2023 Fiscal Year Final Research Report
Representation theory of elliptic quantum toroidal algebras and its application to integrable systems
| Project/Area Number |
21K03191
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 11010:Algebra-related
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| Research Institution | Aichi Institute of Technology |
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Principal Investigator |
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| Project Period (FY) |
2021-04-01 – 2024-03-31
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| Keywords | Elliptic analogue / Quantum toroidal algebra / representaiton / vertex operator / W algebra |
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| Outline of Final Research Achievements |
We introduce an elliptic analogue of quantum toroidal algebras which are an affinization of quantum affine algebras. Quantum affine algebras play an important role in solvable lattice models or integrable systems. We study some applications for integrable systems using the representation theories of the elliptic quantum toroidal algebras.
Quantum toroidal algebras can be defined for any simple Lie algebras, among others, quantum toroidal algebra for type A is most interesting because it admits quntum deformation parameter and one more parameter. Considering an elliptic analogue, the elliptic quantum toroidal algebra for A type has 3 parameter including the elliptic deformation parameter, which we show the relation between the elliptic quantum toroidal algebra and 5 dimensional or 6 dimensional Gauge theories.
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| Free Research Field |
Mathematical Physics
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| Academic Significance and Societal Importance of the Research Achievements |
可解格子模型や可積分系で重要な役割を果たしているアフィン量子群の自然な拡張を考えることは重要である．本研究はアフィン量子群のアフィン化である量子トロイダル代数と，アフィン量子群の楕円化である楕円量子群との両面を併せ持った自然な拡張である．楕円量子トロイダル代数においても，有限次元表現やフォック表現など基本的な表現を構成することが可能であり，それらのテンソル積上に頂点作用素を構成することによりW代数を導出するなど，可積分系への応用が可能であることを示した．
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