Quantum double constructions for compact quantum group

For a non-degenerate pair of compact quantum groups, we first construct the quantum double as an algebraic compact quantum group in an algebraic framework. Then by adopting some completion procedure, we give the universal and reduced quantum double constructions in the correspondence C＊-algebraic settings, which generalize Drinfeld＇s quantum double construction and yield new C＊-algebraic compact quantum groups.     

Direct limits of infinite-dimensional Lie groups compared to direct limits in related categories

Let G be a Lie group which is the union of an ascending sequence G₁⊆G₂⊆? of Lie groups (all of which may be infinite-dimensional). We study the question when in the category of Lie groups, topological groups, smooth manifolds, respectively, topological spaces. Full answers are obtained for G the group Diffc(M) of compactly supported C^∞-diffeomorphisms of a σ-compact smooth manifold M; and for test function groups of compactly supported smooth maps with values in a finite-dimensional Lie group H. We also discuss the cases where G is a direct limit of unit groups of Banach algebras, a Lie group of germs of Lie group-valued analytic maps, or a weak direct product of Lie groups.     

On a Problem Proposed by Kartsatos

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More on positive commutators

Let A and B be positive operators on a Banach lattice E such that the commutator C=AB−BA is also positive. The paper continues the investigation of the spectral properties of C initiated in J. Bra?i? et al. (in press) [3]. If the sum A+B is a Riesz operator and the commutator C is a power compact operator, then C is a quasi-nilpotent operator having a triangularizing chain of closed ideals of E. If we assume that the operator A is compact and the commutator AC−CA is positive, the operator C is quasi-nilpotent as well. We also show that the commutator C is not invertible provided the resolvent set of C is connected.     

Superconvergence of Legendre projection methods for the eigenvalue problem of a compact integral operator

In this paper, we consider the Galerkin and collocation methods for the eigenvalue problem of a compact integral operator with a smooth kernel using the Legendre polynomials of degree ≤n. We prove that the error bounds for eigenvalues are of the order O(n^−2r) and the gap between the spectral subspaces are of the orders O(n^−r) in L₂-norm and O(n^1/2−r) in the infinity norm, where r denotes the smoothness of the kernel. By iterating the eigenvectors we show that the iterated eigenvectors converge with the orders of convergence O(n^−2r) in both L₂-norm and infinity norm. We illustrate our results with numerical examples.     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-singular dichotomy for orbital measures of classical simple Lie algebras

We prove that the G-invariant orbital measures supported on adjoint orbits in the Lie algebra of a classical, compact, connected, simple Lie group satisfy a smoothness dichotomy: Either μ ^k is singular to Lebesgue measure or μ ^k ∈ L ². The minimum k for which μ ^k ∈ L ² is specified and is also the minimum k such that the k-fold sum of the orbit has positive measure. S. K. Gupta appreciates the hospitality of the Department of Pure Mathematics at the University of Waterloo where some of this research was done. K. E. Hare was supported in part by NSERC.     

强Z-连续domain和Z-代数domain的刻画定理

给出强Z-连续domain和Z-代数domain的一个刻画及一个范畴性质--余反射性质.     

A global pinching theorem for compact surfaces inS 3 with constant mean curvature

LetM be a compact minimal surface inS ³. Y. J. Hsu^[5] proved that if S₂22, thenM is either the equatorial sphere or the Clifford torus, whereS is the square of the length of the second fundamental form ofM, ·₂ denotes theL ²-norm onM. In this paper, we generalize Hsu's result to any compact surfaces inS ³ with constant mean curvature.Supported by NSFH.     

Approach Theory in a Category: A Study of Compactness and Hausdorff Separation

In a topological construct endowed with a proper -factorization system and a concrete functor , we study -compactness and -Hausdorff separation, where is a class of “closed morphisms” in the sense of Clementino et al. (A functional approach to general topology. In: Categorical Foundations. Encyclopedia of Mathematics and Its Applications, vol. 97, pp. 103–163. Cambridge University Press, Cambridge, 2004), determined by Λ. In particular, we point out under which conditions on Λ, the notion of -compactness of an object of coincides with 0-compactness of the image in Prap. Our results will be illustrated by some examples: except for some well-known ones, like b-compactness of a topological space, we also capture some compactness notions that were not considered before in the literature. In particular, we obtain a generalization of b-compactness to the setting of approach spaces. This notion is shown to play an important role in the study of uniformizability. The author is research assistant at the Fund of Scientific Research Vlaanderen (FWO).