Infinitedimensional Lie Algebras

Semisimple Lie groups, and their algebraic analogues over fields other than the reals, are of fundamental importance in geometry, analysis, and mathematical physics. Three independent, self-contained volumes, under the general title "Lie Theory," feature survey work and original results by well-established researchers in key areas of semisimple Lie theory.

(Cartan sub Lie algebra, roots, Weyl group, Dynkin diagram, . . . ) and the classification, as found by Killing and Cartan (the list of all semisimple Lie algebras consists of (1) the special- linear ones, i. e. all matrices (of any fixed dimension) with trace 0, (2) the orthogonal ones, i. e. all skewsymmetric ma­ trices (of any fixed dimension), (3) the symplectic ones, i. e. all matrices M (of any fixed even dimension) that satisfy M J = - J MT with a certain non-degenerate skewsymmetric matrix J, and (4) five special Lie algebras G2, F , E , E , E , of dimensions 14,52,78,133,248, the "exceptional Lie 4 6 7 s algebras" , that just somehow appear in the process). There is also a discus­ sion of the compact form and other real forms of a (complex) semisimple Lie algebra, and a section on automorphisms. The third chapter brings the theory of the finite dimensional representations of a semisimple Lie alge­ bra, with the highest or extreme weight as central notion. The proof for the existence of representations is an ad hoc version of the present standard proof, but avoids explicit use of the Poincare-Birkhoff-Witt theorem. Complete reducibility is proved, as usual, with J. H. C. Whitehead's proof (the first proof, by H. Weyl, was analytical-topological and used the exis­ tence of a compact form of the group in question). Then come H.

In several proofs from the theory of finite-dimensional Lie algebras, an essential contribution comes from the Jordan canonical structure of linear maps acting on finite-dimensional vector spaces. On the other hand, there exist classical results concerning Lie algebras which advise us to use infinite-dimensional vector spaces as well. For example, the classical Lie Theorem asserts that all finite-dimensional irreducible representations of solvable Lie algebras are one-dimensional. Hence, from this point of view, the solvable Lie algebras cannot be distinguished from one another, that is, they cannot be classified. Even this example alone urges the infinite-dimensional vector spaces to appear on the stage. But the structure of linear maps on such a space is too little understood; for these linear maps one cannot speak about something like the Jordan canonical structure of matrices. Fortunately there exists a large class of linear maps on vector spaces of arbi­ trary dimension, having some common features with the matrices. We mean the bounded linear operators on a complex Banach space. Certain types of bounded operators (such as the Dunford spectral, Foia§ decomposable, scalar generalized or Colojoara spectral generalized operators) actually even enjoy a kind of Jordan decomposition theorem. One of the aims of the present book is to expound the most important results obtained until now by using bounded operators in the study of Lie algebras.