課程名稱 |
李代數 Lie Algebra |
開課學期 |
107-2 |
授課對象 |
理學院 數學研究所 |
授課教師 |
林惠雯 |
課號 |
MATH5098 |
課程識別碼 |
221 U8430 |
班次 |
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學分 |
2.0 |
全/半年 |
半年 |
必/選修 |
選修 |
上課時間 |
星期二3,4(10:20~12:10) |
上課地點 |
天數102 |
備註 |
總人數上限:30人 |
Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1072MATH5098_Lie |
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 |
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 |
1. Semisimple Lie algebras:
Engel's theorem, Lie's theorem, Cartan's criterion, Semisimplicity of Lie algebras,
Representations of sl(2, F), Orthogonality + Integrality + Rationality.
2. Root systems:
Simple roots, Weyl chambers, Weyl group, Dynkin diagrams, Classification theorem.
3. Isomorphism and conjugacy theorems:
Isomorphism theorem, Cartan subalgebras, Conjugacy theorems, Conjugacy of Borel
subalgebras.
4. Existence theorem:
Universal enveloping algebra, Poincare-Birkhoff-Witt theorem, Serre's theorem, Criterion for
semisimplicity.
5. Representation theory:
Weights, Weight strings and weight diagrams, Casimir element, Freudenthal's formula,
Characters, Harish-Chandra's theorem, Kostant's multiplicity formula, Weyl's formula,.
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課程目標 |
The purpose of this course is to introduce the theory of Lie algebras, specifically on the structure and the (finite dimensional) representations of the semisimple Lie algebras. We hope to equip students with a good foundation in Lie theory.. |
課程要求 |
先修:Linear algebra, Undergraduate algebra. |
預期每週課後學習時數 |
|
Office Hours |
另約時間 |
指定閱讀 |
J. E. Humphreys: Introduction to Lie algebras and Representation theory |
參考書目 |
Anthony W. Knapp, Lie groups beyond an introduction, 2nd edition. |
評量方式 (僅供參考) |
No. |
項目 |
百分比 |
說明 |
1. |
作業 |
30% |
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2. |
期中考 |
35% |
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3. |
期末考 |
35% |
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週次 |
日期 |
單元主題 |
第1週 |
2/19 |
Engel's theorem (Sec.1: 1, 3; Sec.2: 7; Sec.3: 6, 10) |
第2週 |
2/26 |
Lie's theorem and Cartan's criterion (Sec. 4: 1, 3, 5) |
第3週 |
3/05 |
Semisimplicity of Lie algebras (Sec. 5: 2, 4; Sec. 6: 3, 5) |
第4週 |
3/12 |
Representations of sl(2, F) (Sec. 7: 2, 4, 5, 6) |
第5週 |
3/19 |
Root space decomposition : Orthogonality + Integrality + Rationality (Sec. 8: 2, 5, 7, 8, 9) |
第6週 |
3/26 |
Root system (Sec. 9: 4, 6, 9; Sec. 10: 2, 5) |
第7週 |
4/02 |
春假 |
第8週 |
4/09 |
Cartan's matrices and Dynkin diagram (Sec. 10: 10, 11, 13; Sec. 11: 1, 2) |
第9週 |
4/16 |
Classification (Sec. 11: 4,5; Sec. 12: 3, 4) |
第10週 |
4/23 |
期中考 |
第11週 |
4/30 |
Isomorphism theorem (Sec. 14: 4, 6; Sec. 15: 1, 3, 4) |
第12週 |
5/07 |
Conjugacy theorems (Sec. 16: 1, 3, 5, 6 (*7: 加分題)) |
第13週 |
5/14 |
Poincare-Birkhoff-Witt theorem (Sec. 17: 1, 2, 3, 4) |
第14週 |
5/21 |
Serre's theorem (Sec. 18:1; Ex: Construct the simple algebra of type G_2) |
第15週 |
5/28 |
Theorem of the highest weight (作業課堂上提供) |
第16週 |
6/04 |
Characters (作業課堂上提供) |
第17週 |
6/11 |
Related formulas |
第18週 |
6/18 |
期末考 |
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