Representations of Finite Groups: Local Cohomology and Support
Title | Representations of Finite Groups: Local Cohomology and Support PDF eBook |
Author | David J. Benson |
Publisher | Springer Science & Business Media |
Pages | 115 |
Release | 2011-11-15 |
Genre | Mathematics |
ISBN | 3034802609 |
The seminar focuses on a recent solution, by the authors, of a long standing problem concerning the stable module category (of not necessarily finite dimensional representations) of a finite group. The proof draws on ideas from commutative algebra, cohomology of groups, and stable homotopy theory. The unifying theme is a notion of support which provides a geometric approach for studying various algebraic structures. The prototype for this has been Daniel Quillen’s description of the algebraic variety corresponding to the cohomology ring of a finite group, based on which Jon Carlson introduced support varieties for modular representations. This has made it possible to apply methods of algebraic geometry to obtain representation theoretic information. Their work has inspired the development of analogous theories in various contexts, notably modules over commutative complete intersection rings and over cocommutative Hopf algebras. One of the threads in this development has been the classification of thick or localizing subcategories of various triangulated categories of representations. This story started with Mike Hopkins’ classification of thick subcategories of the perfect complexes over a commutative Noetherian ring, followed by a classification of localizing subcategories of its full derived category, due to Amnon Neeman. The authors have been developing an approach to address such classification problems, based on a construction of local cohomology functors and support for triangulated categories with ring of operators. The book serves as an introduction to this circle of ideas.
Cohomology Rings of Finite Groups
Title | Cohomology Rings of Finite Groups PDF eBook |
Author | Jon F. Carlson |
Publisher | Springer Science & Business Media |
Pages | 782 |
Release | 2013-04-17 |
Genre | Mathematics |
ISBN | 9401702152 |
Group cohomology has a rich history that goes back a century or more. Its origins are rooted in investigations of group theory and num ber theory, and it grew into an integral component of algebraic topology. In the last thirty years, group cohomology has developed a powerful con nection with finite group representations. Unlike the early applications which were primarily concerned with cohomology in low degrees, the in teractions with representation theory involve cohomology rings and the geometry of spectra over these rings. It is this connection to represen tation theory that we take as our primary motivation for this book. The book consists of two separate pieces. Chronologically, the first part was the computer calculations of the mod-2 cohomology rings of the groups whose orders divide 64. The ideas and the programs for the calculations were developed over the last 10 years. Several new features were added over the course of that time. We had originally planned to include only a brief introduction to the calculations. However, we were persuaded to produce a more substantial text that would include in greater detail the concepts that are the subject of the calculations and are the source of some of the motivating conjectures for the com putations. We have gathered together many of the results and ideas that are the focus of the calculations from throughout the mathematical literature.
Equivariant Topology and Derived Algebra
Title | Equivariant Topology and Derived Algebra PDF eBook |
Author | Scott Balchin |
Publisher | Cambridge University Press |
Pages | 357 |
Release | 2021-11-18 |
Genre | Mathematics |
ISBN | 1108931944 |
A collection of research papers, both new and expository, based on the interests of Professor J. P. C. Greenlees.
Finite and Locally Finite Groups
Title | Finite and Locally Finite Groups PDF eBook |
Author | B. Hartley |
Publisher | Springer Science & Business Media |
Pages | 469 |
Release | 2012-12-06 |
Genre | Mathematics |
ISBN | 9401103291 |
This volume contains the proceedings of the NATO Advanced Study Institute on Finite and Locally Finite Groups held in Istanbul, Turkey, 14-27 August 1994, at which there were about 90 participants from some 16 different countries. The ASI received generous financial support from the Scientific Affairs Division of NATO. INTRODUCTION A locally finite group is a group in which every finite set of elements is contained in a finite subgroup. The study of locally finite groups began with Schur's result that a periodic linear group is, in fact, locally finite. The simple locally finite groups are of particular interest. In view of the classification of the finite simple groups and advances in representation theory, it is natural to pursue classification theorems for simple locally finite groups. This was one of the central themes of the Istanbul conference and significant progress is reported herein. The theory of simple locally finite groups intersects many areas of group theory and representation theory, so this served as a focus for several articles in the volume. Every simple locally finite group has what is known as a Kegel cover. This is a collection of pairs {(G , Ni) liE I}, where I is an index set, each group Gi is finite, i Ni
Representations of Algebras and Related Topics
Title | Representations of Algebras and Related Topics PDF eBook |
Author | Andrzej Skowroński |
Publisher | European Mathematical Society |
Pages | 744 |
Release | 2011 |
Genre | Algebra |
ISBN | 9783037191019 |
This book, which explores recent trends in the representation theory of algebras and its exciting interaction with geometry, topology, commutative algebra, Lie algebras, combinatorics, quantum algebras, and theoretical field, is conceived as a handbook to provide easy access to the present state of knowledge and stimulate further development. The many topics discussed include quivers, quivers with potential, bound quiver algebras, Jacobian algebras, cluster algebras and categories, Calabi-Yau algebras and categories, triangulated and derived categories, and quantum loop algebras. This book consists of thirteen self-contained expository survey and research articles and is addressed to researchers and graduate students in algebra as well as a broader mathematical community. The articles contain a large number of examples and open problems and give new perspectives for research in the field.
Group Representations: Cohomology, Group Actions and Topology
Title | Group Representations: Cohomology, Group Actions and Topology PDF eBook |
Author | Alejandro Adem |
Publisher | American Mathematical Soc. |
Pages | 549 |
Release | 1998 |
Genre | Mathematics |
ISBN | 0821806580 |
This volume combines contributions in topology and representation theory that reflect the increasingly vigorous interactions between these areas. Topics such as group theory, homotopy theory, cohomology of groups, and modular representations are covered. All papers have been carefully refereed and offer lasting value.
Advances in Representation Theory of Algebras
Title | Advances in Representation Theory of Algebras PDF eBook |
Author | Ibrahim Assem |
Publisher | American Mathematical Soc. |
Pages | 257 |
Release | 2021-01-06 |
Genre | Education |
ISBN | 147045159X |
The Seventh ARTA (“Advances in Representation Theory of Algebras VII”) conference took place at the Instituto de Matemáticas of the Universidad Nacional Autónoma de México, in Mexico City, from September 24–28, 2018, in honor of José Antonio de la Peña's 60th birthday. Papers in this volume cover topics Professor de la Peña worked on, such as covering theory, tame algebras, and the use of quadratic forms in representation theory. Also included are papers on the categorical approach to representations of algebras and relations to Lie theory, Cohen–Macaulay modules, quantum groups and other algebraic structures.