The theory of Jordan algebras has played important roles behind the scenes of several areas of mathematics. Jacobson's book has long been the definitive treatment of the subject. It covers foundational material, structure theory, and representation theory for Jordan algebras. Of course, there are immediate connections with Lie algebras, which Jacobson details in Chapter 8. Of particular continuing interest is the discussion of exceptional Jordan algebras, which serve to explain the exceptional Lie algebras and Lie groups. Jordan algebras originally arose in the attempts by Jordan, von Neumann, and Wigner to formulate the foundations of quantum mechanics. They are still useful and important in modern mathematical physics, as well as in Lie theory, geometry, and certain areas of analysis.
Foundations Elements of representation theory Peirce decompositions and Jordan matrix algebras Jordan algebras with minimum conditions on quadratic ideals Structure theory for finite-dimensional Jordan algebras Generic minimum polynomials, traces and norms Representation theory for separable Jordan algebras Connections with Lie algebras Exceptional Jordan algebras Further results and open questions Bibliography Subject index.