This study has tended to develop in numerous specialized fields within mathematics and physics over the last few decades, particularly in the Langlands program. Here Humphreys (mathematics, U. of Massachusetts) focuses on algebraic methods in the case of the semisimple Lie algebra g over C with universal enveloping algebra U(g) where the prerequisites are modest. Humphreys offers his material first in a textbook style at the level of a second year graduate course at a US university, reviewing semisimple Lie algebras, then covering the basics of category O, characters of finite dimensional modules, Category O methods, highest weight modules, extensions and resolutions, translation functors, and Kazhdan-Lusztig theory. He presents more advanced materials for self-study, including parabolic versions of Category O, projective functors and principal series, tilting modules, twisting and completion functors, and complements. He also includes a well-chosen bibliography for further research. Annotation ©2008 Book News, Inc., Portland, OR (booknews.com)