Finite-Dimensional Division Algebras over Fields by Nathan Jacobson (PDF)

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Here, the eminent algebraist, Nathan Jacobsen, concentrates on those algebras that have an involution. Although they appear in many contexts, these algebras first arose in the study of the so-called “multiplication algebras of Riemann matrices”. Of particular interest are the Jordan algebras determined by such algebras, and thus their structure is discussed in detail. Two important concepts also dealt with are the universal enveloping algebras and the reduced norm. However, the largest part of the book is the fifth chapter, which focuses on involutorial simple algebras of finite dimension over a field.

User’s Reviews

Editorial Reviews: Review “…the author takes us on a tour of division algebras, pointing out the salient facts, often with little-known proofs, but never going on so long as to bore the reader. This makes the book a pleasure to read” Bulletin of the London Mathematical Society From the Back Cover Finite-dimensional division algebras over fields determine, by the Wedderburn Theorem, the semi-simple finite-dimensio= nal algebras over a field. They lead to the definition of the Brauer group and to certain geometric objects, the Brau= er-Severi varieties. The book concentrates on those algebras that have an involution. Algebras with involution appear in many contexts;they arose first in the study of the so-called “multiplication algebras of Riemann matrices”. The largest part of the book is the fifth chapter, dealing with involu= torial simple algebras of finite dimension over a field. Of particular interest are the Jordan algebras determined by these algebras with involution;their structure is discussed. Two important concepts of these algebras with involution are the universal enveloping algebras and the reduced norm.Corrections of the 1st edition (1996) carried out on behalf of N. Jacobson (deceased) by Prof. P.M. Cohn (UC London, UK).

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