Lie Groups, Lie Algebras, and Representations: An Elementary Introduction (Hardcover)
暫譯: 李群、李代數與表示:初步介紹 (精裝版)

Brian Hall

  • 出版商: Springer
  • 出版日期: 2015-05-22
  • 售價: $3,180
  • 貴賓價: 9.5$3,021
  • 語言: 英文
  • 頁數: 453
  • 裝訂: Hardcover
  • ISBN: 3319134663
  • ISBN-13: 9783319134666
  • 海外代購書籍(需單獨結帳)

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商品描述

This textbook treats Lie groups, Lie algebras and their representations in an elementary but fully rigorous fashion requiring minimal prerequisites. In particular, the theory of matrix Lie groups and their Lie algebras is developed using only linear algebra, and more motivation and intuition for proofs is provided than in most classic texts on the subject.

In addition to its accessible treatment of the basic theory of Lie groups and Lie algebras, the book is also noteworthy for including:

  • a treatment of the Baker–Campbell–Hausdorff formula and its use in place of the Frobenius theorem to establish deeper results about the relationship between Lie groups and Lie algebras
  • motivation for the machinery of roots, weights and the Weyl group via a concrete and detailed exposition of the representation theory of sl(3;C)
  • an unconventional definition of semisimplicity that allows for a rapid development of the structure theory of semisimple Lie algebras
  • a self-contained construction of the representations of compact groups, independent of Lie-algebraic arguments

The second edition of Lie Groups, Lie Algebras, and Representations contains many substantial improvements and additions, among them: an entirely new part devoted to the structure and representation theory of compact Lie groups; a complete derivation of the main properties of root systems; the construction of finite-dimensional representations of semisimple Lie algebras has been elaborated; a treatment of universal enveloping algebras, including a proof of the Poincaré–Birkhoff–Witt theorem and the existence of Verma modules; complete proofs of the Weyl character formula, the Weyl dimension formula and the Kostant multiplicity formula.

Review of the first edition:

This is an excellent book. It deserves to, and undoubtedly will, become the standard text for early graduate courses in Lie group theory ... an important addition to the textbook literature ... it is highly recommended.

― The Mathematical Gazette

商品描述(中文翻譯)

這本教科書以基本但完全嚴謹的方式處理李群、李代數及其表示,並要求的先備知識最少。特別是,矩陣李群及其李代數的理論僅使用線性代數進行發展,並提供比大多數經典文本更多的動機和直覺來支持證明。

除了對李群和李代數基本理論的易懂處理外,本書還值得注意的是包含:

- 對貝克-坎貝爾-豪斯多夫公式的處理及其在建立李群與李代數之間關係的更深結果中取代弗羅貝尼烏斯定理的使用
- 通過對 sl(3; C) 表示理論的具體和詳細闡述,提供根、權重和韋爾群機制的動機
- 一個非常規的半簡單性定義,允許快速發展半簡單李代數的結構理論
- 一個獨立於李代數論證的緊湊群表示的自足構造

《李群、李代數及其表示》的第二版包含許多實質性的改進和新增內容,其中包括:專門用於緊湊李群的結構和表示理論的全新部分;根系主要性質的完整推導;對半簡單李代數的有限維表示的構造已被詳細闡述;對普遍包絡代數的處理,包括對龐加萊-比爾科夫-維特定理的證明及維爾瑪模塊的存在;韋爾特徵公式、韋爾維度公式和科斯坦特重數公式的完整證明。

**第一版的評價**:

*這是一本優秀的書籍。它值得成為,並且無疑會成為李群理論早期研究生課程的標準教材……對教科書文獻的重要補充……強烈推薦。*

― 數學公報

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