Constructions of Lie Algebras and their Modules

1. Front Matter

   Pages I-VI

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2. An introductory example: sl(n,D)

   • George B. Seligman

   Pages 1-11

3. General considerations

   • George B. Seligman

   Pages 12-20

4. Involutorial algebras and modules for their skew elements

   • George B. Seligman

   Pages 21-36

5. Construction of modules with prescribed relative highest weights, for the isotropic algebras of chapter 3

   • George B. Seligman

   Pages 37-51

6. Construction of exceptional algebras from quadratic forms

   • George B. Seligman

   Pages 52-83

7. Representations of exceptional algebras constructed from quadratic forms

   • George B. Seligman

   Pages 84-114

8. Non-reduced excepticnal algebras with a one-dimensional root space

   • George B. Seligman

   Pages 115-140

9. Construction of modules for the super-exceptional algebras of rank one

   • George B. Seligman

   Pages 141-178

10. Complements

    • George B. Seligman

    Pages 179-185

11. Back Matter

    Pages 186-195

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This book deals with central simple Lie algebras over arbitrary fields of characteristic zero. It aims to give constructions of the algebras and their finite-dimensional modules in terms that are rational with respect to the given ground field. All isotropic algebras with non-reduced relative root systems are treated, along with classical anisotropic algebras. The latter are treated by what seems to be a novel device, namely by studying certain modules for isotropic classical algebras in which they are embedded. In this development, symmetric powers of central simple associative algebras, along with generalized even Clifford algebras of involutorial algebras, play central roles. Considerable attention is given to exceptional algebras. The pace is that of a rather expansive research monograph. The reader who has at hand a standard introductory text on Lie algebras, such as Jacobson or Humphreys, should be in a position to understand the results. More technical matters arise in some of the detailed arguments. The book is intended for researchers and students of algebraic Lie theory, as well as for other researchers who are seeking explicit realizations of algebras or modules. It will probably be more useful as a resource to be dipped into, than as a text to be worked straight through.

• algebra
• associative algebra
• clifford algebra
• field
• lie algebra
• quadratic form

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