Nathan Jacobson Livres

These algebras determine, by the Sliedderburn Theorem. the semi-simple finite dimensional algebras over a field. They lead to the definition of the Brauer group and to certain geometric objects, the Brauer-Severi varieties. Sie shall be interested in these algebras which have an involution. Algebras with involution arose first in the study of the so-called .'multiplication algebras of Riemann matrices". Albert undertook their study at the behest of Lefschetz. He solved the problem of determining these algebras. The problem has an algebraic part and an arithmetic part which can be solved only by determining the finite dimensional simple algebras over an algebraic number field. We are not going to consider the arithmetic part but will be interested only in the algebraic part. In Albert's classical book (1939). both parts are treated. A quick survey of our Table of Contents will indicate the scope of the present volume. The largest part of our book is the fifth chapter which deals with invo- torial rimple algebras of finite dimension over a field. Of particular interest are the Jordan algebras determined by these algebras with involution. Their structure is determined and two important concepts of these algebras with involution are the universal enveloping algebras and the reduced norm. Of great importance is the concept of isotopy. There are numerous applications of these concepts, some of which are quite old.

"Explores all of the topics typically covered in undergraduate courses including the rudiments of set theory, group theory, rings, modules, Galois theory, polynomials, linear algebra, and associative algebra"--Cover p. 4

Definitive treatment covers split semi-simple Lie algebras, universal enveloping algebras, classification of irreducible modules, automorphisms, simple Lie algebras over an arbitrary field, and more. Classic handbook for researchers and students; useable in graduate courses or for self-study.

Focusing on algebras with involution, the book explores their significance in determining semi-simple finite dimensional algebras over a field, as established by the Sliedderburn Theorem. It delves into the historical context of these algebras, particularly their origins in the study of Riemann matrices. The text emphasizes the algebraic aspects, particularly in the fifth chapter, which covers invotorial simple algebras and their connections to Jordan algebras, universal enveloping algebras, and the concept of isotopy, highlighting their various applications.