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課程名稱 |
李群與李代數
Lie groups and Lie algebras |
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開課學期 |
111-1 |
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授課對象 |
理學院 數學研究所 |
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授課教師 |
王金龍 |
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課號 |
MATH5278 |
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課程識別碼 |
221 U9500 |
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班次 |
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學分 |
4.0 |
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全/半年 |
半年 |
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必/選修 |
選修 |
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上課時間 |
星期一3,4(10:20~12:10)星期三6,7(13:20~15:10) |
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上課地點 |
天數202天數102 |
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備註 |
限學士班二年級以上 且 限本系所學生(含輔系、雙修生)
總人數上限：40人 |
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課程簡介影片 |
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核心能力關聯 |
本課程尚未建立核心能力關連 |
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課程大綱
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為確保您我的權利,請尊重智慧財產權及不得非法影印
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課程概述 |
Basics on Lie groups. Classification of semi-simple Lie algebras, real and complex. Constructions of representations. |
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課程目標 |
Lie theory, either Lie groups or Lie algebras, appears everywhere in modern sciences. The course is intended to provide the basic techniques toward a classification result of Lie theory, as well as computational skills in special functions attached to it, and basic representation theory. Two more geometric aspects will also be discussed: (1) classifications of symmetric spaces, (2) basic geometric representation theory. |
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課程要求 |
Undergraduate Algebra I, II and Geometry, including the basics on manifolds. |
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預期每週課後學習時數 |
6 to 10 hours |
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Office Hours |
另約時間 |
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指定閱讀 |
Hall: Lie Groups, Lie Algebras and Representations
Humphreys: Introduction to Lie Algebras and Representation Theory, GTM 9 |
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參考書目 |
Fulton and Harris: Representation Theory, a first course
Helgason: Differential Geometry, Lie Groups and Symmetry Spaces
Knapp: Lie groups beyond an introduction
Sepanski: Compact Lie groups, GTM 235 |
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評量方式
(僅供參考) |
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針對學生困難提供學生調整方式 |
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上課形式 |
以錄影輔助 |
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作業繳交方式 |
書面報告取代口頭報告 |
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考試形式 |
考試取代書面(口頭)報告 |
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其他 |
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週次 |
日期 |
單元主題 |
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Week 1 |
9/5, 9/7 |
Engel, Lie, and Cartan's criterion for solvability |
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Week 2 |
9/12, 9/14 |
Semisimple Lie algebras and representations of sl(2) |
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Week 3 |
9/19, 9/21 |
Root systems of semisimple Lie algebras |
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Week 4 |
9/26 |
Classifications and constructions of root systems |
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Week 5 |
10/3, 10/5 |
Cartan and Borel sub-algebras, conjugacy theorems |
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Week 6 |
10/12 |
Existence and uniqueness theorem of semisimple Lie algebras |
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Week 7 |
10/17, 10/19 |
Representations via highest weights |
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Week 8 |
10/24, 10/26 |
Character theory and explicit formulas |
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Week 9 |
10/31, 11/2 |
Proof of char formulas, examples of compact Lie groups |
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Week 10 |
11/7, 11/9 |
Basic topology of Lie groups, examples of representations |
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Week 11 |
11/14, 11/16 |
Midterm Exam |
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Week 12 |
11/21, 11/23 |
Schur's lemma, orthogonality and characters |
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Week 13 |
11/28, 11/30 |
The Peter--Weyl theorem |
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Week 14 |
12/5, 12/7 |
Fourier theory, from Lie groups to Lie algebras |
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Week 15 |
12/12, 12/14 |
Abelian Lie subgroups and highest weight theory revisited |
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Week 16 |
12/19, 12/21 |
Borel--Weil theorem, and Final Exam |
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