授業の目的 【日本語】
Goals of the Course(JPN) | | The theme of this course is "Quasi-hereditary algebras".
As the name suggests, they generalise hereditary algebras (rings) - such as path algebra of finite acyclic quivers.
They often appear in algebraic Lie theory - highest weight representations of Lie algebras, polynomial representations of algebraic groups, etc. The most distinctive feature of these algebras is that their homological algebra are `stratified' by the homological-trivial algebras (such as fields and matrix algebra). The stratification data involves a partially ordered set, which has deep tie with the combinatorics appearing in algebraic Lie theory. Moreover, they have finite global dimension - a `smooth property' - and are intimately related to non-commutative resolution of singularity.
The first half of the course aims at explaining the necessary homological algebra for understanding the beauty of quasi-hereditary algebras. This part includes basic of quiver representations which allow us to write down many examples easily. The second half of the course will dive deep into the abstract (ring theoretic) theory of quasi-hereditary algebras, and some results around them. |
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授業の目的 【英語】
Goals of the Course | | The theme of this course is "Quasi-hereditary algebras".
As the name suggests, they generalise hereditary algebras (rings) - such as path algebra of finite acyclic quivers.
They often appear in algebraic Lie theory - highest weight representations of Lie algebras, polynomial representations of algebraic groups, etc. The most distinctive feature of these algebras is that their homological algebra are `stratified' by the homological-trivial algebras (such as fields and matrix algebra). The stratification data involves a partially ordered set, which has deep tie with the combinatorics appearing in algebraic Lie theory. Moreover, they have finite global dimension - a `smooth property' - and are intimately related to non-commutative resolution of singularity.
The first half of the course aims at explaining the necessary homological algebra for understanding the beauty of quasi-hereditary algebras. This part includes basic of quiver representations which allow us to write down many examples easily. The second half of the course will dive deep into the abstract (ring theoretic) theory of quasi-hereditary algebras, and some results around them. |
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到達目標 【日本語】
Objectives of the Course(JPN)) | | (1) Understanding of basic homological algebra,
(2) Being able to compute representations of the path algebras of quivers with relations.
(3) Being able to utilise the inductive homological property of quasi-hereditary algebras. |
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到達目標 【英語】
Objectives of the Course | | (1) Understanding of basic homological algebra,
(2) Being able to compute representations of the path algebras of quivers with relations.
(3) Being able to utilise the inductive homological property of quasi-hereditary algebras. |
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授業の内容や構成
Course Content / Plan | | Part 1:
- Reminder on rings, algebras, modules, representations
- Some basic homological algebra, including tensor-Hom adjunction, exact sequences, Tor, Ext-groups
- possibly some basic category theory
Part 2:
- Representations of quvier and path algebras
- either theory of highest weight representations of complex semisimple algebra, or Schur algebra as endomorphism of permutation modules over symmetric groups
Part 3:
- Quasi-hereditary algebra
- Characteristic tilting and Ringel duality (to be confirmed)
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 | | | Linear algebra is essential. Knowledge of rings-and-modules will be very helpful. |
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関連する科目
Related Courses | | | Any course on rings and modules |
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成績評価の方法と基準
Course Evaluation Method and Criteria | | | Homework assignment (only) |
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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 | | Krause - Homological theory of representations (available online)
Assem-Simson-Skowronski - Elements of the representation theory of associative algebras vol. 1
Klucznik-Koenig - (Lecture notes) Characteristic tilting modules over quasi-hereditary algebras
For detailed treatment of homological algebras:
Rotman - An introduction to homological algebra
Weibel - An introduction to homological algebra
More refrence books will be provided as the course goes on. |
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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) | | Students are expected to revise material after each lecture, and be self-motivated to seek help from the lecturer whenever necessary.
There will be three to four homework assignments to complete outside lecture time. |
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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, representation theory, representation of quivers, quasi-hereditary algebras, highest weight representations |
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履修の際のアドバイス
Advice | | | It is advised to check my webpage http://aaronkychan.github.io/teaching/ weekly (after 1 day after lecture finish) for any information and lecture notes update. |
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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) | | | Any information will be updated on the webpage http://aaronkychan.github.io/teaching/ |
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