Simple Lie Algebras Over Fields Of Positive Characteristic

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Simple Lie Algebras Over Fields of Positive Characteristic: Structure theory

Author : Helmut Strade
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.

Simple Lie algebras over fields of positive characteristic.

Author : Helmut Strade
Publisher :
ISBN 13 : 9783110517613
Total Pages : pages
Book Rating : 4.5/5 (176 download)

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Book Synopsis Simple Lie algebras over fields of positive characteristic. by : Helmut Strade

Download or read book Simple Lie algebras over fields of positive characteristic. written by Helmut Strade and published by . This book was released on 2017 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

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

Author : Helmut Strade
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.

Simple Lie Algebras Over Fields of Positive Characteristics. II. Classifying the Absolute Toral Rank Two Case

Author : Helmut Strade
Publisher :
ISBN 13 : 9783110197013
Total Pages : 0 pages
Book Rating : 4.1/5 (97 download)

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Book Synopsis Simple Lie Algebras Over Fields of Positive Characteristics. II. Classifying the Absolute Toral Rank Two Case by : Helmut Strade

Download or read book Simple Lie Algebras Over Fields of Positive Characteristics. II. Classifying the Absolute Toral Rank Two Case written by Helmut Strade and published by . This book was released on 2009 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Gradings on Simple Lie Algebras

Author : Alberto Elduque
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.

Modular Lie Algebras and their Representations

Author : H. Strade
Publisher : CRC Press
ISBN 13 : 1000103390
Total Pages : 318 pages
Book Rating : 4.0/5 (1 download)

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Book Synopsis Modular Lie Algebras and their Representations by : H. Strade

Download or read book Modular Lie Algebras and their Representations written by H. Strade and published by CRC Press. This book was released on 2020-08-11 with total page 318 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents an introduction to the structure and representation theory of modular Lie algebras over fields of positive characteristic. It introduces the beginner to the theory of modular Lie algebras and is meant to be a reference text for researchers.

Modular Lie Algebras

Author : Geoge B. Seligman
Publisher : Springer Science & Business Media
ISBN 13 : 3642949851
Total Pages : 175 pages
Book Rating : 4.6/5 (429 download)

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Book Synopsis Modular Lie Algebras by : Geoge B. Seligman

Download or read book Modular Lie Algebras written by Geoge B. Seligman and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 175 pages. Available in PDF, EPUB and Kindle. Book excerpt: The study of the structure of Lie algebras over arbitrary fields is now a little more than thirty years old. The first papers, to my know ledge, which undertook this study as an end in itself were those of JACOBSON (" Rational methods in the theory of Lie algebras ") in the Annals, and of LANDHERR ("Uber einfache Liesche Ringe") in the Hamburg Abhandlungen, both in 1935. Over fields of characteristic zero, these thirty years have seen the ideas and results inherited from LIE, KILLING, E. CARTAN and WEYL developed and given new depth, meaning and elegance by many contributors. Much of this work is presented in [47, 64, 128 and 234] of the bibliography. For those who find the rationalization for the study of Lie algebras in their connections with Lie groups, satisfying counterparts to these connections have been found over general non-modular fields, with the substitution of the formal groups of BOCHNER [40] (see also DIEUDONNE [108]), or that of the algebraic linear groups of CHEVALLEY [71], for the usual Lie group. In particular, the relation with algebraic linear groups has stimulated the study of Lie algebras of linear transformations. When one admits to consideration Lie algebras over a base field of positive characteristic (such are the algebras to which the title of this monograph refers), he encounters a new and initially confusing scene.

Lie Algebras and Related Topics

Author : Georgia Benkart
Publisher : American Mathematical Soc.
ISBN 13 : 0821851195
Total Pages : 352 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Lie Algebras and Related Topics by : Georgia Benkart

Download or read book Lie Algebras and Related Topics written by Georgia Benkart and published by American Mathematical Soc.. This book was released on 1990 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt: Discusses the problem of determining the finite-dimensional simple Lie algebras over an algebraically closed field of characteristic $p>7$. This book includes topics such as Lie algebras of prime characteristic, algebraic groups, combinatorics and representation theory, and Kac-Moody and Virasoro algebras.

Representations of Algebraic Groups

Author : Jens Carsten Jantzen
Publisher : American Mathematical Soc.
ISBN 13 : 082184377X
Total Pages : 594 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Representations of Algebraic Groups by : Jens Carsten Jantzen

Download or read book Representations of Algebraic Groups written by Jens Carsten Jantzen and published by American Mathematical Soc.. This book was released on 2003 with total page 594 pages. Available in PDF, EPUB and Kindle. Book excerpt: Gives an introduction to the general theory of representations of algebraic group schemes. This title deals with representation theory of reductive algebraic groups and includes topics such as the description of simple modules, vanishing theorems, Borel-Bott-Weil theorem and Weyl's character formula, and Schubert schemes and lne bundles on them.

Modular Lie Algebras

Author : George B. Seligman
Publisher :
ISBN 13 :
Total Pages : 184 pages
Book Rating : 4.3/5 (91 download)

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Book Synopsis Modular Lie Algebras by : George B. Seligman

Download or read book Modular Lie Algebras written by George B. Seligman and published by . This book was released on 1967 with total page 184 pages. Available in PDF, EPUB and Kindle. Book excerpt: The study of the structure of Lie algebras over arbitrary fields is now a little more than thirty years old. The first papers, to my know ledge, which undertook this study as an end in itself were those of JACOBSON (" Rational methods in the theory of Lie algebras ") in the Annals, and of LANDHERR ("Uber einfache Liesche Ringe") in the Hamburg Abhandlungen, both in 1935. Over fields of characteristic zero, these thirty years have seen the ideas and results inherited from LIE, KILLING, E. CARTAN and WEYL developed and given new depth, meaning and elegance by many contributors. Much of this work is presented in [47, 64, 128 and 234] of the bibliography. For those who find the rationalization for the study of Lie algebras in their connections with Lie groups, satisfying counterparts to these connections have been found over general non-modular fields, with the substitution of the formal groups of BOCHNER [40] (see also DIEUDONNE [108]), or that of the algebraic linear groups of CHEVALLEY [71], for the usual Lie group. In particular, the relation with algebraic linear groups has stimulated the study of Lie algebras of linear transformations. When one admits to consideration Lie algebras over a base field of positive characteristic (such are the algebras to which the title of this monograph refers), he encounters a new and initially confusing scene.

Lie Theory

Author : Jean-Philippe Anker
Publisher : Springer Science & Business Media
ISBN 13 : 0817681922
Total Pages : 341 pages
Book Rating : 4.8/5 (176 download)

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Book Synopsis Lie Theory by : Jean-Philippe Anker

Download or read book Lie Theory written by Jean-Philippe Anker and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 341 pages. Available in PDF, EPUB and Kindle. Book excerpt: * First of three independent, self-contained volumes under the general title, "Lie Theory," featuring original results and survey work from renowned mathematicians. * Contains J. C. Jantzen's "Nilpotent Orbits in Representation Theory," and K.-H. Neeb's "Infinite Dimensional Groups and their Representations." * Comprehensive treatments of the relevant geometry of orbits in Lie algebras, or their duals, and the correspondence to representations. * Should benefit graduate students and researchers in mathematics and mathematical physics.

Lie Algebras and Related Topics

Author : Marina Avitabile
Publisher : American Mathematical Soc.
ISBN 13 : 1470410230
Total Pages : 242 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 242 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.

Simple Lie Algebras Over Fields of Positive Characteristic 02.Classifying the Absolute Toral Rank Two Case

Author : Helmut Strade
Publisher : de Gruyter
ISBN 13 : 9783110516760
Total Pages : 388 pages
Book Rating : 4.5/5 (167 download)

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Book Synopsis Simple Lie Algebras Over Fields of Positive Characteristic 02.Classifying the Absolute Toral Rank Two Case by : Helmut Strade

Download or read book Simple Lie Algebras Over Fields of Positive Characteristic 02.Classifying the Absolute Toral Rank Two Case written by Helmut Strade and published by de Gruyter. This book was released on 2017-04-15 with total page 388 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this second volume, the author presents the state of the art of the structure and classification of Lie algebras over fields of positive characteristic, an important topic in algebra. The contents of the book are leading to the forefront of current research in this field.

Abstract Lie Algebras

Author : David J Winter
Publisher : Courier Corporation
ISBN 13 : 0486783464
Total Pages : 162 pages
Book Rating : 4.4/5 (867 download)

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Book Synopsis Abstract Lie Algebras by : David J Winter

Download or read book Abstract Lie Algebras written by David J Winter and published by Courier Corporation. This book was released on 2013-12-01 with total page 162 pages. Available in PDF, EPUB and Kindle. Book excerpt: Solid but concise, this account of Lie algebra emphasizes the theory's simplicity and offers new approaches to major theorems. Author David J. Winter, a Professor of Mathematics at the University of Michigan, also presents a general, extensive treatment of Cartan and related Lie subalgebras over arbitrary fields. Preliminary material covers modules and nonassociate algebras, followed by a compact, self-contained development of the theory of Lie algebras of characteristic 0. Topics include solvable and nilpotent Lie algebras, Cartan subalgebras, and Levi's radical splitting theorem and the complete reducibility of representations of semisimple Lie algebras. Additional subjects include the isomorphism theorem for semisimple Lie algebras and their irreducible modules, automorphism of Lie algebras, and the conjugacy of Cartan subalgebras and Borel subalgebras. An extensive theory of Cartan and related subalgebras of Lie algebras over arbitrary fields is developed in the final chapter, and an appendix offers background on the Zariski topology.

The Recognition Theorem for Graded Lie Algebras in Prime Characteristic

Author : Georgia Benkart
Publisher : American Mathematical Soc.
ISBN 13 : 0821842269
Total Pages : 164 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis The Recognition Theorem for Graded Lie Algebras in Prime Characteristic by : Georgia Benkart

Download or read book The Recognition Theorem for Graded Lie Algebras in Prime Characteristic written by Georgia Benkart and published by American Mathematical Soc.. This book was released on 2009 with total page 164 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Volume 197, number 920 (second of 5 numbers)."

Advanced Modern Algebra

Author : Joseph J. Rotman
Publisher : American Mathematical Soc.
ISBN 13 : 0821847414
Total Pages : 1026 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Advanced Modern Algebra by : Joseph J. Rotman

Download or read book Advanced Modern Algebra written by Joseph J. Rotman and published by American Mathematical Soc.. This book was released on 2010-08-11 with total page 1026 pages. Available in PDF, EPUB and Kindle. Book excerpt: "This book is designed as a text for the first year of graduate algebra, but it can also serve as a reference since it contains more advanced topics as well. This second edition has a different organization than the first. It begins with a discussion of the cubic and quartic equations, which leads into permutations, group theory, and Galois theory (for finite extensions; infinite Galois theory is discussed later in the book). The study of groups continues with finite abelian groups (finitely generated groups are discussed later, in the context of module theory), Sylow theorems, simplicity of projective unimodular groups, free groups and presentations, and the Nielsen-Schreier theorem (subgroups of free groups are free). The study of commutative rings continues with prime and maximal ideals, unique factorization, noetherian rings, Zorn's lemma and applications, varieties, and Gr'obner bases. Next, noncommutative rings and modules are discussed, treating tensor product, projective, injective, and flat modules, categories, functors, and natural transformations, categorical constructions (including direct and inverse limits), and adjoint functors. Then follow group representations: Wedderburn-Artin theorems, character theory, theorems of Burnside and Frobenius, division rings, Brauer groups, and abelian categories. Advanced linear algebra treats canonical forms for matrices and the structure of modules over PIDs, followed by multilinear algebra. Homology is introduced, first for simplicial complexes, then as derived functors, with applications to Ext, Tor, and cohomology of groups, crossed products, and an introduction to algebraic K-theory. Finally, the author treats localization, Dedekind rings and algebraic number theory, and homological dimensions. The book ends with the proof that regular local rings have unique factorization."--Publisher's description.

Identities of Algebras and their Representations

Author : I︠U︡riĭ Pitrimovich Razmyslov
Publisher : American Mathematical Soc.
ISBN 13 : 9780821846087
Total Pages : 468 pages
Book Rating : 4.8/5 (46 download)

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Book Synopsis Identities of Algebras and their Representations by : I︠U︡riĭ Pitrimovich Razmyslov

Download or read book Identities of Algebras and their Representations written by I︠U︡riĭ Pitrimovich Razmyslov and published by American Mathematical Soc.. This book was released on 1994 with total page 468 pages. Available in PDF, EPUB and Kindle. Book excerpt: During the past forty years, a new trend in the theory of associative algebras, Lie algebras, and their representations has formed under the influence of mathematical logic and universal algebra, namely, the theory of varieties and identities of associative algebras, Lie algebras, and their representations. The last twenty years have seen the creation of the method of 2-words and *a-functions, which allowed a number of problems in the theory of groups, rings, Lie algebras, and their representations to be solved in a unified way. The possibilities of this method are far from exhausted. This book sums up the applications of the method of 2-words and *a-functions in the theory of varieties and gives a systematic exposition of contemporary achievements in the theory of identities of algebras and their representations closely related to this method. The aim is to make these topics accessible to a wider group of mathematicians.