Jordan structures in Lie algebras
This is neither a book on Lie algebras nor on Jordan systems. This
book is just intended to show how Jordan theory can be applied to
Lie algebras. In one sense it is introductory, since very few
prerequisites on Lie algebras are presupposed. In fact, Lie
readers, especially those accustomed to working with root systems
or with the fruitful relationship between Lie groups and Lie
algebras, might be disappointed by the fact that most of the Lie
algebras considered here are over a ring of scalars where only the
invertibility of 2, 3, and sometimes 5, is assumed, and even in
the case of Lie algebras over a field, these are not necessarily
finite-dimensional. Yet in this poor soil a germinal form of root
system can still grow: the grading induced by a compatible family
of idempotents (a notion which is defined in Jordan terms, and
which connects with the Jacobson--Morozov lemma) resembles in a
broad sense the root decomposition associated to a root system. On
the other hand, the book is far from being elementary. We allow
ourselves to use any Jordan result under the sun if this were
required for solving a single Lie question.
(Mathematical Surveys and Monographs of the
American Mathematical Society, Volume 240)
A. Fernández López < emalfer@uma.es >