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課程名稱 |
緊緻李群與其表現
Compact Lie groups and their representations |
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開課學期 |
109-2 |
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授課對象 |
理學院 數學研究所 |
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授課教師 |
蔡政江 |
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課號 |
MATH5258 |
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課程識別碼 |
221 U8970 |
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班次 |
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學分 |
3.0 |
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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 |
星期一8,9(15:30~17:20)星期四5(12:20~13:10) |
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上課地點 |
天數304天數305 |
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備註 |
總人數上限：40人 |
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Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1092MATH5258 |
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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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課程概述 |
Theory of compact Lie groups and their representations, with an eye towards the general theory of reductive groups. |
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課程目標 |
Lie groups are manifolds that are also groups. They are very closely related to Lie algebras in various representation theories, and algebraic groups in algebraic and arithmetic geometry. The relation is particularly strong for the case for a common important sub-class - reductive groups/algebras. Reductive groups/algebras arise naturally in many ways, e.g. as a canonical quotient, as a centralizer of another, etc., and contains natural examples like unitary groups and orthogonal groups. Connected reductive groups/algebras are all governed by fairly simple discrete, combinatorial data called "root data", that is shared in the three categories - connected reductive Lie groups, algebras and connected reductive algebraic groups.
On the other hand, compact Lie groups arise naturally from general Lie groups. All compact Lie groups are reductive groups, and therefore governed by the aforementioned root data. To understand representations of general Lie groups it is necessary to understand that of compact Lie groups first, and (complex) representation theory of compact Lie groups are governed again by root data - a feature that is again shared by finite-dimensional representations of reductive Lie algebras and algebraic representations of reductive algebraic groups.
In this course, our emphasize will not be on the relation between compact Lie groups and general Lie groups nor on its application to geometry. Instead, we wish to use compact Lie groups as a starting point to introduce the yoga of reductive groups and root data. We will cover:
(1) Basic structure theory for Lie groups and its Lie algebras.
(2) The theorem that all maximal tori are conjugate in a connected compact Lie group, and from here the construction of associated root datum.
(3) Peter-Weyl theorem - how all irreducible representations of a compact Lie group G are to be realized in L^2(G).
(4) Classification of irreducible representations of connected compact Lie groups and their character formula in terms of the root data.
(5) Complexification of connected compact Lie groups, and the connection to algebraic groups.
After that, we might discuss some of the following material depending on time allowance:
(6) Classification of connected compact Lie groups and complex semisimple Lie algebras.
(7) Introduction to representation theory of non-compact Lie groups. |
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課程要求 |
Linear Algebra, introductory analysis and differential geometry (at the level of required undergraduate courses). To be precise, the audience are expected to be familiar with vector fields and differential forms on manifolds and their differential calculus.
In addition, after about 1/3 of the course we will use the notion of fundamental group and covering space which is probably covered in the first 15% of an algebraic topology course, and the audience are expected to either take the course or self-learn the subject. |
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預期每週課後學習時數 |
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Office Hours |
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指定閱讀 |
待補 |
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參考書目 |
Representations of compact Lie groups, Theodor Brocker and Tammo tom Dieck.
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評量方式
(僅供參考) |
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