Simple Lie Algebras Over Fields of Positive Characteristic: Classifying the absolute toral rank two case

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Publisher : Walter de Gruyter
ISBN 13 : 3110197014
Total Pages : 392 pages
Book Rating : 4.1/5 (11 download)

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Book Synopsis Simple Lie Algebras Over Fields of Positive Characteristic: Classifying the absolute toral rank two case by : Helmut Strade

Download or read book Simple Lie Algebras Over Fields of Positive Characteristic: Classifying the absolute toral rank two case written by Helmut Strade and published by Walter de Gruyter. This book was released on 2004 with total page 392 pages. Available in PDF, EPUB and Kindle. Book excerpt: The problem of classifying the finite-dimensional simple Lie algebras over fields of characteristic p > 0 is a long-standing one. Work on this question during the last 45 years has been directed by the Kostrikin-Shafarevich Conjecture of 1966, which states that over an algebraically closed field of characteristic p > 5 a finite-dimensional restricted simple Lie algebra is classical or of Cartan type. This conjecture was proved for p > 7 by Block and Wilson in 1988. The generalization of the Kostrikin-Shafarevich Conjecture for the general case of not necessarily restricted Lie algebras and p > 7 was announced in 1991 by Strade and Wilson and eventually proved by Strade in 1998. The final Block-Wilson-Strade-Premet Classification Theorem is a landmark result of modern mathematics and can be formulated as follows: Every finite-dimensional simple Lie algebra over an algebraically closed field of characteristic p > 3 is of classical, Cartan, or Melikian type. In the three-volume book, the author is assembling the proof of the Classification Theorem with explanations and references. The goal is a state-of-the-art account on the structure and classification theory of Lie algebras over fields of positive characteristic leading to the forefront of current research in this field. This is the second part of the three-volume book about the classification of the simple Lie algebras over algebraically closed fields of characteristics > 3. The first volume contains the methods, examples, and a first classification result. This second volume presents insight in the structure of tori of Hamiltonian and Melikian algebras. Based on sandwich element methods due to Aleksei. I. Kostrikin and Alexander A. Premet and the investigation of absolute toral rank 2 simple Lie algebras over algebraically closed fields of characteristics > 3 is given.

Classifying the Absolute Toral Rank Two Case

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Publisher : Walter de Gruyter GmbH & Co KG
ISBN 13 : 3110517604
Total Pages : 394 pages
Book Rating : 4.1/5 (15 download)

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Book Synopsis Classifying the Absolute Toral Rank Two Case by : Helmut Strade

Download or read book Classifying the Absolute Toral Rank Two Case written by Helmut Strade and published by Walter de Gruyter GmbH & Co KG. This book was released on 2017-04-10 with total page 394 pages. Available in PDF, EPUB and Kindle. Book excerpt: The problem of classifying the finite dimensional simple Lie algebras over fields of characteristic p > 0 is a long standing one. Work on this question has been directed by the Kostrikin Shafarevich Conjecture of 1966, which states that over an algebraically closed field of characteristic p > 5 a finite dimensional restricted simple Lie algebra is classical or of Cartan type. This conjecture was proved for p > 7 by Block and Wilson in 1988. The generalization of the Kostrikin-Shafarevich Conjecture for the general case of not necessarily restricted Lie algebras and p > 7 was announced in 1991 by Strade and Wilson and eventually proved by Strade in 1998. The final Block-Wilson-Strade-Premet Classification Theorem is a landmark result of modern mathematics and can be formulated as follows: Every simple finite dimensional simple Lie algebra over an algebraically closed field of characteristic p > 3 is of classical, Cartan, or Melikian type. This is the second part of a three-volume book about the classification of the simple Lie algebras over algebraically closed fields of characteristic > 3. The first volume contains the methods, examples and a first classification result. This second volume presents insight in the structure of tori of Hamiltonian and Melikian algebras. Based on sandwich element methods due to A. I. Kostrikin and A. A. Premet and the investigations of filtered and graded Lie algebras, a complete proof for the classification of absolute toral rank 2 simple Lie algebras over algebraically closed fields of characteristic > 3 is given. Contents Tori in Hamiltonian and Melikian algebras 1-sections Sandwich elements and rigid tori Towards graded algebras The toral rank 2 case

Simple Lie Algebras Over Fields of Positive Characteristic: Structure theory

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Publisher : Walter de Gruyter
ISBN 13 : 3110142112
Total Pages : 548 pages
Book Rating : 4.1/5 (11 download)

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Book Synopsis Simple Lie Algebras Over Fields of Positive Characteristic: Structure theory by : Helmut Strade

Download or read book Simple Lie Algebras Over Fields of Positive Characteristic: Structure theory written by Helmut Strade and published by Walter de Gruyter. This book was released on 2004 with total page 548 pages. Available in PDF, EPUB and Kindle. Book excerpt: The problem of classifying the finite-dimensional simple Lie algebras over fields of characteristic p > 0 is a long-standing one. Work on this question during the last 45 years has been directed by the Kostrikin-Shafarevich Conjecture of 1966, which states that over an algebraically closed field of characteristic p > 5 a finite-dimensional restricted simple Lie algebra is classical or of Cartan type. This conjecture was proved for p > 7 by Block and Wilson in 1988. The generalization of the Kostrikin-Shafarevich Conjecture for the general case of not necessarily restricted Lie algebras and p > 7 was announced in 1991 by Strade and Wilson and eventually proved by Strade in 1998. The final Block-Wilson-Strade-Premet Classification Theorem is a landmark result of modern mathematics and can be formulated as follows: Every finite-dimensional simple Lie algebra over an algebraically closed field of characteristic p > 3 is of classical, Cartan, or Melikian type. In the three-volume book, the author is assembling the proof of the Classification Theorem with explanations and references. The goal is a state-of-the-art account on the structure and classification theory of Lie algebras over fields of positive characteristic leading to the forefront of current research in this field. This first volume is devoted to preparing the ground for the classification work to be performed in the second and third volume. The concise presentation of the general theory underlying the subject matter and the presentation of classification results on a subclass of the simple Lie algebras for all odd primesmake this volume an invaluable source and reference for all research mathematicians and advanced graduate students in albegra.

Gradings on Simple Lie Algebras

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Author :
Publisher : American Mathematical Soc.
ISBN 13 : 0821898469
Total Pages : 355 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Gradings on Simple Lie Algebras by : Alberto Elduque

Download or read book Gradings on Simple Lie Algebras written by Alberto Elduque and published by American Mathematical Soc.. This book was released on 2013 with total page 355 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is a self-contained exposition of the classification of gradings by arbitrary groups on classical simple Lie algebras over algebraically closed fields of characteristic not equal to 2 as well as on some non-classical simple Lie algebras in positive characteristic. Other important algebras also enter the stage: matrix algebras, the octonions, and the Albert algebra. Most of the presented results are recent and have not yet appeared in book form.

Lie Algebras and Related Topics

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Publisher : American Mathematical Soc.
ISBN 13 : 1470410230
Total Pages : 258 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Lie Algebras and Related Topics by : Marina Avitabile

Download or read book Lie Algebras and Related Topics written by Marina Avitabile and published by American Mathematical Soc.. This book was released on 2015-11-30 with total page 258 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the Workshop on Lie Algebras, in honor of Helmut Strade's 70th Birthday, held from May 22-24, 2013, at the Università degli Studi di Milano-Bicocca, Milano, Italy. Lie algebras are at the core of several areas of mathematics, such as, Lie groups, algebraic groups, quantum groups, representation theory, homogeneous spaces, integrable systems, and algebraic topology. The first part of this volume combines research papers with survey papers by the invited speakers. The second part consists of several collections of problems on modular Lie algebras, their representations, and the conjugacy of their nilpotent elements as well as the Koszulity of (restricted) Lie algebras and Lie properties of group algebras or restricted universal enveloping algebras.

Completion of the Classification

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Publisher : Walter de Gruyter
ISBN 13 : 3110263017
Total Pages : 249 pages
Book Rating : 4.1/5 (12 download)

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Book Synopsis Completion of the Classification by : Helmut Strade

Download or read book Completion of the Classification written by Helmut Strade and published by Walter de Gruyter. This book was released on 2012-12-19 with total page 249 pages. Available in PDF, EPUB and Kindle. Book excerpt: The problem of classifying the finite-dimensional simple Lie algebras over fields of characteristic p > 0 is a long-standing one. Work on this question during the last 45 years has been directed by the Kostrikin–Shafarevich Conjecture of 1966, which states that over an algebraically closed field of characteristic p > 5 a finite-dimensional restricted simple Lie algebra is classical or of Cartan type. This conjecture was proved for p > 7 by Block and Wilson in 1988. The generalization of the Kostrikin–Shafarevich Conjecture for the general case of not necessarily restricted Lie algebras and p > 7 was announced in 1991 by Strade and Wilson and eventually proved by Strade in 1998. The final Block–Wilson–Strade–Premet Classification Theorem is a landmark result of modern mathematics and can be formulated as follows: Every finite-dimensional simple Lie algebra over an algebraically closed field of characteristic p > 3 is of classical, Cartan, or Melikian type. In the three-volume book, the author is assembling the proof of the Classification Theorem with explanations and references. The goal is a state-of-the-art account on the structure and classification theory of Lie algebras over fields of positive characteristic leading to the forefront of current research in this field. This is the last of three volumes. In this monograph the proof of the Classification Theorem presented in the first volume is concluded. It collects all the important results on the topic which can be found only in scattered scientific literature so far.

Complementation of Normal Subgroups

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Publisher : Walter de Gruyter GmbH & Co KG
ISBN 13 : 3110480212
Total Pages : 156 pages
Book Rating : 4.1/5 (14 download)

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Book Synopsis Complementation of Normal Subgroups by : Joseph Kirtland

Download or read book Complementation of Normal Subgroups written by Joseph Kirtland and published by Walter de Gruyter GmbH & Co KG. This book was released on 2017-09-11 with total page 156 pages. Available in PDF, EPUB and Kindle. Book excerpt: Starting with the Schur-Zassenhaus theorem, this monograph documents a wide variety of results concerning complementation of normal subgroups in finite groups. The contents cover a wide range of material from reduction theorems and subgroups in the derived and lower nilpotent series to abelian normal subgroups and formations. Contents Prerequisites The Schur-Zassenhaus theorem: A bit of history and motivation Abelian and minimal normal subgroups Reduction theorems Subgroups in the chief series, derived series, and lower nilpotent series Normal subgroups with abelian sylow subgroups The formation generation Groups with specific classes of subgroups complemented

Representations of Algebraic Groups, Quantum Groups, and Lie Algebras

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Publisher : American Mathematical Soc.
ISBN 13 : 0821839241
Total Pages : 270 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Representations of Algebraic Groups, Quantum Groups, and Lie Algebras by : Georgia Benkart

Download or read book Representations of Algebraic Groups, Quantum Groups, and Lie Algebras written by Georgia Benkart and published by American Mathematical Soc.. This book was released on 2006 with total page 270 pages. Available in PDF, EPUB and Kindle. Book excerpt: Covers various aspects of the representation theory of Lie algebras, finite groups of Lie types, Hecke algebras, and Lie super algebras. This book outlines connections among irreducible representations of certain blocks of reduced enveloping algebras of semi-simple Lie algebras in positive characteristic.

Developments and Retrospectives in Lie Theory

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Publisher : Springer
ISBN 13 : 3319098047
Total Pages : 403 pages
Book Rating : 4.3/5 (19 download)

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Book Synopsis Developments and Retrospectives in Lie Theory by : Geoffrey Mason

Download or read book Developments and Retrospectives in Lie Theory written by Geoffrey Mason and published by Springer. This book was released on 2014-10-31 with total page 403 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Lie Theory Workshop, founded by Joe Wolf (UC, Berkeley), has been running for over two decades. These workshops have been sponsored by the NSF, noting the talks have been seminal in describing new perspectives in the field covering broad areas of current research. At the beginning, the top universities in California and Utah hosted the meetings which continue to run on a quarterly basis. Experts in representation theory/Lie theory from various parts of the US, Europe, Asia (China, Japan, Singapore, Russia), Canada, and South and Central America were routinely invited to give talks at these meetings. Nowadays, the workshops are also hosted at universities in Louisiana, Virginia, and Oklahoma. The contributors to this volume have all participated in these Lie theory workshops and include in this volume expository articles which cover representation theory from the algebraic, geometric, analytic, and topological perspectives with also important connections to math physics. These survey articles, review and update the prominent seminal series of workshops in representation/Lie theory mentioned-above, and reflects the widespread influence of those workshops in such areas as harmonic analysis, representation theory, differential geometry, algebraic geometry, number theory, and mathematical physics. Many of the contributors have had prominent roles in both the classical and modern developments of Lie theory and its applications.

Products of Finite Groups

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Publisher : Walter de Gruyter
ISBN 13 : 311022061X
Total Pages : 347 pages
Book Rating : 4.1/5 (12 download)

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Book Synopsis Products of Finite Groups by : Adolfo Ballester-Bolinches

Download or read book Products of Finite Groups written by Adolfo Ballester-Bolinches and published by Walter de Gruyter. This book was released on 2010-10-19 with total page 347 pages. Available in PDF, EPUB and Kindle. Book excerpt: The study of finite groups factorised as a product of two or more subgroups has become a subject of great interest during the last years with applications not only in group theory, but also in other areas like cryptography and coding theory. It has experienced a big impulse with the introduction of some permutability conditions. The aim of this book is to gather, order, and examine part of this material, including the latest advances made, give some new approach to some topics, and present some new subjects of research in the theory of finite factorised groups. Some of the topics covered by this book include groups whose subnormal subgroups are normal, permutable, or Sylow-permutable, products of nilpotent groups, and an exhaustive structural study of totally and mutually permutable products of finite groups and their relation with classes of groups. This monograph is mainly addressed to graduate students and senior researchers interested in the study of products and permutability of finite groups. A background in finite group theory and a basic knowledge of representation theory and classes of groups is recommended to follow it.

Representations of Algebras

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Publisher : American Mathematical Soc.
ISBN 13 : 1470435764
Total Pages : 294 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Representations of Algebras by : Graham J. Leuschke

Download or read book Representations of Algebras written by Graham J. Leuschke and published by American Mathematical Soc.. This book was released on 2018 with total page 294 pages. Available in PDF, EPUB and Kindle. Book excerpt: Contains the proceedings of the 17th Workshop and International Conference on Representations of Algebras (ICRA 2016), held in August 2016, at Syracuse University. This volume includes three survey articles based on short courses in the areas of commutative algebraic groups, modular group representation theory, and thick tensor ideals of bounded derived categories.

Stability Analysis of Impulsive Functional Differential Equations

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Publisher : Walter de Gruyter
ISBN 13 : 3110221829
Total Pages : 241 pages
Book Rating : 4.1/5 (12 download)

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Book Synopsis Stability Analysis of Impulsive Functional Differential Equations by : Ivanka Stamova

Download or read book Stability Analysis of Impulsive Functional Differential Equations written by Ivanka Stamova and published by Walter de Gruyter. This book was released on 2009-10-16 with total page 241 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to impulsive functional differential equations which are a natural generalization of impulsive ordinary differential equations (without delay) and of functional differential equations (without impulses). At the present time the qualitative theory of such equations is under rapid development. After a presentation of the fundamental theory of existence, uniqueness and continuability of solutions, a systematic development of stability theory for that class of problems is given which makes the book unique. It addresses to a wide audience such as mathematicians, applied researches and practitioners.

Yakov G. Berkovich; Lev S. Kazarin; Emmanuel M. Zhmud': Characters of Finite Groups. Volume 1

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Publisher : Walter de Gruyter GmbH & Co KG
ISBN 13 : 3110224070
Total Pages : 624 pages
Book Rating : 4.1/5 (12 download)

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Book Synopsis Yakov G. Berkovich; Lev S. Kazarin; Emmanuel M. Zhmud': Characters of Finite Groups. Volume 1 by : Yakov G. Berkovich

Download or read book Yakov G. Berkovich; Lev S. Kazarin; Emmanuel M. Zhmud': Characters of Finite Groups. Volume 1 written by Yakov G. Berkovich and published by Walter de Gruyter GmbH & Co KG. This book was released on 2017-12-18 with total page 624 pages. Available in PDF, EPUB and Kindle. Book excerpt: This updated edition of this classic book is devoted to ordinary representation theory and is addressed to finite group theorists intending to study and apply character theory. It contains many exercises and examples, and the list of problems contains a number of open questions.

Moufang Sets and Structurable Division Algebras

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Publisher : American Mathematical Soc.
ISBN 13 : 1470435543
Total Pages : 102 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Moufang Sets and Structurable Division Algebras by : Lien Boelaert

Download or read book Moufang Sets and Structurable Division Algebras written by Lien Boelaert and published by American Mathematical Soc.. This book was released on 2019-06-10 with total page 102 pages. Available in PDF, EPUB and Kindle. Book excerpt: A Moufang set is essentially a doubly transitive permutation group such that each point stabilizer contains a normal subgroup which is regular on the remaining vertices; these regular normal subgroups are called the root groups, and they are assumed to be conjugate and to generate the whole group. It has been known for some time that every Jordan division algebra gives rise to a Moufang set with abelian root groups. The authors extend this result by showing that every structurable division algebra gives rise to a Moufang set, and conversely, they show that every Moufang set arising from a simple linear algebraic group of relative rank one over an arbitrary field k of characteristic different from 2 and 3 arises from a structurable division algebra. The authors also obtain explicit formulas for the root groups, the τ-map and the Hua maps of these Moufang sets. This is particularly useful for the Moufang sets arising from exceptional linear algebraic groups.

Structure Theory

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Publisher : Walter de Gruyter GmbH & Co KG
ISBN 13 : 3110515237
Total Pages : 686 pages
Book Rating : 4.1/5 (15 download)

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Book Synopsis Structure Theory by : Helmut Strade

Download or read book Structure Theory written by Helmut Strade and published by Walter de Gruyter GmbH & Co KG. This book was released on 2017-04-24 with total page 686 pages. Available in PDF, EPUB and Kindle. Book excerpt: The problem of classifying the finite dimensional simple Lie algebras over fields of characteristic p > 0 is a long-standing one. Work on this question has been directed by the Kostrikin-Shafarevich Conjecture of 1966, which states that over an algebraically closed field of characteristic p > 5 a finite dimensional restricted simple Lie algebra is classical or of Cartan type. This conjecture was proved for p > 7 by Block and Wilson in 1988. The generalization of the Kostrikin-Shafarevich Conjecture for the general case of not necessarily restricted Lie algebras and p > 7 was announced in 1991 by Strade and Wilson and eventually proved by Strade in 1998. The final Block-Wilson-Strade-Premet Classification Theorem is a landmark result of modern mathematics and can be formulated as follows: Every simple finite dimensional simple Lie algebra over an algebraically closed field of characteristic p > 3 is of classical, Cartan, or Melikian type. In the three-volume book, the author is assembling the proof of the Classification Theorem with explanations and references. The goal is a state-of-the-art account on the structure and classification theory of Lie algebras over fields of positive characteristic. This first volume is devoted to preparing the ground for the classification work to be performed in the second and third volumes. The concise presentation of the general theory underlying the subject matter and the presentation of classification results on a subclass of the simple Lie algebras for all odd primes will make this volume an invaluable source and reference for all research mathematicians and advanced graduate students in algebra. The second edition is corrected. Contents Toral subalgebras in p-envelopes Lie algebras of special derivations Derivation simple algebras and modules Simple Lie algebras Recognition theorems The isomorphism problem Structure of simple Lie algebras Pairings of induced modules Toral rank 1 Lie algebras

Classifying the Absolute Toral Rank Two Case

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Author :
Publisher : Walter de Gruyter
ISBN 13 : 3110203057
Total Pages : 392 pages
Book Rating : 4.1/5 (12 download)

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Book Synopsis Classifying the Absolute Toral Rank Two Case by : Helmut Strade

Download or read book Classifying the Absolute Toral Rank Two Case written by Helmut Strade and published by Walter de Gruyter. This book was released on 2009-09-04 with total page 392 pages. Available in PDF, EPUB and Kindle. Book excerpt: The problem of classifying the finite-dimensional simple Lie algebras over fields of characteristic p > 0 is a long-standing one. Work on this question during the last 45 years has been directed by the Kostrikin–Shafarevich Conjecture of 1966, which states that over an algebraically closed field of characteristic p > 5 a finite-dimensional restricted simple Lie algebra is classical or of Cartan type. This conjecture was proved for p > 7 by Block and Wilson in 1988. The generalization of the Kostrikin–Shafarevich Conjecture for the general case of not necessarily restricted Lie algebras and p > 7 was announced in 1991 by Strade and Wilson and eventually proved by Strade in 1998. The final Block–Wilson–Strade–Premet Classification Theorem is a landmark result of modern mathematics and can be formulated as follows: Every finite-dimensional simple Lie algebra over an algebraically closed field of characteristic p > 3 is of classical, Cartan, or Melikian type. In the three-volume book, the author is assembling the proof of the Classification Theorem with explanations and references. The goal is a state-of-the-art account on the structure and classification theory of Lie algebras over fields of positive characteristic leading to the forefront of current research in this field. This is the second part of the three-volume book about the classification of the simple Lie algebras over algebraically closed fields of characteristics > 3. The first volume contains the methods, examples, and a first classification result. This second volume presents insight in the structure of tori of Hamiltonian and Melikian algebras. Based on sandwich element methods due to Aleksei. I. Kostrikin and Alexander A. Premet and the investigation of absolute toral rank 2 simple Lie algebras over algebraically closed fields of characteristics > 3 is given.

Mathematical Reviews

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Publisher :
ISBN 13 :
Total Pages : 1234 pages
Book Rating : 4.X/5 (2 download)

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Book Synopsis Mathematical Reviews by :

Download or read book Mathematical Reviews written by and published by . This book was released on 1990 with total page 1234 pages. Available in PDF, EPUB and Kindle. Book excerpt: