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Lie algebras of finite and affine type. (English) Zbl 1110.17001
Cambridge Studies in Advanced Mathematics 96. Cambridge: Cambridge University Press (ISBN 0-521-85138-6/hbk). xvii, 632 p. (2005).
The structure and representation theories of finite-dimensional simple (complex) Lie algebras and affine Kac-Moody algebras occupy an important place in modern mathematics. While the outer regions of Lie theory are experiencing rapid development, the core theory is now quite classical. In this book, the author provides a clear and focused exposition of this core theory, building from the beginning in patient steps, requiring only a ‘sound knowledge of linear algebra’.
The first thirteen chapters develop the theory of finite-dimensional simple Lie algebras starting from definitions of algebras, ideals, homomorphisms and nilpotent and solvable algebras. After proving the Lie and Engel theorems, the author introduces Cartan subalgebras and proves their conjugacy in simple algebras. Next, roots and root systems are explored, the Cartan matrix and Dynkin diagrams are brought to play and the simple Lie algebras are classified. With the structural classification in place the author turns to representations, classifying finite-dimensional representations for the semisimple Lie algebras and giving the various character, denominator and multiplicity formulas. The last chapter on the finite-dimensional case gives the construction of fundamental modules.
Turning to the infinite-dimensional case, the author introduces generalized Cartan matrices and shows how to construct a realization of a generalized Cartan matrix (GCM) and then a Kac-Moody algebra by factoring. Next, the author proves the trichotomy of indecomposable GCMs into finite, affine and indefinite types, introduces symmetrizable GCMs and then gives the classification of affine Kac-Moody algebras. All this has come from the GCM. The next three chapters are devoted to the construction of affine Kac-Moody algebras from loop algebras, using the theory of roots and Weyl groups, including the multiplicities of imaginary roots and the construction of twisted affine algebras as fixed point algebras. The author next introduces category $$\mathcal O$$ and the basics of representation theory for symmetrizable algebras, including again, character, denominator and multiplicity formulas. Specializing to affine algebras, the author gives a nice description of the weights of irreducible modules for affine algebras and fundamental modules. With the realization of basic representations via vertex operators, the author states, but does not prove the character formulas. Up until this point, every result has been proved in full. A final chapter concerns Borcherd algebras and the Monster.
Rounding out the book is a sequence of appendices and indexes. A handy 50-page appendix collects all the basic data on simple finite-dimensional and affine algebras; there is a thoughtful index of notation that gives the meaning as well as location of definition, a select bibliography and a general index.
The book provides a clear, patient and readable introduction to Lie algebras. The author and publishers are to be commended for generous layout and use of white space that enhances the lucid narrative. Every result is clearly and patiently proved, and the author continually signals where a proof is going. There are few examples and no exercises. The author has remained firmly focused on his main topic: there are no mentions of real forms, characteristic $$p$$, or Lie groups for example. However, as a text bringing together both the finite-dimensional and affine theories with minimal pre-requisites and no distraction, this book is an expository gem.

MSC:
 17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras