2020 Fiscal Year Final Research Report
Representation theory of affine Yangians and integrable systems
| Project/Area Number |
18K13390
|
|---|
| Research Category |
Grant-in-Aid for Early-Career Scientists
|
| Allocation Type | Multi-year Fund |
|---|
| Review Section |
Basic Section 11010:Algebra-related
|
|---|
| Research Institution | Chiba University (2020)
Kobe University (2018-2019) |
|---|
Principal Investigator |
Kodera Ryosuke 千葉大学, 大学院理学研究院, 准教授 (20634512)
|
|---|
| Project Period (FY) |
2018-04-01 – 2021-03-31
|
| Keywords | アファインヤンギアン / 表現論 / 可積分系 / W代数 / トロイダルLie代数 / シフト量子アファイン代数 |
|---|
| Outline of Final Research Achievements |
We studied the structure theory and the representation theory of affine Yangians. We also studied the representation theory of related algebras.
For the affine Yangians, we studied properties of the evaluation map, and constructed algebra homomorphisms to affine W-algebras of rectangular type by composing it with the coproduct. They are significant results. Since the affine W-algebras are closely related to integrable systems, it is expected that we can apply the representation theory of the affine Yangians to the study of the integrable systems via the homomorphisms.
The following results were obtained as the studies on related algebras. We derived the characters of the level 1 Weyl modules of toroidal Lie algebras. We classified the finite-dimensional irreducible representations of (q,Q)-current algebras when q is not a root of unity.
|
| Free Research Field |
表現論
|
| Academic Significance and Societal Importance of the Research Achievements |
本研究によってアファインヤンギアンの構造論の理解が進んだことに伴って，長方形型アファインW代数との関係が明らかになった．特に，アファインヤンギアンのテンソル積表現と長方形型アファインW代数の放物誘導が対応することを示したことで，テンソル積表現の研究の重要性が再確認された．この成果は，今後可積分系の研究への応用を考えるうえでも重要だと考えている．
トロイダルLie代数のWeyl加群の研究は，アファインヤンギアンの表現論の理解のために役立つことが期待されるとともに，それ自体が特殊函数論的な観点からも興味深い．本研究の成果はWeyl加群の研究の第一歩となるものである．
|
