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Kac Algebras Arising From Composition Of Subfactors General Theory And Classification
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Book Synopsis Kac Algebras Arising from Composition of Subfactors: General Theory and Classification by : Masaki Izumi
Download or read book Kac Algebras Arising from Composition of Subfactors: General Theory and Classification written by Masaki Izumi and published by American Mathematical Soc.. This book was released on 2002 with total page 198 pages. Available in PDF, EPUB and Kindle. Book excerpt: We deal with a map $\alpha$ from a finite group $G$ into the automorphism group $Aut({\mathcal L})$ of a factor ${\mathcal L}$ satisfying: $G=N \rtimes H$ is a semi-direct product, the induced map $g \in G \to [\alpha_g] \in Out({\mathcal L})=Aut({\mathcal L})/Int({\mathcal L})$ is an injective homomorphism, and the restrictions $\alpha\!\!\mid_N,\alpha\!\!\mid_H$ are genuine actions of the subgroups on the factor ${\mathcal L}$. The pair ${\mathcal M}={\mathcal L} \rtimes_{\alpha} H \supseteq {\mathcal N}={\mathcal L}^{\alpha\mid_N}$ (of the crossed product ${\mathcal L} \rtimes_{\alpha} H$ and the fixed-point algebra ${\mathcal L}^{\alpha\mid_N}$) gives us an irreducible inclusion of factors with Jones index $\ No. G$. The inclusion ${\mathcal M} \supseteq {\mathcal N}$ is of depth $2$ and hence known to correspond to a Kac algebra of dimension $\ No. G$.A Kac algebra arising in this way is investigated in detail, and in fact the relevant multiplicative unitary (satisfying the pentagon equation) is described. We introduce and analyze a certain cohomology group (denoted by $H^2((N,H),{\mathbf T})$) providing complete information on the Kac algebra structure, and we construct an abundance of non-trivial examples by making use of various cocycles. The operator algebraic meaning of this cohomology group is clarified, and some related topics are also discussed. Sector technique enables us to establish structure results for Kac algebras with certain prescribed underlying algebra structure.They guarantee that 'most' Kac algebras of low dimension (say less than $60$) actually arise from inclusions of the form ${\mathcal L} \rtimes_{\alpha} H \supseteq {\mathcal L}^{\alpha\mid_N}$, and consequently their classification can be carried out by determining $H^2((N,H),{\mathbf T})$. Among other things we indeed classify Kac algebras of dimension $16$ and $24$, which (together with previously known results) gives rise to the complete classification of Kac algebras of dimension up to $31$. Partly to simplify classification procedure and hopefully for its own sake, we also study 'group extensions' of general (finite-dimensional) Kac algebras with some discussions on related topics.
Book Synopsis Kac Algebras Arising from Composition of Subfactors by : Masaki Izumi
Download or read book Kac Algebras Arising from Composition of Subfactors written by Masaki Izumi and published by . This book was released on 2014-09-11 with total page 198 pages. Available in PDF, EPUB and Kindle. Book excerpt: This title deals with a map $\alpha$ from a finite group $G$ into the automorphism group $Aut({\mathcal L})$ of a factor ${\mathcal L}$ satisfying (i) $G=N \rtimes H$ is a semi-direct product, (ii) the induced map $g \in G \to [\alpha_g] \in Out({\mathcal L})=Aut({\mathcal L})/Int({\mathcal L})$ is an injective homomorphism, and (iii) the restrictions $\alpha \! \! \mid_N, \alpha \! \! \mid_H$ are genuine actions of the subgroups on the factor ${\mathcal L}$. The pair ${\mathcal M}={\mathcal L} \rtimes_{\alpha} H \supseteq {\mathcal N}={\mathcal L} DEGREES{\alpha\mid_N}$ (of the crossed product ${\mathcal L} \rtimes_{\alpha} H$ and the fixed-point algebra ${\mathcal L} DEGREES{\alpha\mid_N}$) gives an irreducible inclusion of factors with Jones index $\# G$. The inclusion ${\mathcal M} \supseteq {\mathcal N}$ is of depth $2$ and hence known to correspond to a Kac algebra of dim
Book Synopsis Classification of Actions of Discrete Kac Algebras on Injective Factors by : Toshihiko Masuda
Download or read book Classification of Actions of Discrete Kac Algebras on Injective Factors written by Toshihiko Masuda and published by American Mathematical Soc.. This book was released on 2017-01-18 with total page 118 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors study two kinds of actions of a discrete amenable Kac algebra. The first one is an action whose modular part is normal. They construct a new invariant which generalizes a characteristic invariant for a discrete group action, and we will present a complete classification. The second is a centrally free action. By constructing a Rohlin tower in an asymptotic centralizer, the authors show that the Connes–Takesaki module is a complete invariant.
Book Synopsis Quantum Symmetries in Theoretical Physics and Mathematics by : M A Lifshits
Download or read book Quantum Symmetries in Theoretical Physics and Mathematics written by M A Lifshits and published by American Mathematical Soc.. This book was released on 2002 with total page 266 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume presents articles from several lectures presented at the school on 'Quantum Symmetries in Theoretical Physics and Mathematics' held in Bariloche, Argentina. The various lecturers provided significantly different points of view on several aspects of Hopf algebras, quantum group theory, and noncommutative differential geometry, ranging from analysis, geometry, and algebra to physical models, especially in connection with integrable systems and conformal field theories. Primary topics discussed in the text include subgroups of quantum $SU(N)$, quantum ADE classifications and generalized Coxeter systems, modular invariance, defects and boundaries in conformal field theory, finite dimensional Hopf algebras, Lie bialgebras and Belavin-Drinfeld triples, real forms of quantum spaces, perturbative and non-perturbative Yang-Baxter operators, braided subfactors in operator algebras and conformal field theory, and generalized ($d^N$) cohomologies.
Book Synopsis New Directions in Hopf Algebras by : Susan Montgomery
Download or read book New Directions in Hopf Algebras written by Susan Montgomery and published by Cambridge University Press. This book was released on 2002-05-06 with total page 502 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains survey articles for graduate students and researchers on various topics surrounding Hopf algebras.
Book Synopsis On the Classification of Polish Metric Spaces Up to Isometry by : Su Gao
Download or read book On the Classification of Polish Metric Spaces Up to Isometry written by Su Gao and published by American Mathematical Soc.. This book was released on 2003 with total page 78 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis The Role of the Spectrum in the Cyclic Behavior of Composition Operators by : Eva A. Gallardo-Gutieŕrez
Download or read book The Role of the Spectrum in the Cyclic Behavior of Composition Operators written by Eva A. Gallardo-Gutieŕrez and published by American Mathematical Soc.. This book was released on 2004 with total page 98 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduction and preliminaries Linear fractional maps with an interior fixed point Non elliptic automorphisms The parabolic non automorphism Supercyclic linear fractional composition operators Endnotes Bibliography.
Book Synopsis Groups, Rings, Lie and Hopf Algebras by : Y. Bahturin
Download or read book Groups, Rings, Lie and Hopf Algebras written by Y. Bahturin and published by Springer Science & Business Media. This book was released on 2013-12-01 with total page 240 pages. Available in PDF, EPUB and Kindle. Book excerpt: The volume is almost entirely composed of the research and expository papers by the participants of the International Workshop "Groups, Rings, Lie and Hopf Algebras", which was held at the Memorial University of Newfoundland, St. John's, NF, Canada. All four areas from the title of the workshop are covered. In addition, some chapters touch upon the topics, which belong to two or more areas at the same time. Audience: The readership targeted includes researchers, graduate and senior undergraduate students in mathematics and its applications.
Book Synopsis Lie Algebras Graded by the Root Systems BC$_r$, $r\geq 2$ by : Bruce Normansell Allison
Download or read book Lie Algebras Graded by the Root Systems BC$_r$, $r\geq 2$ written by Bruce Normansell Allison and published by American Mathematical Soc.. This book was released on 2002 with total page 158 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduction The $\mathfrak{g}$-module decomposition of a $\mathrm{BC}_r$-graded Lie algebra, $r\ge 3$ (excluding type $\mathrm{D}_3)$ Models for $\mathrm{BC}_r$-graded Lie algebras, $r\ge 3$ (excluding type $\mathrm{D}_3)$ The $\mathfrak{g}$-module decomposition of a $\mathrm{BC}_r$-graded Lie algebra with grading subalgebra of type $\mathrm{B}_2$, $\mathrm{C}_2$, $\mathrm{D}_2$, or $\mathrm{D}_3$ Central extensions, derivations and invariant forms Models of $\mathrm{BC}_r$-graded Lie algebras with grading subalgebra of type $\mathrm{B}_2$, $\mathrm{C}_2$, $\mathrm{D}_2$, or $\mathrm{D}_3$ Appendix: Peirce decompositions in structurable algebras References.
Book Synopsis Dualities on Generalized Koszul Algebras by : Edward L. Green
Download or read book Dualities on Generalized Koszul Algebras written by Edward L. Green and published by American Mathematical Soc.. This book was released on 2002 with total page 90 pages. Available in PDF, EPUB and Kindle. Book excerpt: Koszul rings are graded rings which have played an important role in algebraic topology, noncommutative algebraic geometry and in the theory of quantum groups. One aspect of the theory is to compare the module theory for a Koszul ring and its Koszul dual. There are dualities between subcategories of graded modules; the Koszul modules.
Book Synopsis From Representation Theory to Homotopy Groups by : Donald M. Davis
Download or read book From Representation Theory to Homotopy Groups written by Donald M. Davis and published by American Mathematical Soc.. This book was released on 2002 with total page 50 pages. Available in PDF, EPUB and Kindle. Book excerpt: A formula for the odd-primary v1-periodic homotopy groups of a finite H-space in terms of its K-theory and Adams operations has been obtained by Bousfield. This work applies this theorem to give explicit determinations of the v1-periodic homotopy groups of (E8,5) and (E8,3), thus completing the determination of all odd-primary v1-periodic homotopy groups of all compact simple Lie groups, a project suggested by Mimura in 1989. The method is different to that used by the author in previous works. There is no homotopy theoretic input, and no spectral sequence calculation. The input is the second exterior power operation in the representation ring of E8, which we determine using specialized software. This can be interpreted as giving the Adams operation psi^2 in K(E8). Eigenvectors of psi^2 must also be eigenvectors of psi^k for any k. The matrix of these eigenvectors is the key to the analysis. Its determinant is closely related to the homotopy decomposition of E8 localized at each prime. By taking careful combinations of eigenvectors, a set of generators of K(E8) can be obtained on which there is a nice formula for all Adams operations. Bousfield's theorem (and considerable Maple computation) allows the v1-periodic homotopy groups to be obtained from this.
Book Synopsis Descriptive Set Theory and Definable Forcing by : Jindřich Zapletal
Download or read book Descriptive Set Theory and Definable Forcing written by Jindřich Zapletal and published by American Mathematical Soc.. This book was released on 2004 with total page 158 pages. Available in PDF, EPUB and Kindle. Book excerpt: Focuses on the relationship between definable forcing and descriptive set theory; the forcing serves as a tool for proving independence of inequalities between cardinal invariants of the continuum.
Book Synopsis The Connective K-Theory of Finite Groups by : Robert Ray Bruner
Download or read book The Connective K-Theory of Finite Groups written by Robert Ray Bruner and published by American Mathematical Soc.. This book was released on 2003 with total page 144 pages. Available in PDF, EPUB and Kindle. Book excerpt: Includes a paper that deals the connective K homology and cohomology of finite groups $G$. This title uses the methods of algebraic geometry to study the ring $ku DEGREES*(BG)$ where $ku$ denotes connective complex K-theory. It describes the variety in terms of the category of abelian $p$-subgroups of $G$ for primes $p$ dividing the group
Book Synopsis Equivariant Orthogonal Spectra and S-Modules by : M. A. Mandell
Download or read book Equivariant Orthogonal Spectra and S-Modules written by M. A. Mandell and published by American Mathematical Soc.. This book was released on 2002 with total page 108 pages. Available in PDF, EPUB and Kindle. Book excerpt: The last few years have seen a revolution in our understanding of the foundations of stable homotopy theory. Many symmetric monoidal model categories of spectra whose homotopy categories are equivalent to the stable homotopy category are now known, whereas no such categories were known before 1993. The most well-known examples are the category of $S$-modules and the category of symmetric spectra. We focus on the category of orthogonal spectra, which enjoys some of the best features of $S$-modules and symmetric spectra and which is particularly well-suited to equivariant generalization. We first complete the nonequivariant theory by comparing orthogonal spectra to $S$-modules. We then develop the equivariant theory.For a compact Lie group $G$, we construct a symmetric monoidal model category of orthogonal $G$-spectra whose homotopy category is equivalent to the classical stable homotopy category of $G$-spectra. We also complete the theory of $S_G$-modules and compare the categories of orthogonal $G$-spectra and $S_G$-modules. A key feature is the analysis of change of universe, change of group, fixed point, and orbit functors in these two highly structured categories for the study of equivariant stable homotopy theory.
Book Synopsis Desingularization of Nilpotent Singularities in Families of Planar Vector Fields by : Daniel Panazzolo
Download or read book Desingularization of Nilpotent Singularities in Families of Planar Vector Fields written by Daniel Panazzolo and published by American Mathematical Soc.. This book was released on 2002 with total page 108 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this work, we prove a desingularization theorem for analytic families of two-dimensional vector fields, under the hypothesis that all its singularities have a non-vanishing first jet. Application to problems of Singular Perturbations and Finite Cyclicity are discussed in the last chapter.
Book Synopsis Homogeneous Spaces, Tits Buildings, and Isoparametric Hypersurfaces by : Linus Kramer
Download or read book Homogeneous Spaces, Tits Buildings, and Isoparametric Hypersurfaces written by Linus Kramer and published by American Mathematical Soc.. This book was released on 2002 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt: This title classifys 1-connected compact homogeneous spaces which have the same rational cohomology as a product of spheres $\mahtbb{S} DEGREES{n_1}\times\mathbb{S} DEGREES{n_2}$, with $3\leq n_1\leq n_2$ and $n_2$ odd. As an application, it classifys compact generalized quadrangles (buildings of type $C_2)$ which admit a point transitive automorphism group, and isoparametric hypersurfaces which admit a transitive isometry group on one f
Book Synopsis Dynamics of Topologically Generic Homeomorphisms by : Ethan Akin
Download or read book Dynamics of Topologically Generic Homeomorphisms written by Ethan Akin and published by American Mathematical Soc.. This book was released on 2003 with total page 146 pages. Available in PDF, EPUB and Kindle. Book excerpt: The goal of this work is to describe the dynamics of generic homeomorphisms of certain compact metric spaces $X$. Here ``generic'' is used in the topological sense -- a property of homeomorphisms on $X$ is generic if the set of homeomorphisms with the property contains a residual subset (in the sense of Baire category) of the space of all homeomorphisms on $X$. The spaces $X$ we consider are those with enough local homogeneity to allow certain localized perturbations of homeomorphisms; for example, any compact manifold is such a space. We show that the dynamics of a generic homeomorphism is quite complicated, with a number of distinct dynamical behaviors coexisting (some resemble subshifts of finite type, others, which we call `generalized adding machines', appear strictly periodic when viewed to any finite precision, but are not actually periodic). Such a homeomorphism has infinitely many, intricately nested attractors and repellors, and uncountably many distinct dynamically-connected components of the chain recurrent set. We single out several types of these ``chain components'', and show that each type occurs densely (in an appropriate sense) in the chain recurrent set. We also identify one type that occurs generically in the chain recurrent set. We also show that, at least for $X$ a manifold, the chain recurrent set of a generic homeomorphism is a Cantor set, so its complement is open and dense. Somewhat surprisingly, there is a residual subset of $X$ consisting of points whose limit sets are chain components of a type other than the type of chain components that are residual in the space of all chain components. In fact, for each generic homeomorphism on $X$ there is a residual subset of points of $X$ satisfying a stability condition stronger than Lyapunov stability.
