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Author : Pierre-Emmanuel Caprace
Publisher : American Mathematical Soc.
Page : 109 pages
File Size : 48,9 Mb
Release : 2009-01-01
Category : Mathematics
ISBN : 9780821866658
Author : Pierre-Emmanuel Caprace
Publisher : American Mathematical Soc.
Page : 84 pages
File Size : 48,8 Mb
Release : 2009-03-06
Category : Mathematics
ISBN : 9780821842584
This work is devoted to the isomorphism problem for split Kac-Moody groups over arbitrary fields. This problem turns out to be a special case of a more general problem, which consists in determining homomorphisms of isotropic semisimple algebraic groups to Kac-Moody groups, whose image is bounded. Since Kac-Moody groups possess natural actions on twin buildings, and since their bounded subgroups can be characterized by fixed point properties for these actions, the latter is actually a rigidity problem for algebraic group actions on twin buildings. The author establishes some partial rigidity results, which we use to prove an isomorphism theorem for Kac-Moody groups over arbitrary fields of cardinality at least $4$. In particular, he obtains a detailed description of automorphisms of Kac-Moody groups. This provides a complete understanding of the structure of the automorphism group of Kac-Moody groups over ground fields of characteristic $0$. The same arguments allow to treat unitary forms of complex Kac-Moody groups. In particular, the author shows that the Hausdorff topology that these groups carry is an invariant of the abstract group structure. Finally, the author proves the non-existence of cocentral homomorphisms of Kac-Moody groups of indefinite type over infinite fields with finite-dimensional target. This provides a partial solution to the linearity problem for Kac-Moody groups.
Author : Olivier Frécon
Publisher : American Mathematical Soc.
Page : 99 pages
File Size : 49,8 Mb
Release : 2018-10-03
Category : Electronic
ISBN : 9781470429232
The author analyzes the abstract structure of algebraic groups over an algebraically closed field . For of characteristic zero and a given connected affine algebraic Q -group, the main theorem describes all the affine algebraic Q -groups such that the groups and are isomorphic as abstract groups. In the same time, it is shown that for any two connected algebraic Q -groups and , the elementary equivalence of the pure groups and implies that they are abstractly isomorphic. In the final section, the author applies his results to characterize the connected algebraic groups, all of whose abstract automorphisms are standard, when is either Q or of positive characteristic. In characteristic zero, a fairly general criterion is exhibited.
Author : Christian Bär,Joachim Lohkamp,Matthias Schwarz
Publisher : Springer Science & Business Media
Page : 520 pages
File Size : 54,5 Mb
Release : 2011-12-18
Category : Mathematics
ISBN : 9783642228421
This volume contains a collection of well-written surveys provided by experts in Global Differential Geometry to give an overview over recent developments in Riemannian Geometry, Geometric Analysis and Symplectic Geometry. The papers are written for graduate students and researchers with a general interest in geometry, who want to get acquainted with the current trends in these central fields of modern mathematics.
Author : Huaxin Lin
Publisher : American Mathematical Soc.
Page : 131 pages
File Size : 44,5 Mb
Release : 2010
Category : Mathematics
ISBN : 9780821851944
In this paper the author proves Generalized Homotopy Lemmas. These type of results play an important role in the classification theory of $*$-homomorphisms up to asymptotic unitary equivalence. Table of Contents: Prelude; The basic homotopy lemma for higher dimensional spaces; Purely infinite simple $C^*$-algebras; Approximate homotopy; Super homotopy; Postlude; Bibliography. (MEMO/205/963)
Author : Skip Garibaldi
Publisher : American Mathematical Soc.
Page : 102 pages
File Size : 53,8 Mb
Release : 2009-06-05
Category : Mathematics
ISBN : 9780821844045
This volume concerns invariants of $G$-torsors with values in mod $p$ Galois cohomology--in the sense of Serre's lectures in the book Cohomological invariants in Galois cohomology--for various simple algebraic groups $G$ and primes $p$. The author determines the invariants for the exceptional groups $F_4$ mod 3, simply connected $E_6$ mod 3, $E_7$ mod 3, and $E_8$ mod 5. He also determines the invariants of $\mathrm{Spin}_n$ mod 2 for $n \leq 12$ and constructs some invariants of $\mathrm{Spin}_{14}$. Along the way, the author proves that certain maps in nonabelian cohomology are surjective. These surjectivities give as corollaries Pfister's results on 10- and 12-dimensional quadratic forms and Rost's theorem on 14-dimensional quadratic forms. This material on quadratic forms and invariants of $\mathrm{Spin}_n$ is based on unpublished work of Markus Rost. An appendix by Detlev Hoffmann proves a generalization of the Common Slot Theorem for 2-Pfister quadratic forms.
Author : Adam R. Thomas
Publisher : American Mathematical Soc.
Page : 191 pages
File Size : 55,9 Mb
Release : 2021-06-18
Category : Education
ISBN : 9781470443375
This paper is a contribution to the study of the subgroup structure of excep-tional algebraic groups over algebraically closed fields of arbitrary characteristic. Following Serre, a closed subgroup of a semisimple algebraic group G is called irreducible if it lies in no proper parabolic subgroup of G. In this paper we com-plete the classification of irreducible connected subgroups of exceptional algebraic groups, providing an explicit set of representatives for the conjugacy classes of such subgroups. Many consequences of this classification are also given. These include results concerning the representations of such subgroups on various G-modules: for example, the conjugacy classes of irreducible connected subgroups are determined by their composition factors on the adjoint module of G, with one exception. A result of Liebeck and Testerman shows that each irreducible connected sub-group X of G has only finitely many overgroups and hence the overgroups of X form a lattice. We provide tables that give representatives of each conjugacy class of connected overgroups within this lattice structure. We use this to prove results concerning the subgroup structure of G: for example, when the characteristic is 2, there exists a maximal connected subgroup of G containing a conjugate of every irreducible subgroup A1 of G.
Author : Drew Armstrong
Publisher : American Mathematical Soc.
Page : 176 pages
File Size : 42,8 Mb
Release : 2009-10-08
Category : Mathematics
ISBN : 9780821844908
This memoir is a refinement of the author's PhD thesis -- written at Cornell University (2006). It is primarily a desription of new research but also includes a substantial amount of background material. At the heart of the memoir the author introduces and studies a poset $NC^{(k)}(W)$ for each finite Coxeter group $W$ and each positive integer $k$. When $k=1$, his definition coincides with the generalized noncrossing partitions introduced by Brady and Watt in $K(\pi, 1)$'s for Artin groups of finite type and Bessis in The dual braid monoid. When $W$ is the symmetric group, the author obtains the poset of classical $k$-divisible noncrossing partitions, first studied by Edelman in Chain enumeration and non-crossing partitions.
Author : Martin W. Liebeck,Cheryl E. Praeger,Jan Saxl
Publisher : American Mathematical Soc.
Page : 87 pages
File Size : 55,9 Mb
Release : 2010
Category : Finite simple groups
ISBN : 9780821846544
Addresses the classical problem of determining finite primitive permutation groups G with a regular subgroup B.
Author : Irina D. Suprunenko
Publisher : American Mathematical Soc.
Page : 168 pages
File Size : 46,8 Mb
Release : 2009-06-05
Category : Mathematics
ISBN : 9780821843697
The minimal polynomials of the images of unipotent elements in irreducible rational representations of the classical algebraic groups over fields of odd characteristic are found. These polynomials have the form $(t-1)^d$ and hence are completely determined by their degrees. In positive characteristic the degree of such polynomial cannot exceed the order of a relevant element. It occurs that for each unipotent element the degree of its minimal polynomial in an irreducible representation is equal to the order of this element provided the highest weight of the representation is large enough with respect to the ground field characteristic. On the other hand, classes of unipotent elements for which in every nontrivial representation the degree of the minimal polynomial is equal to the order of the element are indicated. In the general case the problem of computing the minimal polynomial of the image of a given element of order $p^s$ in a fixed irreducible representation of a classical group over a field of characteristic $p>2$ can be reduced to a similar problem for certain $s$ unipotent elements and a certain irreducible representation of some semisimple group over the field of complex numbers. For the latter problem an explicit algorithm is given. Results of explicit computations for groups of small ranks are contained in Tables I-XII. The article may be regarded as a contribution to the programme of extending the fundamental results of Hall and Higman (1956) on the minimal polynomials from $p$-solvable linear groups to semisimple groups.
Author : Thomas Lam
Publisher : American Mathematical Soc.
Page : 82 pages
File Size : 51,7 Mb
Release : 2010
Category : Mathematics
ISBN : 9780821846582
The authors study combinatorial aspects of the Schubert calculus of the affine Grassmannian ${\rm Gr}$ associated with $SL(n,\mathbb{C})$.Their main results are: Pieri rules for the Schubert bases of $H^*({\rm Gr})$ and $H_*({\rm Gr})$, which expresses the product of a special Schubert class and an arbitrary Schubert class in terms of Schubert classes. A new combinatorial definition for $k$-Schur functions, which represent the Schubert basis of $H_*({\rm Gr})$. A combinatorial interpretation of the pairing $H^*({\rm Gr})\times H_*({\rm Gr}) \rightarrow\mathbb Z$ induced by the cap product.
Author : Dillon Mayhew,Gordon Royle,Geoff Whittle
Publisher : American Mathematical Soc.
Page : 110 pages
File Size : 50,6 Mb
Release : 2010
Category : Combinatorial designs and configurations
ISBN : 9780821848265
The authors give a characterization of the internally $4$-connected binary matroids that have no minor isomorphic to $M(K_{3,3})$. Any such matroid is either cographic, or is isomorphic to a particular single-element extension of the bond matroid of a cubic or quartic Mobius ladder, or is isomorphic to one of eighteen sporadic matroids.
Author : Cdric Villani
Publisher : American Mathematical Soc.
Page : 154 pages
File Size : 49,5 Mb
Release : 2009-10-08
Category : Mathematics
ISBN : 9780821844984
This memoir attempts at a systematic study of convergence to stationary state for certain classes of degenerate diffusive equations, taking the general form ${\frac{\partial f}{\partial t}}+ L f =0$. The question is whether and how one can overcome the degeneracy by exploiting commutators.
Author : Pierre Magal,Shigui Ruan
Publisher : American Mathematical Soc.
Page : 84 pages
File Size : 46,8 Mb
Release : 2009
Category : Bifurcation theory
ISBN : 9780821846537
Several types of differential equations, such as delay differential equations, age-structure models in population dynamics, evolution equations with boundary conditions, can be written as semilinear Cauchy problems with an operator which is not densely defined in its domain. The goal of this paper is to develop a center manifold theory for semilinear Cauchy problems with non-dense domain. Using Liapunov-Perron method and following the techniques of Vanderbauwhede et al. in treating infinite dimensional systems, the authors study the existence and smoothness of center manifolds for semilinear Cauchy problems with non-dense domain. As an application, they use the center manifold theorem to establish a Hopf bifurcation theorem for age structured models.
Author : Alfonso Castro,Victor Padron
Publisher : American Mathematical Soc.
Page : 87 pages
File Size : 45,7 Mb
Release : 2010
Category : Differential equations, Elliptic
ISBN : 9780821847268
The authors provide a complete classification of the radial solutions to a class of reaction diffusion equations arising in the study of thermal structures such as plasmas with thermal equilibrium or no flux at the boundary. In particular, their study includes rapidly growing nonlinearities, that is, those where an exponent exceeds the critical exponent. They describe the corresponding bifurcation diagrams and determine existence and uniqueness of ground states, which play a central role in characterizing those diagrams. They also provide information on the stability-unstability of the radial steady states.
