인공고관절의 세라믹 볼 헤드의 기계적 안정성 평가를 위한 3 차원 유한요소 해석

Using a variety of methods developed in the literature (in particular, the theory of weak Hopf algebras), we prove a number of general results about fusion categories in characteristic zero. We show that the global dimension of a fusion category is always positive, and that the S-matrix of any (not necessarily hermitian) modular category is unitary. We also show that the category of module functors between two module categories over a fusion category is semisimple, and that fusion categories and tensor functors between them are undeformable (generalized Ocneanu rigidity). In particular the number of such categories (functors) realizing a given fusion datum is finite. Finally, we develop the theory of Frobenius-Perron dimensions in an arbitrary fusion category. At the end of the paper we generalize some of these results to positive characteristic.

http://annals.math.princeton.edu/wp-content/uploads/annals-v162-n2-p01.pdf

On fusion categories

In this chapter we will embark upon the task of systematically identifying important specific phenomena associated with the asymptotic behavior of smooth dynamical systems We will build upon the results of our survey of specific examples in Chapter 1 as well as on the insights gained from the general structural approach outlined and illustrated in Chapter 2 Most of the properties discussed in the present chapter are in fact topological invariants and can be defined for broad classes of topological dynamical systems, including symbolic ones The predominance of topological invariants fits well with the picture that emerges from the considerations of Sections 21, 23, 24, and 26 The considerations of the previous chapter make it very plausible that smooth dynamical systems are virtually never differentiably stable and can only rarely be classified locally up to smooth conjugacy In contrast, structural and the related topological stability seem to be fairly widespread phenomena We will consider three broad classes of asymptotic invariants: (i) growth of the numbers of orbits of various kinds and of the complexity of orbit families, (ii) types of recurrence, and (iii) asymptotic distribution and statistical behavior of orbits The first two classes are of a purely topological nature; they are discussed in the present chapter The last class is naturally related to ergodic theory and hence we will provide an introduction to key aspects of that subject This will require some space so we put that material into a separate chapter The two chapters are intimately connected

Introduction to the Modern Theory of Dynamical Systems: PRINCIPAL CLASSES OF ASYMPTOTIC TOPOLOGICAL INVARIANTS

Using a variety of methods developed in the literature (in particular, the theory of weak Hopf algebras), we prove a number of general results about fusion categories in characteristic zero. We show that the global dimension of a fusion category is always positive, and that the S-matrix of any modular category (not necessarily hermitian) is unitary. We also show that the category of module functors between two module categories over a fusion category is semisimple, and that fusion categories and tensor functors between them are undeformable (generalized Ocneanu rigidity). In particular the number of such categories (functors) realizing a given fusion datum is finite. Finally, we develop the theory of Frobenius-Perron dimensions in an arbitrary fusion category and classify categories of prime dimension.

I used to keep the entire spin networks literature in a small folder on my shelf. The recent explosion of interest in the subject has made this impossible; more is probably now written every week than the contents of my folder in 1991. This brings the need to consolidate the subject and collect together results and formulae in one coherent place. Kauffman and Lins' book is a splendid contribution, as the well thumbed reference copy can testify. The renewed interest in the subject came from two areas. Firstly there was the q-revolution which spawned the representation theory of the q-deformation of SU(2) and its application to 3-manifold theory by combinatorial techniques. Secondly, spin networks came to be applied in quantum field theories in places where discrete methods began to seem more fundamental. There are many overlaps between these, and many of the tools developed for specific 3-manifold purposes have, in fact, a much wider application. Perhaps recognizing this, the authors made the first half of the book a review of the recoupling theory of SU(2), leaving the application to 3-manifolds strictly to the second part. The development of the recoupling theory makes no reference to SU(2) itself, as the invariant theory can be developed entirely by the means of diagrams. This is a continuation of the theme of the spin networks and strand networks developed by Penrose, extended to the q-deformed case. The use of diagrams was originally conceived as an aid to understanding algebraic formulae; now the connection runs deeper. We now understand that a planar diagram for the recoupling of representations of SU(2) is an indispensible part of the theory. Without it, the fine detail of the formulae become impossible to keep track of in a coherent way. Thus basing the theory on diagrams is probably the most useful and certainly the quickest way of grasping the subject. The book starts with the Jones - Wenzl projectors, which are the diagrammatic version of the projection onto irreducible representations of SU(2). Then it quickly moves on to calculating the value of q-spin networks with these irreducible representations on the edges. Particular networks are calculated in detail, namely the diagram and the tetrahedron. This then allows the properties of the 6j symbol to be developed. Having all the formulae in one place, and they have proved remarkably reliable, makes this book invaluable, as the literature is plagued by slightly different versions of the same formulae caused by differences in convention. One unfortunate departure from the conventions established in the physics literature is that the curly bracket symbol for the 6j symbol has been assigned a different normalization, which makes the formulae related to associativity simple but not symmetrical under permutations of the tetrahedron. Most of the recoupling theory is developed for general q, and the particular case of q as a root of unity. However, the reader interested only in classical spin networks need not be lost; one can put q = 1 everywhere and still enjoy reading the book. The calculations of q-spin network evaluations are long but tantalizingly similar to the formulae for the classical q = 1 case. General formulae are known only for the classical case. Chapter 8 describes the chromatic method for q = 1 due to Penrose, leaving the generalization to the q-deformation as an unsolved problem. The combinatorics of the chromatic method are somewhat delicate to describe, and the informal method of presentation, which serves the book well elsewhere, is stretched to the limit at this point. The second part of the book is much more specialized. There are reviews of the Turaev - Viro invariant, the Witten - Reshetikhin - Turaev invariant and the `shadow world' description of graph invariants due to Kirillov and Resheatikhin. All of these have been condensed and simplified from their original presentations, but some of the theory, for example, of the topology, has been omitted. The culmination of the book is a series of tables of the WRT invariants for some 3-manifolds, based on the formulae given in the first part of the book.

http://iopscience.iop.org/0264-9381/13/12/024/pdf/0264-9381_13_12_024.pdf

Temperley - Lieb Recoupling Theory and Invariants of 3-Manifolds

Basic properties of finite group actions with the Rohlin property on unital C*-algebras are investigated. A characterization of finite group actions with the Rohlin property on the Cuntz algebra 02 is given in terms of central sequences, which may be considered as an equivariant version of E. Kirchberg and N. C. Phillips's characterization of 02 . A large class of symmetries on 02 are classified in terms of the fixed-point algebras for conjugacy and the crossed products for cocycle conjugacy. Model actions of symmetries of 02 are constructed for given K-theoretical invariants.

Finite group actions on C*-algebras with the Rohlin property, I

As an application of the general theory established in the first part, we determine the structure of Longo–Rehren inclusions for several systems of sectors arising from endomorphisms of the Cuntz algebras. The E6 subfactor and the Haagerup subfactor are included among these examples, and the dual principal graphs and the S and T-matrices for their Longo–Rehren inclusions are obtained. We also construct several new subfactors using endomorphisms of the Cuntz algebras, and determine their tube algebra structure.

The structure of sectors associated with longo–rehren inclusions ii: examples

We investigate the structure of the Longo–Rehren inclusion for a finite closed system of endomorphisms of factors, whose categorical structure is known to be the same as the asymptotic inclusion of A. Ocneanu. In particular, we obtain a precise description of the sectors associated with the Longo–Rehren inclusions in terms of half braidings, which do not necessarily satisfy the usual condition of braidings. In doing so, we give new proofs to most of the known statements concerning asymptotic inclusions. We construct a complete system of matrix units of the tube algebra using the half braidings, which will be used in the second part to describe concrete examples of the Longo–Rehren inclusions arising from the Cuntz algebra endomorphisms. We also discuss the case where the original system has a braiding, and generalize Ocneanu and Evans–Kawahigashi's method for the analysis of the asymptotic inclusions of the Hecke algebra subfactors.

Masaki Izumi

Papers

Finite group actions on C*-algebras with the Rohlin property, I

The structure of sectors associated with longo–rehren inclusions ii: examples

The Structure of Sectors¶Associated with Longo–Rehren Inclusions¶I. General Theory

Non-commutative Poisson Boundaries and Compact Quantum Group Actions

Finite group actions on C∗-algebras with the Rohlin property—II