This textbook treats Lie teams, Lie algebras and their representations in an hassle-free yet totally rigorous type requiring minimum must haves. specifically, the speculation of matrix Lie teams and their Lie algebras is built utilizing purely linear algebra, and extra motivation and instinct for proofs is supplied than in such a lot vintage texts at the subject.
In addition to its available remedy of the fundamental concept of Lie teams and Lie algebras, the publication can be noteworthy for including:
- a remedy of the Baker–Campbell–Hausdorff formulation and its use instead of the Frobenius theorem to set up deeper effects in regards to the courting among Lie teams and Lie algebras
- motivation for the equipment of roots, weights and the Weyl crew through a concrete and targeted exposition of the illustration conception of sl(3;C)
- an unconventional definition of semisimplicity that permits for a fast improvement of the constitution conception of semisimple Lie algebras
- a self-contained development of the representations of compact teams, autonomous of Lie-algebraic arguments
The moment variation of Lie teams, Lie Algebras, and Representations includes many vast advancements and additions, between them: a completely new half dedicated to the constitution and illustration idea of compact Lie teams; an entire derivation of the most homes of root platforms; the development of finite-dimensional representations of semisimple Lie algebras has been elaborated; a remedy of common enveloping algebras, together with an explanation of the Poincaré–Birkhoff–Witt theorem and the lifestyles of Verma modules; whole proofs of the Weyl personality formulation, the Weyl size formulation and the Kostant multiplicity formula.
Review of the 1st edition:
This is a wonderful e-book. It merits to, and certainly will, turn into the traditional textual content for early graduate classes in Lie team conception ... a major addition to the textbook literature ... it really is hugely recommended.
― The Mathematical Gazette