授業の目的 【日本語】
Goals of the Course(JPN) | | The theme of this course is "quiver representations".
A quiver (directed graph) provides a powerful tool in visualising an algebra (ring) and computing its modules.
The major part of the course aims at explaining the necessary ring-theoretic, categorical, and homological algebra working with quiver representations. The last part of the course will dive deep into the homological algebra, namely, of tilting modules and associated categories. We aim to develop a basic understanding of tilting theory, which has become a central technology in study homoloigcal algebra across various fields in modern days. |
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授業の目的 【英語】
Goals of the Course | | The theme of this course is "quiver representations".
A quiver (directed graph) provides a powerful tool in visualising an algebra (ring) and computing its modules.
The major part of the course aims at explaining the necessary ring-theoretic, categorical, and homological algebra working with quiver representations. The last part of the course will dive deep into the homological algebra, namely, of tilting modules and associated categories. We aim to develop a basic understanding of tilting theory, which has become a central technology in study homoloigcal algebra across various fields in modern days. |
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到達目標 【日本語】
Objectives of the Course(JPN)) | | 1. Understanding of fundamental ring-theoretic results, category theory, and homological algebra, that are used in the representation theory.
2. Being able to compute representations of the path algebras of quivers with relations, as well as their homological algebras (for example, computation of resolution complexes, Ext-groups, etc.) |
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到達目標 【英語】
Objectives of the Course | | 1. Understanding of fundamental ring-theoretic results, category theory, and homological algebra, that are used in the representation theory.
2. Being able to compute representations of the path algebras of quivers with relations, as well as their homological algebras (for example, computation of resolution complexes, Ext-groups, etc.) |
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授業の内容や構成
Course Content / Plan | | Part 1:
- Reminder on rings, algebras, modules, representations
- Introduction on the language of basic category theory, focusing on its use in representation theory.
Part 2:
- Representations of quvier and path algebras
- Basic homological algebra, including tensor-Hom adjunction, exact sequences, Tor, Ext-groups, etc.
Part 3:
- Computation of homological algebras using quiver representations
- Introduction to tilting modules
Plan is tentative and subject to change. All changes will be announced on my webpage http://aaronkychan.github.io/teaching/ |
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履修条件
Course Prerequisites | | | Essential knowledge includes linear algebra, ring, field, module. |
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関連する科目
Related Courses | | | Any course on rings and modules and homological algebras |
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成績評価の方法と基準
Course Evaluation Method and Criteria | | |
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教科書・テキスト
Textbook | | | There is no textbook, but lecture notes will be provided. |
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参考書
Reference Book | | Assem-Simson-Skowronski - Elements of the representation theory of associative algebras vol. 1
Erdmann-Holm - Algebras and Representation Theory |
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課外学習等（授業時間外学習の指示）
Study Load(Self-directed Learning Outside Course Hours) | | | Read the reference books if possible. Feel free to contact me for any consultation. |
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注意事項
Notice for Students | | |
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他学科聴講の可否
Propriety of Other department student's attendance | | |
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他学科聴講の条件
Conditions of Other department student's attendance | | |
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レベル
Level | | |
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キーワード
Keyword | | | Algebra, homological algebra, category theory, representation theory, representation of quivers, tilting theory |
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履修の際のアドバイス
Advice | | |
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授業開講形態等
Lecture format, etc. | | |
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遠隔授業（オンデマンド型）で行う場合の追加措置
Additional measures for remote class (on-demand class) | | |
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