George B. Seligman's Books

Rational Constructions of Modules for Simple Lie Algebras

By: George B. Seligman

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Rational Constructions of Modules for Simple Lie Algebras

By: George B. Seligman

Suitable for researchers in Lie theory and in the theory of linear algebra, associative or otherwise, and to graduate students who have had some background in one or more of these areas.

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Page Count

203

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821850083

ISBN 13

9780821850084

Constructions of Lie Algebras and their Modules

By: George B. Seligman

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Constructions of Lie Algebras and their Modules

By: George B. Seligman

This book deals with central simple Lie algebras over arbitrary fields of characteristic zero. It aims to give constructions of the algebras and their finite-dimensional modules in terms that are rational with respect to the given ground field. All isotropic algebras with non-reduced relative root systems are treated, along with classical anisotropic algebras. The latter are treated by what seems to be a novel device, namely by studying certain modules for isotropic classical algebras in which they are embedded. In this development, symmetric powers of central simple associative algebras, along with generalized even Clifford algebras of involutorial algebras, play central roles. Considerable attention is given to exceptional algebras. The pace is that of a rather expansive research monograph. The reader who has at hand a standard introductory text on Lie algebras, such as Jacobson or Humphreys, should be in a position to understand the results. More technical matters arise in some of the detailed arguments. The book is intended for researchers and students of algebraic Lie theory, as well as for other researchers who are seeking explicit realizations of algebras or modules. It will probably be more useful as a resource to be dipped into, than as a text to be worked straight through.

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203

Publisher

Springer

Published Date

ISBN 10

3540388648

ISBN 13

9783540388647

Infinite Algebraic Extensions of Finite Fields

By: Joel V. Brawley,George E. Schnibben

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Infinite Algebraic Extensions of Finite Fields

By: Joel V. Brawley,George E. Schnibben

Over the last several decades there has been a renewed interest in finite field theory, partly as a result of important applications in a number of diverse areas such as electronic communications, coding theory, combinatorics, designs, finite geometries, cryptography, and other portions of discrete mathematics. In addition, a number of recent books have been devoted to the subject. Despite the resurgence in interest, it is not widely known that many results concerning finite fields have natural generalizations to abritrary algebraic extensions of finite fields. The purpose of this book is to describe these generalizations. After an introductory chapter surveying pertinent results about finite fields, the book describes the lattice structure of fields between the finite field $GF(q)$ and its algebraic closure $\Gamma (q)$. The authors introduce a notion, due to Steinitz, of an extended positive integer $N$ which includes each ordinary positive integer $n$ as a special case. With the aid of these Steinitz numbers, the algebraic extensions of $GF(q)$ are represented by symbols of the form $GF(q^N)$. When $N$ is an ordinary integer $n$, this notation agrees with the usual notation $GF(q^n)$ for a dimension $n$ extension of $GF(q)$. The authors then show that many of the finite field results concerning $GF(q^n)$ are also true for $GF(q^N)$. One chapter is devoted to giving explicit algorithms for computing in several of the infinite fields $GF(q^N)$ using the notion of an explicit basis for $GF(q^N)$ over $GF(q)$. Another chapter considers polynomials and polynomial-like functions on $GF(q^N)$ and contains a description of several classes of permutation polynomials, including the $q$-polynomials and the Dickson polynomials. Also included is a brief chapter describing two of many potential applications. Aimed at the level of a beginning graduate student or advanced undergraduate, this book could serve well as a supplementary text for a course in finite field theory.

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Page Count

126

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821851012

ISBN 13

9780821851012

Book's by George B. Seligman

Rational Constructions of Modules for Simple Lie Algebras

Rational Constructions of Modules for Simple Lie Algebras

Constructions of Lie Algebras and their Modules

Constructions of Lie Algebras and their Modules

Infinite Algebraic Extensions of Finite Fields

Infinite Algebraic Extensions of Finite Fields

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