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Jordan Structures in Lie Algebras

Antonio Fernández López
Publication Date: 
Number of Pages: 
Mathematical Surveys and Monographs
[Reviewed by
Mark Hunacek
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Many years ago, when my wife and I got engaged, a number of our mathematically-minded friends joked about the difficulties we would face in making a mixed marriage work. Our marriage was “mixed” not because we were of different races or nationalities, but because I was writing my doctoral thesis in the area of Lie algebras and she was writing hers in the area of Jordan algebras, these being the two major different (and, our friends joked, incompatible) kinds of nonassociative algebras. 

Had I the benefit of the book now under review back then, I would have realized that the two subjects are not so incompatible after all. Indeed, that is the subject matter of the book now under review: to use Jordan theory to gain information about Lie algebras. The “Jordan structures” of the title include not just Jordan algebras but also things called “Jordan pairs”; one finds both of these things lurking in Lie algebras, and they can be used to study the Lie algebra itself.

Let me attempt to elaborate briefly on these vague comments. A Lie algebra is a vector space over a field on which there is defined a “bracket operation” [x, y] satisfying certain identities; a Jordan algebra is a vector space on which there is defined an operation x ∘y, satisfying different identities. (Actually, in this book, the scalars need not form a field, but rather a commutative ring with identity in which certain elements are invertible). If L is a Lie algebra, certain elements called Jordan elements are defined, and, to each one, a Jordan algebra is associated. It turns out that this Jordan algebra reflects certain properties of the Lie algebra and can be used to say interesting things about it.

In addition to Jordan algebras, there are, as noted above, also objects called Jordan pairs, which consist of a pair of modules and two trilinear mappings satisfying yet more technical conditions. It turns out that certain kinds of submodules of a Lie algebra (the technical term is “abelian inner ideal”) can have associated to them a Jordan pair, which again manages to capture properties of the Lie algebra. 

This sounds like pretty technical stuff, and in fact it is: a lot of the material covered here comes straight from research papers, many of them written by the author and/or his collaborators. There is also an extensive (eight pages of small type) bibliography, much of the references being to journal articles. However, the author has made a real effort to make this material as accessible as possible to an audience of non-specialists. The definitions of Lie algebras and Jordan algebras are provided, rather than assumed, and early chapters provide background information. These chapters are written at what I would estimate to be the level of a second or third year graduate student; obviously a one-year graduate algebra course is a prerequisite for the book, and some prior exposure to Lie algebras would be useful as well. The book is written in the style of a textbook, rather than a research monograph; it even comes complete with exercises. For somebody contemplating entering this area, this book should prove very valuable.

To summarize and conclude: this book is not going to attract a very broad, general audience, but it will likely garner an appreciative one. 

Mark Hunacek ( teaches mathematics at Iowa State University.