### 2018　Special lectures on advanced topics in Mathematics B

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Instructor(s)
Naito Satoshi  Naoi Katsuyuki
Course component(s)
Lecture
Day/Period(Room No.)
Intensive ()
Group
-
Course number
MTH.E432
Credits
2
2018
Offered quarter
3Q
Syllabus updated
2018/3/20
Lecture notes updated
-
Language used
Japanese
Access Index

### Course description and aims

A current algebra is a Lie algebra defined to be the tensor product of a finite-dimensional simple Lie algebra and the ring of polynomials in one indeterminate. The original motivation for the study of the representation theory of current algebras is the application to that of quantum loop algebras, but nowadays it is studied in various contexts such as the structure of an affine highest weight category, the relation with Macdonald polynomials, and so on.

In this lecture, we give a talk about some topics on the representation theory of current algebras; representations of current algebras are not completely reducible, and hence the study of these is not easy. The main goal of this lecture is to give a detailed exposition on local/global Weyl modules, which are important examples of representations of current algebras, and also on the affine highest weight structure of the category of graded representations of current algebras. The purpose of this lecture is to introduce several methods for studying representations of current algebras through the topics above.

### Student learning outcomes

・Understand the method for studying representations of current algebras by using generators and relations.

・Understand the structure of local/global Weyl modules.

・Understand the affine highest weight　structure of the category of　graded representations of current algebras.

### Keywords

current algebra, local Weyl module, global Weyl module, Demazure module, affine highest weight category

### Competencies that will be developed

Intercultural skills Communication skills Specialist skills Critical thinking skills Practical and/or problem-solving skills
- - - -

### Class flow

This is a standard lecture course; there will be some assignments.

### Course schedule/Required learning

Course schedule Required learning
Class 1 Basics on current algebras and their representations 1 Details will be provided during each class session.
Class 2 Basics on current algebras and their representations 2 Details will be provided during each class session.
Class 3 Basics on current algebras and their representations 3 Details will be provided during each class session.
Class 4 Local Weyl modules 1 Details will be provided during each class session.
Class 5 Local Weyl modules 2 Details will be provided during each class session.
Class 6 Local Weyl modules 3 Details will be provided during each class session.
Class 7 Global Weyl modules 1 Details will be provided during each class session.
Class 8 Global Weyl modules 2 Details will be provided during each class session.
Class 9 Global Weyl modules 3 Details will be provided during each class session.
Class 10 Affine highest weight category 1 Details will be provided during each class session.
Class 11 Affine highest weight category 2 Details will be provided during each class session.
Class 12 Affine highest weight category 3 Details will be provided during each class session.
Class 13 Affine highest weight structure of the category of graded representations 1 Details will be provided during each class session.
Class 14 Affine highest weight structure of the category of graded representations 2 Details will be provided during each class session.
Class 15 Affine highest weight structure of the category of graded representations 3 Details will be provided during each class session.

### Textbook(s)

None in particular.

### Reference books, course materials, etc.

None in particular.

### Assessment criteria and methods

By assignments (100%); details will be provided in the class.

### Related courses

• MTH.A201 ： Introduction to Algebra I
• MTH.A202 ： Introduction to Algebra II
• MTH.A203 ： Introduction to Algebra III
• MTH.A204 ： Introduction to Algebra IV
• MTH.A301 ： Algebra I
• MTH.A302 ： Algebra II

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

None in particular.