Projective Modules over Lie Algebras of Cartan Type

This monograph focuses on extending theorems for the classical Lie algebras in order to determine the structure and representation theory for Lie algebras of Cartan type. More specifically, Nakano investigates the block theory for the restricted universal enveloping algebras of the lie algebras of Cartan type. The first section employs techniques developed by Holmes and nakano to prove a Brauer-Humphreys reciprocity law for graded restricted lie algebras and also to find the decompositions for the intermediate (Verma) modules used in the reciprocity law. The second section uses this information to investigate the structure of projective modules for the Lie algebras of types W and K. The restricted enveloping algebras for these Lie algebras are shown to have one block. Furthermore, Nakano provides a procedure for computing the Cartan invariants for Lie algebras of types W and K, given knowledge about the decomposition of the generalized Verma modules and about the jantzen matrix of the classical/reductive zero component. Noteworthy for its readability and the continuity of its them and purpose, this monograph appeals to graduate students and researchers interested in Lie algebras.