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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.

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.

v4 is the final version; International audience; In an earlier paper we established that every second countable, completely normal spectral space is homeomorphic to the ℓ-spectrum of some Abelian ℓ-group. We extend that result to ℓ-spectra of vector lattices over any countable totally ordered division ring k. Extending our original machinery, about finite lattices of polyhedra, from linear to affine and allowing relativizations to convex subsets, then invoking Baro's Normal Triangulation Theorem, we obtain the following result:Theorem. For every countable formally real field k, every second countable, completely normal spectral space is homeomorphic to the real spectrum of some commutative unital k-algebra.The countability assumption on k is necessary: there exists a second countable, completely normal spectral space that cannot be embedded, as a spectral subspace, into either the ℓ-spectrum of any right vector lattice over an uncountable directed partially ordered division ring, or the real spectrum of any commutative unital algebra over an uncountable field.

We prove a categorical duality between a class of abstract algebras of partial functions and a class of (small) topological categories. The algebras are the isomorphs of collections of partial functions closed under the operations of composition, antidomain, range, and preferential union (or 'override'). The topological categories are those whose space of objects is a Stone space, source map is a local homeomorphism, target map is open, and all of whose arrows are epimorphisms., Comment: 25 pages. Very minor changes

Let $r$ and $n$ be positive integers such that $r, Comment: 41 pages (version 2, minor errors corrected)

We define the fundamental group of a Hopf algebra over a field. For this purpose we first consider gradings of Hopf algebras and Galois coverings. The latter are given by linear categories with new additional structure which we call Hopf linear categories over a finite group. We compare this invariant to the fundamental group of the underlying linear category, and we compute those groups for families of examples., Computations of the fundamental group of some Hopf algebras are added. The relation with the fundamental group of the underlying associative structure is now considered. We also analyse the situation when universal covers and/or gradings exist. Dedicated to Eduardo N. Marcos for his 60th birthday. 24 pages

Resume Nous utilisons des resultats classiques sur les zeros des fonctions L de Dirichlet pour prouver la non-nullite de deux determinants analogues au determinant de Maillet. Nous en deduisons un theoreme d'extension pour les isometries de Lee et euclidienne des codes lineaires sur un corps premier.

An important problem on almost perfect nonlinear (APN) functions is the existence of APN permutations on even-degree extensions of F 2 larger than 6. Browning et al. (2010) gave the first known example of an APN permutation on the degree-6 extension of F 2 . The APN permutation is CCZ-equivalent to the previously known quadratic Kim κ-function (Browning et al. (2009)). Aside from the computer based CCZ-inequivalence results on known APN functions on even-degree extensions of F 2 with extension degrees less than 12, no theoretical CCZ-inequivalence result on infinite families is known. In this paper, we show that Gold and Kasami APN functions are not CCZ-equivalent to permutations on infinitely many even-degree extensions of F 2 . In the Gold case, we show that Gold APN functions are not equivalent to permutations on any even-degree extension of F 2 , whereas in the Kasami case we are able to prove inequivalence results for every doubly-even-degree extension of F 2 .

International audience; In this study, we consider Lie algebras that admit para-Kählerand hyper-para-Kähler structures. We provide new charac-terizations of these Lie algebras and develop many methodsfor building large classes of examples. Previously, Bai consid-ered para-Kähler Lie algebras as left symmetric bialgebras.We reconsider this viewpoint and make improvements in or-der to obtain some new results. The study of para-Kähler andhyper-para-Kähler is intimately linked to the study of leftsymmetric algebras, particularly those that admit invariantsymplectic forms. In this study, we provide many new classesof left symmetric algebras and complete descriptions of all theassociative algebras that admit an invariant symplectic form.We also describe all four-dimensional hyper-para-Kähler Liealgebras.

International audience; It is shown that a ring is left semihereditary if and only each homomorphic image of its injective hull as left module is FP-injective. It is also proven that a commutative ring R is reduced and arithmetical if and only if E/U if FP-injective for any FP-injective R-module E and for any submodule U of finite Goldie dimension. A characterization of commutative rings for which each module of finite Goldie dimension is of injective dimension at most one is given. Let R be a chain ring and Z its subset of zerodivisors. It is proven that E/U is FP-injective for each FP-injective R-module E and each pure polyserial submodule U of E if R/I is complete in its f.c. topology for each ideal I whose the top prime ideal is Z. The converse holds if each indecomposable injective module whose the bottom prime ideal is Z contains a pure uniserial submodule. For some chain ring R we show that E/U is FP-injective for any FP-injective module E and any its submodule U of finite Goldie dimension, even if R is not coherent. It follows that any Archimedean chain ring is either coherent or maximal if and only if each factor of any injective module of finite Goldie dimension modulo a pure submodule is injective.

This paper deals with the problem of conjugacy of Cartan subalgebras for affine Kac-Moody Lie algebras. Unlike the methods used by Peterson and Kac, our approach is entirely cohomological and geometric. It is deeply rooted on the theory of reductive group schemes developed by Demazure and Grothendieck, and on the work of J. Tits on buildings., 24 pages

International audience; Laplace presented an algebraic proof of the fundamental theorem of algebra which relied on the existence of a splitting field. This proof can be generalized by replacing real numbers by any real closed field. Although it is possible to build a splitting field in a classical context, it is not true anymore in an constructive context. We present a constructive version of Laplace's proof as the result of a general method to make constructive sense of the notion of splitting fields.

International audience; Para-Kähler Lie algebras which decompose as the sum of two abelian Lagrangian sub-algebras are studied. We propose several constructions and provide an inductive descriptionof such Lie algebras. The curvatures of the para-Kähler metric are computed and sufficientconditions to ensure flatness or Ricci-flatness are given. The Lie algebras for which thepara-Kähler metric is Einstein and non-Ricci-flat are completely characterized.

Let G be a finite group, let p be a prime number, and let K be a field of characteristic 0 and k be a field of characteristic p, both large enough. In this note we state explicit formulae for the primitive idempotents of K\otimes pp_k(G), where pp_k(G) is the ring of p-permutation kG-modules.

Let G be a connected reductive algebraic group and let G ′ = [ G , G ] be its derived subgroup. Let ( G , V ) be a multiplicity free representation with a one-dimensional quotient (see definition below). We prove that the algebra D ( V ) G ′ of G ′ -invariant differential operators with polynomial coefficients on V , is a quotient of a so-called Smith algebra over its center. Over C this class of algebras was introduced by S.P. Smith (1990) as a class of algebras similar to U ( sl 2 ) . Our result generalizes the case of the Weil representation, where the associative algebra generated by Q ( x ) and Q ( ∂ ) ( Q being a non-degenerate quadratic form on V ) is a quotient of U ( sl 2 ) . Other structure results are obtained when ( G , V ) is a regular prehomogeneous vector space of commutative parabolic type. To cite this article: H. Rubenthaler, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

The object of this Note is to define the generalized flag geometry of a graded Lie algebra which corresponds to the generalized projective geometry in the case of 3-gradings. Then we construct a structure of manifold on this generalized flag geometry. This result generalizes a result known for 3-graded Lie algebras to the more general case of ( 2 k + 1 ) -graded Lie algebras. To cite this article: J. Chenal, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

The present article takes advantage of the properties of algebras in the category of S-modules (twisted algebras) to investigate further the fine algebraic structure of Hopf operads. We prove that any Hopf operad P carries naturally the structure of twisted Hopf P-algebra. Many properties of classical Hopf algebraic structures are then shown to be encapsulated in the twisted Hopf algebraic structure of the corresponding Hopf operad. In particular, various classical theorems of Lie theory relating Lie polynomials to words (i.e. elements of the tensor algebra) are lifted to arbitrary Hopf operads., Comment: 23 pages. Using xyPic

A $\mu$-algebra is a model of a first order theory that is an extension of the theory of bounded lattices, that comes with pairs of terms $(f,\mu_{x}.f)$ where $\mu_{x}.f$ is axiomatized as the least prefixed point of $f$, whose axioms are equations or equational implications. Standard $\mu$-algebras are complete meaning that their lattice reduct is a complete lattice. We prove that any non trivial quasivariety of $\mu$-algebras contains a $\mu$-algebra that has no embedding into a complete $\mu$-algebra. We focus then on modal $\mu$-algebras, i.e. algebraic models of the propositional modal $\mu$-calculus. We prove that free modal $\mu$-algebras satisfy a condition -- reminiscent of Whitman's condition for free lattices -- which allows us to prove that (i) modal operators are adjoints on free modal $\mu$-algebras, (ii) least prefixed points of $\Sigma_{1}$-operations satisfy the constructive relation $\mu_{x}.f = \bigvee_{n \geq 0} f^{n}(\bot)$. These properties imply the following statement: {\em the MacNeille-Dedekind completion of a free modal $\mu$-algebra is a complete modal $\mu$-algebra and moreover the canonical embedding preserves all the operations in the class $Comp(\Sigma_{1},\Pi_{1})$ of the fixed point alternation hierarchy.}, Comment: 36 pages, extended abstract appears in LICS 2005 proceedings

In this paper we investigate the class of the connected graded algebras which are finitely generated in degree 1, which are finitely presented with relations of degrees greater or equal to 2 and which are of finite global dimension D and Gorenstein. For D greater or equal to 4 we add the condition that these algebras are homogeneous and Koszul. It is shown that each such algebra is completely characterized by a multilinear form satisfying a twisted cyclicity condition and some other nondegeneracy conditions depending on the global dimension D. This multilinear form plays the role of a volume form and canonically identifies in the quadratic case to a nontrivial Hochschild cycle of maximal degree. Several examples including the Yang-Mills algebra and the extended 4-dimensional Sklyanin algebra are analyzed in this context. Actions of quantum groups are also investigated., Comment: 39 pages, slight reformulation of Theorem 11, conjecture of Section 9 replaced by a counterexample, references added

We introduce non-abelian cohomology sets of Hopf algebras with coefficients in Hopf modules. We prove that these sets generalize Serre's non-abelian group cohomology theory. Using descent techniques, we establish that our construction enables to classify as well twisted forms for modules over Hopf-Galois extensions as torsors over Hopf-modules., 19 pages

24 pages, accepted for publication in Advances in mathematics.; the double Lie algebra L^D of rooted trees decorated by a set D is introduced, generalising the construction of Connes and Kreimer. It is shown that it is a simple Lie algebra. Its derivations and its automorphisms are described, as well as some central extensions. Finally, the category of lowest weight modules is introduced and studied.

For a positive integer n we introduce quadratic Lie algebras tr_n qtr_n and discrete groups Tr_n, QTr_n naturally associated with the classical and quantum Yang-Baxter equation, respectively. We prove that the universal enveloping algebras of the Lie algebras tr_n, qtr_n are Koszul, and find their Hilbert series. We also compute the cohomology rings of these Lie algebras (which by Koszulity are the quadratic duals of the enveloping algebras). We construct cell complexes which are classifying spaces of the groups Tr_n and QTr_n, and show that the boundary maps in them are zero, which allows us to compute the integral cohomology of these groups. We show that the Lie algebras tr_n, qtr_n map onto the associated graded algebras of the Malcev Lie algebras of the groups Tr_n, QTr_n, respectively. We conjecture that this map is actually an isomorphism (this is now a theorem due to P. Lee). At the same time, we show that the groups Tr_n and QTr_n are not formal for n>3., Errors pointed out by P. Lee corrected. Proposition 5.1 in the previous version was incorrect, so Theorems 2.3 and 8.5 and Proposition 6.1 were proved incorrectly. In the current version, Propositions 5.1 and 6.1 are deleted and Theorems 2.3, 8.5 are stated as conjectures. They were proved by Peter Lee, along with Conjectures 2.4 and 8.6 of the previous version

International audience; We develop a purely ordinal model for aggregation functionals for lattice valued functions, comprising as special cases quantiles, the Ky Fan metric and the Sugeno integral. For modeling findings of psychological experiments like the reflection effect in decision behaviour under risk or uncertainty, we introduce reflection lattices. These are complete linear lattices endowed with an order reversing bijection like the reflection at $0$ on the real interval $[-1,1]$. Mathematically we investigate the lattice of non-void intervals in a complete linear lattice, then the class of monotone interval-valued functions and

It is shown that each almost maximal valuation ring R, such that every indecomposable injective R-module is countably generated, satisfies the following condition (C): each fp-injective R-module is locally injective. The converse holds if R is a domain. Moreover, it is proved that a valuation ring R that satisfies this condition (C) is almost maximal. The converse holds if Spec(R) is countable. When this last condition is satisfied it is also proved that every ideal of R is countably generated. New criteria for a valuation ring to be almost maximal are given. They generalize the criterion given by E. Matlis in the domain case. Necessary and sufficient conditions for a valuation ring to be an IF-ring are also given.

We suggest a possible programme to associate geometric "flag-like" data to an arbitrary simple quantum group, in the spirit of the noncommutative algebraic geometry developed by Artin, Tate, and Van den Bergh. We then carry out this programme for the standard quantum SL(n) of Drinfel'd and Jimbo, where the varieties involved are certain T-stable subvarieties of the (ordinary) flag variety., 27 pages; needs LaTeX2e, AMS-LaTeX, XY-pic, and PSTricks (v2: several minor corrections)

The aim of this paper is to present remarkable classes of Lie-admissible algebras containing in particular the associative algebras, the Vinberg algebras and pre-Lie algebras. We determine the associated quadratic operads and their dual operads., Comment: 5 pages

We study the Lie algebra structure of the first Hochschild cohomology group of a finite dimensional monomial algebra A, in terms of the combinatorics of its quiver, in any characteristic. This allows us also to examine the identity component of the algebraic group of outer automorphisms of A in characteristic zero. Criteria for the (semi-)simplicity, the solvability, the reductivity, the commutativity and the nilpotency are given., 22 pages Correction of typing errors

We study Lie algebras generated by extremal elements (i.e., elements spanning inner ideals of L) over a field of characteristic distinct from 2. We prove that any Lie algebra generated by a finite number of extremal elements is finite dimensional. The minimal number of extremal generators for the Lie algebras of type An, Bn (n>2), Cn (n>1), Dn (n>3), En (n=6,7,8), F4 and G2 are shown to be n+1, n+1, 2n, n, 5, 5, and 4 in the respective cases. These results are related to group theoretic ones for the corresponding Chevalley groups., 28 pages

The quantum multiparametric torus is the algebra generated over a field $k$ by the $2N$ variables $x_1,\ldots,x_N$ and $x_1^{-1},\ldots,x_N^{-1}$ and the relations $ x_ix_i^{-1}=1=x_i^{-1} x_i$ and $x_ix_j=q_{ij}x_jx_i$ for every $1\le i,j\le N$ and where $\{q_{ij}\}_{1\le i,j\le N}$ is a family of non-zero scalars of $k$ satisfying the relations $q_{ii}=1$ and $q_{ij}q_{ji}=1$ for every $1\le i,j,\le N$. We explicitly compute its Hochschild homology groups, using previously constructed ``quantum Koszul complexes''. We deduce the corresponding cyclic homology groups.

In this paper, we study algebraic structures defined by a presentation of the form 〈 A; ab = ba for (a, b) ∈ I 〉. We show that these relators are the only one that can be interpreted as both monoid and Lie relators. We also study the free partially commutative monoids, Lie algebras, and groups: in all of these cases, we show how to obtain decomposition results for these structures into the corresponding free structures.

In this paper, we study the free partially commutative Lie K-algebra L(A,θ) defined by a commutation relation θ on an alphabet A. Its behavior is very similar to that of the free Lie algebra. Indeed, we obtain in particular a partially commutative version of Lazard's elimination process which allows us to prove that the K-module L(A, θ) is free and to construct explicitly K-bases for it. We show also how the classical Witt's calculus can be extended to L(A, θ).

In this paper, we study the generalization of Hankel-like results for formal pseudo-differential operators, skew formal series and Malcev series. We show how to obtain good notions of Hankel determinants in the two first cases.