth with respect to each fixed indeterminate is uniformly at most geometric in all Hall bases., Fixed typos

International audience; We give a description of the duals of linearized Reed-Solomon codes in terms of codes obtained by taking residues of Ore rational functions. Our construction shows in particular that, under some assumptions on the base field, the class of linearized Reed-Solomon codes is stable under duality. As a byproduct of our work, we develop a theory of residues in the Ore setting.

We study the symplectic Howe duality using two new and independent combinatorial methods: via determinantal formulae on the one hand, and via (bi)crystals on the other hand. The first approach allows us to establish a generalised version where weight multiplicities are replaced by branching coefficients. In turn, this generalised Howe duality is used to prove the injectivity of induction for Levi branchings as previously conjectured by the last two authors.

Typed decorated trees are used by Bruned, Hairer and Zambotti to give a description of a renormalisation processon stochastic PDEs. We here study the algebraic structures on these objects: multiple prelie algebrasand related operads (generalizing a result by Chapoton and Livernet), noncommutative and cocommutative Hopf algebras (generalizing Grossman and Larson's construction),commutative and noncocommutative Hopf algebras (generalizing Connes and Kreimer's construction),bialgebras in cointeraction (generalizing Calaque, Ebrahimi-Fard and Manchon's result). We also define families of morphisms and in particular we prove that any Connes-Kreimer Hopf algebraof typed and decorated trees is isomorphic to a Connes-Kreimer Hopf algebra of non--typed and decoratedtrees (the set of decorations of vertices being bigger), through a contraction process,and finally obtain the Bruned-Hairer-Zambotti construction as a subquotient., Comment: 28 pages

We build, using the notion of zinbiel algebra, some commutative subalgebras $C_{u,v}$ inside an algebra of formal iterated integrals. There is a quotient map from this algebra of formal iterated integrals to the algebra of motivic multiple zeta values. Restricting this quotient map to the subalgebras $C_{u,v}$ of A$_{1,0}$ gives a morphism of graded commutative algebras with the same graded dimension. This is conjectured to be generically an isomorphism. When u+v = 0, the image is instead a sub-algebra of the algebra of motivic multiple zeta values., Comment: 11 pages, 1 figure

International audience; We identify the Baker-Campbell-Hausdorff recursion driven by a weight$\lambda=1$ Rota-Baxter operator with the Magnus expansion relativeto the post-Lie structure naturally associated to the correspondingRota-Baxter algebra. Post-Lie Magnus expansion and BCH-recursionare reviewed before the proof of the main result.

International audience; A fundamental result of Springer says that a quadratic form over a field of characteristic not 2 is isotropic if it is so after an odd degree extension. In this paper we generalize Springer's theorem as follows. Let R be a an arbitrary semilocal ring, let S be a finite R-algebra of constant odd degree, which is étale or generated by one element, and let q be a nonsingular R-quadratic form whose base ring extension q S is isotropic. We show that then q is already isotropic.

We make available some results about model theory cyclically ordered groups. We start with a classification of complete theories of divisible abelian cyclically ordered groups. Most of this paper is dedicated to the study of the model theory of torsion-free divisible abelian cyclically ordered groups, since we first show that the theory of divisible abelian cyclically ordered groups with non-trivial torsion elements is complete. Then we look at the cyclically ordered groups where the only parametrically definable subsets are finite unions of singletons and open intervals, and those where the definable subsets are finite union of singletons and c-convex subsets, where being c-convex is the analogue of being convex in the linearly ordered case.

35 pages; We give a general account of family algebras over a finitely presented linear operad. In a family algebra, each operation of arity $n$ is replaced by a family of operations indexed by $\Omega^n$, where $\Omega$ is a set of parameters. We show that the operad, together with its presentation, naturally defines an algebraic structure on the set of parameters, which in turn is used in the description of the family version of the relations between operations. The examples of dendriform and duplicial family algebras (hence with two parameters) and operads are treated in detail, as well as the pre-Lie family case. Finally, free one-parameter duplicial family algebras are described, together with the extended duplicial semigroup structure on the set of parameters.

We study extended associative semigroups (briefly, EAS), an algebraic structure used to define generalizations of the operad of associative algebras, and the subclass of commutative extended diassociative semigroups (briefly, CEDS), which are used to define generalizations of the operad of pre-Lie algebras. We give families of examples based on semigroups or on groups, as well as a classification of EAS of cardinality two. We then define linear extended associative semigroups as linear maps satisfying a variation of the braid equation. We explore links between linear EAS and bialgebras and Hopf algebras. We also study the structure of nondegenerate finite CEDS and show that they are obtained by semidirect and direct products involving two groups.

International audience; Let $k$ be a field of characteristic not two or three, let $\mathfrak{g}$ be a finite-dimensional colour Lie algebra and let V be a finite-dimensional representation of $\mathfrak{g}$. In this article we give various ways of constructing a colour Lie algebra $\tilde{\mathfrak{g}}$ whose bracket in some sense extends both the bracket of $\mathfrak{g}$ and the action of $\mathfrak{g}$ on V. Colour Lie algebras, originally introduced by R. Ree ([Ree60]), generalise both Lie algebras and Lie superalgebras, and in those cases our results imply many known results ([Kos99], [Kos01], [CK15], [SS15]). For a class of representations arising in this context we show there are covariants satisfying identities analogous to Mathews identities for binary cubics.

We study diverse parametrized versions of the operad of associative algebra, where the parameter are taken in an associative semigroup $\Omega$ (generalization of matching or family associative algebras) or in its cartesian square (two-parameters associative algebras). We give a description of the free algebras on these operads, study their formal series and prove that they are Koszul when the set of parameters is finite. We also study operadic morphisms between the operad of classical associative algebras and these objects, and links with other types of algebras (diassociative, dendriform, post-Lie...).

International audience; For a finite group, we present three algorithms to compute a generating set of invariant simultaneously to generating sets of basic equivariants, i.e., equivariants for the irreducible representations of the group. The main novelty resides in the exploitation of the orthogonal complement of the ideal generated by invariants; Its symmetry adapted basis delivers the fundamental equivariants. Fundamental equivariants allow to assemble symmetry adapted bases of polynomial spaces of higher degrees, and these are essential ingredients in exploiting and preserving symmetry in computations. They appear within algebraic computation and beyond, in physics, chemistry and engineering. Our first construction applies solely to reflection groups and consists in applying symmetry preserving interpolation, as developed by the same authors, along an orbit in general position. The fundamental invariants can be read off the H-basis of the ideal of the orbit while the fundamental equivariants are obtained from a symmetry adapted basis of an invariant direct complement to this ideal in the polynomial ring. The second algorithm takes as input primary invariants and the output provides not only the secondary invariants but also free bases for the modules of basic equivariants. These are constructed as the components of a symmetry adapted basis of the orthogonal complement, in the polynomial ring, to the ideal generated by primary invariants.The third algorithm proceeds degree by degree, determining the fundamental invariants as forming a H-basis of the Hilbert ideal, i.e., the polynomial ideal generated by the invariants of positive degree. The fundamental equivariants are simultaneously computed degree by degree as the components of a symmetry adapted basis of the orthogonal complement of the Hilbert ideal.

We study generalizations of pre-Lie algebras, where the free objects are based on rooted trees which edges are typed, instead of usual rooted trees, and with generalized pre-Lie products formed by graftings. Working with a discrete set of types, we show how to obtain such objects when this set is given an associative commutative product and a second product making it a commutative extended semigroup. Working with a vector space of types, these two products are replaced by a bilinear map $\Phi$ which satisfies a braid equation and a commutation relation. Examples of such structures are defined on sets, semigroups, or groups. These constructions define a family of operads PreLie $\Phi$ which generalize the operad of pre-Lie algebras PreLie. For any embedding from PreLie into PreLie $\phi$ , we construct a family of pairs of cointeracting bialgebras, based on typed and decorated trees: the first coproduct is given by an extraction and contraction process, the types being modified by the action of $\Phi$; the second coproduct is given by admissible cuts, in the Connes and Kreimer's way, with again types modified by the action of $\Phi$. We also study the Koszul dual of PreLie $\Phi$ , which gives generalizations of permutative algebras.

We introduce and study a new class of modules and rings we call n/2-perspective, which can either be described in terms of perspective direct summands, associate idempotents, or generalized inverses. When n is small (≤ 2), we recover existing class of modules and rings: (endo)abelian, strongly IC and perspective ones. And 3/2-perspective rings are characterized by all their regular elements being special clean. Standard constructions are also discussed and examples are provided.

In this short note, we study two specific generalized inverses, the (b, c)-inverse of Drazin and the inverse along an element of the present author, in a categorical way. Precisely, we prove that the (b, c)-inverse and the inverse along an element in a semigroup are actually genuine inverse when considered as morphisms in the Schutzenberger category of the semigroup. This category was first defined and studied by Costa and Steinberg, and its definition is based on Green's preoders and relations. In the last section, applications to the Reverse Order Law are given and some new formulas are provided. Examples are also given.

International audience; We show that a reductive group scheme over a base scheme S admits a faithful linear representation if and only if its radical torus is isotrivial, that is, it splits after a finite étale cover.

We provide a constructive treatment of basic results in the theory of central simple algebras. One main issue is the fact that one starting result, Wedderburn's Theorem stating that a simple algebra is a matrix algebra over a skew field, is not constructively valid. We solve this problem by proving instead a dynamical version of this theorem. One can use this to give constructive proofs of basic results of the theory of central simple algebras, such as the Skolem-Noether Theorem. We illustrate this development by giving an elementary constructive proof of a theorem of Becher (which is itself a consequence of a celebrated theorem of Merkurjev).

In this paper, we describe an axiom-free Coq formalization that there does not exists a general method for solving by radicals polynomial equations of degree greater than 4. This development includes a proof of Galois' Theorem of the equivalence between solvable extensions and extensions solvable by radicals. The unsolvability of the general quintic follows from applying this theorem to a well chosen polynomial with unsolvable Galois group., LIPIcs, Vol. 193, 12th International Conference on Interactive Theorem Proving (ITP 2021), pages 8:1-8:18

We discuss the question of whether the global dimension is a monoidal invariant for Hopf algebras, in the sense that if two Hopf algebras have equivalent monoidal categories of comodules, then their global dimensions should be equal. We provide several positive new answers to this question, under various assumptions of smoothness, cosemisimplicity or finite dimension. We also discuss the comparison between the global dimension and the Gerstenhaber-Schack cohomological dimension in the cosemisimple case, obtaining equality in the case the latter is finite. One of our main tools is the new concept of twisted separable functor., 20 pages. V2: 23 pages, changes in the redaction, no new result

International audience; This work is devoted to the algebraic and arithmetic properties of Rankin-Cohen brackets allowing to define and study them in several natural situations of number theory. It focuses on the property of these brackets to be formal deformations of the algebras on which they are defined, with related questions on restriction-extension methods. The general algebraic results developed here are applied to the study of formal deformations of the algebra of weak Jacobi forms and their relation with the Rankin-Cohen brackets on modular and quasimodular forms.; Ce travail est consacré aux propriétés algébriques et arithmétiques des crochets de Rankin-Cohen permettant de les définir et de les étudier dans plusieurs situations naturelles de la théorie des nombres. Il se concentre sur la propriété qu'ont ces crochets d'être des déformations formelles des algèbres sur lesquelles ils sont définis, avec des questions connexes sur les méthodes de restriction-extension. Les résultats algébriques généraux développés ici sont appliqués à l'étude des déformations formelles de l'algèbre des formes de Jacobi faibles et leur relation avec les crochets de Rankin-Cohen pour les formes modulaires et quasimodulaires.

We here both unify and generalize nonassociative structures on typed binary trees, that is to say plane binary trees which edges are decorated by elements of a set Ω. We prove that we obtain such a structure, called an Ω-dendriform structure, if Ω has four products satisfying certain axioms (EDS axioms), including the axioms of a diassociative semigroup. This includes matching dendriform algebras introduced by Zhang, Gao and Guo and family dendriform algebras associated to a semigroup introduced by Zhang, Gao and Manchon, and of course dendriform algebras when Ω is reduced to a single element. We also give examples of EDS, including all the EDS of cardinality two; a combinatorial description of the products of such a structure on typed binary trees, but also on words; a study of the Koszul dual of the associated operads; and considerations on the existence of a coproduct, in order to obtain dendriform bialgebras.

International audience; In this paper, we study the operad of C-enriched pre-Lie algebras, defined for any Hopf cooperad C; it slightly generalises a similar notion defined by Calaque and Willwacher to produce conceptual constructions of the operads acting on various deformation complexes. Maps between Hopf cooperads lead to maps between the corresponding operads of enriched pre-Lie algebras; we prove a criterion for the module action of the domain on the codomain to be free, on the left and on the right. In particular, this implies a new functorial Poincaré--Birkhoff--Witt type theorem for universal enveloping brace algebras of pre-Lie algebras.

This paper extends the study of rank-metric codes in extension fields $\mathbb{L}$ equipped with an arbitrary Galois group $G = \mathrm{Gal}(\mathbb{L}/\mathbb{K})$. We propose a framework for studying these codes as subspaces of the group algebra $\mathbb{L}[G]$, and we relate this point of view with usual notions of rank-metric codes in $\mathbb{L}^N$ or in $\mathbb{K}^{N\times N}$, where $N = [\mathbb{L} : \mathbb{K}]$. We then adapt the notion of error-correcting pairs to this context, in order to provide a non-trivial decoding algorithm for these codes. We then focus on the case where $G$ is abelian, which leads us to see codewords as elements of a multivariate skew polynomial ring. We prove that we can bound the dimension of the vector space of zeroes of these polynomials, depending of their degree. This result can be seen as an analogue of Alon-F\"uredi theorem -- and by means, of Schwartz-Zippel lemma -- in the rank metric. Finally, we construct the counterparts of Reed-Muller codes in the rank metric, and we give their parameters. We also show the connection between these codes and classical Reed-Muller codes in the case where $\mathbb{L}$ is a Kummer extension., Comment: 26 pages, 1 figure

International audience; Under suitable conditions the skewfield of fractions of a su-peralgebra which is a noetherian domain is canonically provided with a structure of superalgebra. This gives rise to a notion of rational equivalence in the category of superalgebras. We study from the point of view of this rational equivalence some low dimensional examples of quantum enveloping algebras of Lie superalgebras.

International audience; This paper constitutes a first attempt to do analysis with skew polynomials. Precisely, our main objective is to develop a theory of residues for skew rational functions (which are, by definition, the quotients of two skew polynomials). We prove in particular a skew analogue of the residue formula and a skew analogue of the classical formula of change of variables for residues.

Shuffle operads were introduced to forget the symmetric group actions on symmetric operads while preserving all possible operadic compositions. Rewriting methods were then applied to symmetric operads via shuffle operads: in particular, a notion of Gr\"obner basis was introduced for shuffle operads with respect to a total order on tree monomials. In this article, we introduce the structure of shuffle polygraphs as a categorical model for rewriting in shuffle operads, which generalizes the Gr\"obner bases approach by removing the constraint of a monomial order for the orientation of the rewriting rules. We define w-operads as internal w-categories in the category of shuffle operads. We show how to extend a convergent shuffle polygraph into a shuffle polygraphic resolution generated by the overlapping branchings of the original polygraph. Finally, we prove that a shuffle operad presented by a quadratic convergent shuffle polygraph is Koszul.

With this paper we present an extension of our recent ISSAC paper about computations of Groebner(-Shirshov) bases over free associative algebras Z. We present all the needed proofs in details, add a part on the direct treatment of the ring Z/mZ as well as new examples and applications to e.g. Iwahori-Hecke algebras.The extension of Groebner bases concept from polynomial algebras over fields to polynomial rings over rings allows to tackle numerous applications, both of theoretical and of practical importance.Groebner and Groebner-Shirshov bases can be defined for various non-commutative and even non-associative algebraic structures. We study the case of associative rings and aim at free algebras over principal ideal rings. We concentrate ourselves on the case of commutative coefficient rings without zero divisors (i.e. a domain). Even working over Z allows one to do computations, which can be treated as universal for fields of arbitrary characteristic. By using the systematic approach, we revisit the theory and present the algorithms in the implementable form. We show drastic differences in the behavior of Groebner bases between free algebras and algebras, close to commutative.Even the process of the formation of critical pairs has to be reengineered, together with the implementing the criteria for their quick discarding.We present an implementation of algorithms in the Singular subsystem called Letterplace, which internally uses Letterplace techniques (and Letterplace Groebner bases), due to La Scala and Levandovskyy. Interesting examples and applications accompany our presentation.