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Author : Darryl McCullough,Andy Miller
Publisher : Unknown
Page : 97 pages
File Size : 50,6 Mb
Release : 1996
Category : Automorphisms
ISBN : 1470401673
Author : Darryl McCullough,Andy Miller
Publisher : American Mathematical Soc.
Page : 97 pages
File Size : 46,5 Mb
Release : 1996
Category : Mathematics
ISBN : 9780821804599
This memoir examines the automorphism group of a group $G$ with a fixed free product decomposition $G_1*\cdots *G_n$. An automorphism is called symmetric if it carries each factor $G_i$ to a conjugate of a (possibly different) factor $G_j$. The symmetric automorphisms form a group $\Sigma Aut(G)$ which contains the inner automorphism group $Inn(G)$. The quotient $\Sigma Aut(G)/Inn(G)$ is the symmetric outer automorphism group $\Sigma Out(G)$, a subgroup of $Out(G)$. It coincides with $Out(G)$ if the $G_i$ are indecomposable and none of them is infinite cyclic. To study $\Sigma Out(G)$, the authors construct an $(n-2)$-dimensional simplicial complex $K(G)$ which admits a simplicial action of $Out(G)$. The stabilizer of one of its components is $\Sigma Out(G)$, and the quotient is a finite complex. The authors prove that each component of $K(G)$ is contractible and describe the vertex stabilizers as elementary constructs involving the groups $G_i$ and $Aut(G_i)$. From this information, two new structural descriptions of $\Sigma Aut (G)$ are obtained. One identifies a normal subgroup in $\Sigma Aut(G)$ of cohomological dimension $(n-1)$ and describes its quotient group, and the other presents $\Sigma Aut (G)$ as an amalgam of some vertex stabilizers. Other applications concern torsion and homological finiteness properties of $\Sigma Out (G)$ and give information about finite groups of symmetric automorphisms. The complex $K(G)$ is shown to be equivariantly homotopy equivalent to a space of $G$-actions on $\mathbb R$-trees, although a simplicial topology rather than the Gromov topology must be used on the space of actions.
Author : Roland Speicher
Publisher : American Mathematical Soc.
Page : 105 pages
File Size : 46,5 Mb
Release : 1998
Category : Combinatorial analysis
ISBN : 9780821806937
Free probability theory, introduced by Voiculescu, has developed very actively in the last few years and has had an increasing impact on quite different fields in mathematics and physics. Whereas the subject arose out of the field of von Neumann algebras, presented here is a quite different view of Voiculescu's amalgamated free product. This combinatorial description not only allows re-proving of most of Voiculescu's results in a concise and elegant way, but also opens the way for many new results. Unlike other approaches, this book emphasizes the combinatorial structure of the concept of ``freeness''. This gives an elegant and easily accessible description of freeness and leads to new results in unexpected directions. Specifically, a mathematical framework for otherwise quite ad hoc approximations in physics emerges.
Author : Samson Adepoju Adeleke,P. M. Neumann
Publisher : American Mathematical Soc.
Page : 125 pages
File Size : 48,6 Mb
Release : 1998
Category : Mathematics
ISBN : 9780821806234
This volume is about tree-like structures, namely semilinear ordering, general betweenness relations, $C$-relations and $D$-relations. It contains a systematic study of betweenness and introduces $C$ and $D$-relations to describe the behavior of points at infinity ('leaves' or 'ends' or 'directions') of trees. The focus is on structure theorems and on automorphism groups, with applications to the theory of infinite permutation groups. This work: offers the first systematic treatment of betweenness relations; introduces important new concepts of $C$-relations and $D$-relations; elucidates the close relationships between semilinear orderings, betweenness relations, $C$ and $D$-relations; and, considers their automorphism groups as important examples of Jordan permutation groups.
Author : Igor Fulman
Publisher : American Mathematical Soc.
Page : 107 pages
File Size : 50,5 Mb
Release : 1997
Category : Mathematics
ISBN : 9780821805572
In this book, the author introduces and studies the construction of the crossed product of a von Neumann algebra $M = \int _X M(x)d\mu (x)$ by an equivalence relation on $X$ with countable cosets. This construction is the generalization of the construction of the crossed product of an abelian von Neumann algebra by an equivalence relation introduced by J. Feldman and C. C. Moore. Many properties of this construction are proved in the general case. In addition, the generalizations of the Spectral Theorem on Bimodules and of the theorem on dilations are proved.
Author : Siegfried Echterhoff
Publisher : American Mathematical Soc.
Page : 134 pages
File Size : 46,7 Mb
Release : 1996
Category : Mathematics
ISBN : 9780821805633
The importance of separable continuous trace $C^*$-algebras arises from the following facts: Firstly, their stable isomorphism classes are completely classifiable by topological data and, secondly, continuous-trace $C^*$-algebras form the building blocks of the more general type I $C^*$-algebras. This memoir presents an extensive study of strongly continuous actions of abelian locally compact groups on $C^*$-algebras with continuous trace. Under some natural assumptions on the underlying system $(A,G,\alpha )$, necessary and sufficient conditions are given for the crossed product $A{\times }_{\alpha }G$ to have continuous trace, and some relations between the topological data of $A$ and $A{\times }_{\alpha }G$ are obtained. The results are applied to investigate the structure of group $C^*$-algebras of some two-step nilpotent groups and solvable Lie groups. For readers' convenience, expositions of the Mackey-Green-Rieffel machine of induced representations and the theory of Morita equivalent $C^*$-dynamical systems are included. There is also an extensive elaboration of the representation theory of crossed products by actions of abelian groups on type I $C^*$-algebras, resulting in a new description of actions leading to type I crossed products. Features: The most recent results on the theory of crossed products with continuous trace. Applications to the representation theory of locally compact groups and structure of group $C^*$-algebras. An exposition on the modern theory of induced representations. New results on type I crossed products.
Author : James Damon
Publisher : American Mathematical Soc.
Page : 130 pages
File Size : 53,6 Mb
Release : 1996
Category : Holomorphic mappings
ISBN : 9780821804810
Almost free divisors and complete intersections form a general class of nonisolated hypersurface and completer intersection singularities. They also include discriminants of mappings, bifurcation sets, and certain types of arrangements of hyperplanes such as Coxeter arrangements and generic arrangements. Associated to the singularities of this class is a "singular Milnor fibration" which has the same homotopy properties as the Milnor fibration for isolated singularities. This memoir deduces topological properties of singularities in a number of situations including: complements of hyperplane arrangements, various nonisolated complete intersections, nonlinear arrangements of hypersurfaces, functions on discriminants, singularities defined by compositions of functions, and bifurcation sets.
Author : Richard Warren
Publisher : American Mathematical Soc.
Page : 166 pages
File Size : 49,9 Mb
Release : 1997
Category : Mathematics
ISBN : 9780821806227
The class of cycle-free partial orders (CFPOs) is defined, and the CFPOs fulfilling a natural transitivity assumption, called $k$-connected set transitivity ($k$-$CS$-transitivity), are analyzed in some detail. Classification in many of the interesting cases is given. This work generalizes Droste's classification of the countable $k$-transitive trees ($k \geq 2$). In a CFPO, the structure can branch downwards as well as upwards, and can do so repeatedly (though it never returns to the starting point by a cycle). Mostly it is assumed that $k \geq 3$ and that all maximal chains are finite. The main classification splits into the sporadic and skeletal cases. The former is complete in all cardinalities. The latter is performed only in the countable case. The classification is considerably more complicated than for trees, and skeletal CFPOs exhibit rich, elaborate and rather surprising behavior. Features: Lucid exposition of an important generalization of Droste's work Extended introduction clearly explaining the scope of the memoir Visually attractive topic with copious illustrations Self-contained material, requiring few prerequisites
Author : Bruce Normansell Allison,Saeid Azam,Stephen Berman,Arturo Pianzola,Yun Gao
Publisher : American Mathematical Soc.
Page : 122 pages
File Size : 48,5 Mb
Release : 1997
Category : Mathematics
ISBN : 9780821805947
This work is about extended affine Lie algebras (EALA's) and their root systems. EALA's were introduced by Hoegh-Krohn and Torresani under the name irreducible quasi-simple Lie algebras. The major objective is to develop enough theory to provide a firm foundation for further study of EALA's. The first chapter of the paper is devoted to establishing some basic structure theory. It includes a proof of the fact that, as conjectured by Kac, the invariant symmetric bilinear form on an EALA can be scaled so that its restriction to the real span of the root system is positive semi-definite. The second chapter studies extended affine root systems (EARS) which are an axiomatized version of the root systems arising from EALA's. The concept of a semilattice is used to give a complete description of EARS. In the final chapter, a number of new examples of extended affine Lie algebras are given. The concluding appendix contains an axiomatic characterization of the nonisotropic roots in an EARS in a more general context than the one used in the rest of the paper. Features: Provides a foundation for the study of an important class of Lie algebras that generalizes the class of affine Kac-Moody Lie algebras Includes material on Lie algebras and on root systems that can be read independently.
Author : H. Andréka,Steven R. Givant,I. Németi
Publisher : American Mathematical Soc.
Page : 126 pages
File Size : 47,5 Mb
Release : 1997
Category : Mathematics
ISBN : 9780821805954
This work presents a systematic study of decision problems for equational theories of algebras of binary relations (relation algebras). For example, an easily applicable but deep method, based on von Neumann's coordinatization theorem, is developed for establishing undecidability results. The method is used to solve several outstanding problems posed by Tarski. In addition, the complexity of intervals of equational theories of relation algebras with respect to questions of decidability is investigated. Using ideas that go back to Jonsson and Lyndon, the authors show that such intervals can have the same complexity as the lattice of subsets of the set of the natural numbers. Finally, some new and quite interesting examples of decidable equational theories are given. The methods developed in the monograph show promise of broad applicability. They provide researchers in algebra and logic with a new arsenal of techniques for resolving decision questions in various domains of algebraic logic.
Author : Mikhail Borovoi
Publisher : American Mathematical Soc.
Page : 50 pages
File Size : 50,9 Mb
Release : 1998
Category : Mathematics
ISBN : 9780821806500
In this volume, a new functor $H^2_{ab}(K,G)$ of abelian Galois cohomology is introduced from the category of connected reductive groups $G$ over a field $K$ of characteristic $0$ to the category of abelian groups. The abelian Galois cohomology and the abelianization map$ab^1:H^1(K,G) \rightarrow H^2_{ab}(K,G)$ are used to give a functorial, almost explicit description of the usual Galois cohomology set $H^1(K,G)$ when $K$ is a number field.
Author : Solomon Friedberg,Hervé Jacquet
Publisher : American Mathematical Soc.
Page : 149 pages
File Size : 47,5 Mb
Release : 1996
Category : Mathematics
ISBN : 9780821805404
The authors establish the fundamental lemma for a relative trace formula. The trace formula compares generic automorphic representations of [italic capitals]GS[italic]p(4) with automorphic representations of [italic capitals]GS(4) which are distinguished with respect to a character of the Shalika subgroup, the subgroup of matrices of 2 x 2 block form ([superscript italic]g [over] [subscript capital italic]X [and] 0 [over] [superscript italic]g). The fundamental lemma, giving the equality of the orbital integrals of the unit elements of the respective Hecke algebras, amounts to a comparison of certain exponential sums arising from these two different groups.
Author : Tai-Ping Liu,Yanni Zeng
Publisher : American Mathematical Soc.
Page : 135 pages
File Size : 51,7 Mb
Release : 1997
Category : Conservation laws
ISBN : 9780821805459
We are interested in the time-asymptotic behavior of solutions to viscous conservation laws. Through the pointwise estimates for the Green's function of the linearized system and the analysis of coupling of nonlinear diffusion waves, we obtain explicit expressions of the time-asymptotic behavior of the solutions. This yields optimal estimates in the integral norms. For most physical models, the viscosity matrix is not positive definite and the system is hyperbolic-parabolic, and not uniformly parabolic. This implies that the Green's function may contain Dirac [lowercase Greek]Delta-functions. When the corresponding inviscid system is non-strictly hyperbolic, the time-asymptotic state contains generalized Burgers solutions. These are illustrated by applying our general theory to the compressible Navier-Stokes equations and the equations of magnetohydrodynamics.
Author : Edward Cline,Brian Parshall,Leonard Scott,Leonard L. Scott
Publisher : American Mathematical Soc.
Page : 119 pages
File Size : 48,6 Mb
Release : 1996
Category : Mathematics
ISBN : 9780821804889
Suppose that $R$ is a finite dimensional algebra and $T$ is a right $R$-module. Let $A = \mathrm{ End}_R(T)$ be the endomorphism algebra of $T$. This memoir presents a systematic study of the relationships between the representation theories of $R$ and $A$, especially those involving actual or potential structures on $A$ which ''stratify'' its homological algebra. The original motivation comes from the theory of Schur algebras and the symmetric group, Lie theory, and the representation theory of finite dimensional algebras and finite groups. The book synthesizes common features of many of the above areas, and presents a number of new directions. Included are an abstract ''Specht/Weyl module'' correspondence, a new theory of stratified algebras, and a deformation theory for them. The approach reconceptualizes most of the modular representation theory of symmetric groups involving Specht modules and places that theory in a broader context. Finally, the authors formulate some conjectures involving the theory of stratified algebras and finite Coexeter groups, aiming toward understanding the modular representation theory of finite groups of Lie type in all characteristics.
Author : Kazuyoshi Kiyohara
Publisher : American Mathematical Soc.
Page : 143 pages
File Size : 48,8 Mb
Release : 1997
Category : Mathematics
ISBN : 9780821806401
In this work, two classes of manifolds whose geodesic flows are integrable are defined, and their global structures are investigated. They are called Liouville manifolds and Kahler-Liouville manifolds respectively. In each case, the author finds several invariants with which they are partly classified. The classification indicates, in particular, that these classes contain many new examples of manifolds with integrable geodesic flow.
