finite dimensional division algebras, nathan jacobson [pdf]
Finite Dimensional Division Algebras by Nathan Jacobson
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Finite Dimensional Division Algebras written by
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 involutorial 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. In preparing this volume we have been assisted by our friends, notably Jean-Pierre Tignol and John Faulkner. Also, I arn greatly indebted to my secretary. Donna Belli, and to my wife, Florie. I wish to thank all of them for their help.
Book Detail :-
Title: Finite Dimensional Division Algebras
Author(s): Nathan Jacobson
Publisher: Springer-Verlag Berlin Heidelberg
Series: Grundlehren der Mathematischen Wissenschaften
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About Author :-
Author Nathan Jacobson (1910 – 1999) was an American mathematician. He was born Warsaw, Congress Poland, Russian Empire and emigrated to America with his family in 1918.
He was graduated from the University of Alabama in 1930 and was awarded a doctorate in mathematics from Princeton University on his thesis Non-commutative polynomials and cyclic algebras in 1934.
Jacobson start his career as teacher at Bryn Mawr College in 1935. He join University of Chicago in 1936 and then University of North Carolina at Chapel Hill in 1937 and Johns Hopkins University in 1943 and at then end Yale University in 1947 until his retirement. He was a member of the National Academy of Sciences and the American Academy of Arts and Sciences. He also served as president of the American Mathematical Society from 1971 to 1973, and was awarded their highest honour, the Leroy P. Steele prize for lifetime achievement, in 1998. He was also vice-president of the International Mathematical Union from 1972 to 1974.
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Book Contents :-
Finite Dimensional Division Algebras written by
cover the following topics.
1. Skew Polynominals and Division Algebras
2. Brauer Factor Sets and Noether Factor Sets
3. Galois Descent and Generic Splitting Fields
5. Simple Algebras with Involution
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