By Brian Hall
This textbook treats Lie teams, Lie algebras and their representations in an ordinary yet totally rigorous model requiring minimum must haves. specifically, the idea of matrix Lie teams and their Lie algebras is constructed utilizing simply linear algebra, and extra motivation and instinct for proofs is supplied than in so much vintage texts at the subject.
In addition to its available remedy of the elemental idea of Lie teams and Lie algebras, the booklet can be noteworthy for including:
- a therapy of the Baker–Campbell–Hausdorff formulation and its use as opposed to the Frobenius theorem to set up deeper effects in regards to the dating among Lie teams and Lie algebras
- motivation for the equipment of roots, weights and the Weyl workforce through a concrete and exact exposition of the illustration thought of sl(3;C)
- an unconventional definition of semisimplicity that enables for a speedy improvement of the constitution thought of semisimple Lie algebras
- a self-contained development of the representations of compact teams, autonomous of Lie-algebraic arguments
The moment version of Lie teams, Lie Algebras, and Representations includes many sizeable advancements and additions, between them: a wholly new half dedicated to the constitution and illustration idea of compact Lie teams; an entire derivation of the most homes of root structures; 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; entire proofs of the Weyl personality formulation, the Weyl measurement formulation and the Kostant multiplicity formula.
Review of the 1st edition:
This is a superb e-book. It merits to, and unquestionably will, develop into the traditional textual content for early graduate classes in Lie staff thought ... a huge addition to the textbook literature ... it's hugely recommended.
— The Mathematical Gazette