Quantifications des algèbres de Hopf d'arbres plans décorés et lien avec les groupes quantiques

We introduce a functor from the category of braided spaces into the category of braided Hopf algebras which associates to a braided space V a braided Hopf algebra of planar rooted trees . We show that the Nichols algebra of V is a subquotient of . We construct a Hopf pairing between and , generalising one of the results of [Bull. Sci. Math. 126 (2002) 193-239]. When the braiding of c is given by c(v_i⊗v_j)=q_i,jv_j⊗v_i, we obtain a quantification of the Hopf algebras introduced in [Bull. Sci. Math. 126 (2002) 193-239; 126 (2002) 249-288]. When q_i,j=q^a_i,j, with q an indeterminate and (a_i,j)_i,j the Cartan matrix of a semi-simple Lie algebra , then is a subquotient of . In this case, we construct the crossed product of with a torus and then the Drinfel'd quantum double of this Hopf algebra. We show that is a subquotient of .     

Polynomial identities and noncommutative versal torsors

To any cleft Hopf Galois object, i.e., any algebra obtained from a Hopf algebra H by twisting its multiplication with a two-cocycle α, we attach two “universal algebras” and . The algebra is obtained by twisting the multiplication of H with the most general two-cocycle σ formally cohomologous to α. The cocycle σ takes values in the field of rational functions on H. By construction, is a cleft H-Galois extension of a “big” commutative algebra . Any “form” of can be obtained from by a specialization of and vice versa. If the algebra is simple, then is an Azumaya algebra with center . The algebra is constructed using a general theory of polynomial identities that we set up for arbitrary comodule algebras; it is the universal comodule algebra in which all comodule algebra identities of are satisfied. We construct an embedding of into ; this embedding maps the center of into when the algebra is simple. In this case, under an additional assumption, , thus turning into a central localization of . We completely work out these constructions in the case of the four-dimensional Sweedler algebra.     

Comodules and Landweber exact homology theories

We show that, if E is a commutative MU-algebra spectrum such that is Landweber exact over , then the category of -comodules is equivalent to a localization of the category of -comodules. This localization depends only on the heights of E at the integer primes p. It follows, for example, that the category of -comodules is equivalent to the category of -comodules. These equivalences give simple proofs and generalizations of the Miller-Ravenel and Morava change of rings theorems. We also deduce structural results about the category of -comodules. We prove that every -comodule has a primitive, we give a classification of invariant prime ideals in , and we give a version of the Landweber filtration theorem.     

A new majorization between functions, polynomials, and operator inequalities

Let P₊ be the set of all non-negative operator monotone functions defined on [0,∞), and put . Then and . For a function and a strictly increasing function h we write if is operator monotone. If and and if and , then . We will apply this result to polynomials and operator inequalities. Let and be non-increasing sequences, and put for t≧a₁ and for t≧b₁. Then v₊?u₊ if m≦n and : in particular, for a sequence of orthonormal polynomials, (p_n-1)₊?(p_n)₊. Suppose 0<r,p and s=0 or 1≦s≦1+p/r. Then 0≦A≦B implies for 0<α≦r/(p+r).     

Genericity of wild holomorphic functions and common hypercyclic vectors

Let T_α be the translation operator by α in the space of entire functions defined by . We prove that there is a residual set G of entire functions such that for every f∈G and every the sequence is dense in , that is, G is a residual set of common hypercyclic vectors ( functions) for the family . Also, we prove similar results for many families of operators as: multiples of differential operator, multiples of backward shift, weighted backward shifts.     

The symmetry of the CS condition on one-sided ideals in a prime ring

Let R be a prime ring and e∈R be an idempotent. We show that eR_R is nonsingular, CS and if and only if is nonsingular, CS and .     

On duality between étale groupoids and Hopf algebroids

For any étale Lie groupoid G over a smooth manifold M, the groupoid convolution algebra of smooth functions with compact support on G has a natural coalgebra structure over the commutative algebra which makes it into a Hopf algebroid. Conversely, for any Hopf algebroid A over we construct the associated spectral étale Lie groupoid over M such that is naturally isomorphic to G. Both these constructions are functorial, and is fully faithful left adjoint to . We give explicit conditions under which a Hopf algebroid is isomorphic to the Hopf algebroid of an étale Lie groupoid G.     

Finiteness of graded local cohomology modules

Let R=?_n∈N0R_n be a Noetherian homogeneous ring with local base ring (R₀,m₀) and irrelevant ideal R₊, let M be a finitely generated graded R-module. In this paper we show that is Artinian and is Artinian for each i in the case where R₊ is principal. Moreover, for the case where , we prove that, for each i∈N₀, is Artinian if and only if is Artinian. We also prove that is Artinian, where and c is the cohomological dimension of M with respect to R₊. Finally we present some examples which show that and need not be Artinian.     

Exponentiable monomorphisms in categories of domains

We give a characterization of exponentiable monomorphisms in the categories of ω-complete posets, of directed complete posets and of continuous directed complete posets as those monotone maps f that are convex and that lift an element (and then a queue) of any directed set (ω-chain in the case of ) whose supremum is in the image of f (Theorem 1.9). Using this characterization, we obtain that a monomorphism f:X→B in (, ) exponentiable in w.r.t. the Scott topology is exponentiable also in (, ). We prove that the converse is true in the category , but neither in , nor in .     

Compactness via symmetrization

Consider two types of translation-invariant functionals and on , and a sequence of functions f_n whose corresponding symmetric rearrangements are convergent. We show that f_n themselves converge up to translations if either or . These compactness results lead to applications in variational problems and stability problems in stellar dynamics.