2011 Fiscal Year Final Research Report
Geometric study of quantum groups and its application to representation theory of algebras
| Project/Area Number |
20540009
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | The University of Tokyo |
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Principal Investigator |
SAITO Yoshihisa 東京大学, 大学院・数理科学研究科, 准教授 (20294522)
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| Co-Investigator(Renkei-kenkyūsha) |
IYAMA Osamu 名古屋大学, 大学院・多元数理研究科, 教授 (70347532)
SAITO Kyoji 東京大学, 数物連携宇宙機構, 主任研究員 (20012445)
TANISAKI Toshiyuki 大阪市立大学, 理学研究科, 教授 (70142916)
NAITO Satoshi 東京工業大学, 大学院・理工学研究科, 教授 (60252160)
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| Project Period (FY) |
2008 – 2011
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| Keywords | 量子群 / 結晶基底 |
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| Research Abstract |
Around 2000, Mirkovic and Vilonen defined a new class of algebraic cycles in affine Grassmanians, called Mirkovic-Vilonen(MV for short) cycles. By the definition, these cycles has an action of a real maximal Torus, and their moment map image are plopytoles in a real Cartan subalgebra, called MV polytopes. After their work, Kamnitzer defined a crstal structure on the set of all MV polytopes, and proved that it is isomorphic to the crystal basis of the negative half of quantum universal enveloping algebras of finite type. Moreover, it is known that the followings are eqivalent : to give a MV polytope, and to give a collection of nonnegative integer, called Berenstein-Zelevinsky(BZ for short) data. In other words, he introduced a new realization of the crystal basis of the negative half of quantum universal enveloping algebras of finite type, by using BZ data.
In this study, we generalize his result to the case of affine type A. More precisely, we define a notion of affine BZ data, and prove that the set of all affine BZ data has a crystal structure which are isomorphic to the negative half of quantum universal enveloping algebras of affine type A. Namely, we get a new realization of the crystal basis of the negative half of quantum universal enveloping algebras of affine type A, by using affine BZ data. Our main results are stated by combinatorial language, but in our proof, a geometric construction of crystal basis due to Kashiwara and the author plays a crucial role.
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