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 .     

Wavelet frames with irregular matrix dilations and their stability

Let , B and A_j () be real nonsingular n×n matrices, λ_k () be real numbers. In this paper we present a sufficient condition for the system to be a frame for . This sufficient condition also shows the stability of the system with respect to the perturbation of matrix dilation parameters and the perturbation of translation parameters .     

Nilpotent orbits and some small unitary representations of indefinite orthogonal groups

For 2?m?l/2, let G be a simply connected Lie group with as Lie algebra, let be the complexification of the usual Cartan decomposition, let K be the analytic subgroup with Lie algebra , and let be the universal enveloping algebra of . This work examines the unitarity and K spectrum of representations in the “analytic continuation” of discrete series of G, relating these properties to orbits in the nilpotent radical of a certain parabolic subalgebra of .The roots with respect to the usual compact Cartan subalgebra are all ±e_i±e_j with 1?i<j?l. In the usual positive system of roots, the simple root e_m−e_m+1 is noncompact and the other simple roots are compact. Let be the parabolic subalgebra of for which e_m−e_m+1 contributes to and the other simple roots contribute to , let L be the analytic subgroup of G with Lie algebra , let , let be the sum of the roots contributing to , and let be the parabolic subalgebra opposite to .The members of are nilpotent members of . The group acts on with finitely many orbits, and the topological closure of each orbit is an irreducible algebraic variety. If Y is one of these varieties, let R(Y) be the dual coordinate ring of Y; this is a quotient of the algebra of symmetric tensors on that carries a fully reducible representation of .For , let . Then λ_s defines a one-dimensional module . Extend this to a module by having act by 0, and define . Let be the unique irreducible quotient of . The representations under study are and , where and Π_S is the Sth derived Bernstein functor.For s>2l−2, it is known that π_s=π_s′ and that π_s′ is in the discrete series. Enright, Parthsarathy, Wallach, and Wolf showed for m?s?2l−2 that π_s=π_s′ and that π_s′ is still unitary. The present paper shows that π_s′ is unitary for 0?s?m−1 even though π_s≠π_s′, and it relates the K spectrum of the representations π_s′ to the representation of on a suitable R(Y) with Y depending on s. Use of a branching formula of D. E. Littlewood allows one to obtain an explicit multiplicity formula for each K type in π_s′; the variety Y is indispensable in the proof. The chief tools involved are an idea of B. Gross and Wallach, a geometric interpretation of Littlewood's theorem, and some estimates of norms.It is shown further that the natural invariant Hermitian form on π_s′ does not make π_s′ unitary for s<0 and that the K spectrum of π_s′ in these cases is not related in the above way to the representation of on any R(Y).A final section of the paper treats in similar fashion the simply connected Lie group with Lie algebra , 2?m?l/2.     

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).     

Evaluation maps in rational homotopy

In the rational category of nilpotent complexes, let E be an H-space acting on a space X. With mild hypotheses we show that the action on the base point factors through a map Γ_E:S_E→X, where S_E is a finite product of odd-dimensional spheres and Γ_E is a homotopy monomorphism. Among others, the following consequences are obtained: if and only if is essential and if and only if X satisfies a strong splitting condition.     

Non-linear numerical radius isometries on atomic nest algebras and diagonal algebras

A nonlinear map φ between operator algebras is said to be a numerical radius isometry if w(φ(T−S))=w(T−S) for all T, S in its domain algebra, where w(T) stands for the numerical radius of T. Let and be two atomic nests on complex Hilbert spaces H and K, respectively. Denote the nest algebra associated with and the diagonal algebra. We give a thorough classification of weakly continuous numerical radius isometries from onto and a thorough classification of numerical radius isometries from onto .