Study on the fusion products and crystalbases of level-zero representations of quantum affine algebras

量子仿射代数零级表示的融合积和晶基研究

• 批准号: 17540008
• 负责人: NAITO Satoshi
• 金额: $ 2.35万
• 依托单位: University of Tsukuba
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17540008/
• 关键词: quantum affine algebra; level-zero representation; extremal weight module; crystal basis; Lakshmibai-Seshadri path; path model; energy function; Kostka polynomial; 量子群; 既約最高ウェイト加群; Mirkovic-Vilonen多面体; tropical Plucker関係式; Anderson-Mirkovic予想

Let g be an affine Lie algebra over the complex numbers, and let λ be a level-zero integral weight that is a sum of level-zero fundamental weights π_I, I=1, . N, with repetitions allowed. We consider the quantum Weyl module W_{q} (λ) over the quantum affine algebra U_{q} (g) associated to certain Drinfeld polynomials corresponding to λ. It is known that the classical limit (I. e., "q=1" limit) of W_{q} (λ) becomes the Weyl module W (λ) over the affine Lie algebra g. Also, it is known that the Weyl module W (λ) is isomorphic (as a module over the Current algebra corresponding to g) to a fusion product of the Weyl modules W (π_I), I=1, . N.In our previous works, we showed that the crystal B (λ)_{cl} of Lakshmibai-Seshadri paths (modulo the null root δ of g) of shape λ is isomorphic as a crystal to the crystal basis of the quantum Weyl module W_{q} (λ). Moreover, we showed that the crystal B (λ)_{cl} is isomorphic as a crystal to a tensor product B of the crystals B (π_I), I=1, ., nIn our series of works from 2005 to 2007, we defined a certain (nonnegative) integer-valued function (which we call the degree function) on the crystal B (λ)_{cl} above, and proved that this degree function can be identified (through the isomorphism between B (λ)_{cl} and B) with the "energy function" on B, which arose from the study of solvable lattice models in statistical mechanics. In particular, by restricting ourselves to the case of affine Lie algebras of type A, we obtain a description of Kostka polynomials in terms of Lakshmibai-Seshadri paths.

Let g be an affine Lie algebra over the complex numbers, and let λ be a level-zero integral weight that is a sum of level-zero fundamental weights π_I, I=1, . N, with repetitions allowed. We consider the quantum Weyl module W_{q} (λ) over the quantum affine algebra U_{q} (g) associated to certain Drinfeld polynomials corresponding to λ. It is known that the classical limit (I. e., "q=1" limit) of W_{q} (λ) becomes the Weyl module W (λ) over the affine Lie algebra g. Also, it is known that the Weyl module W (λ) is isomorphic (as a module over the Current algebra corresponding to g) to a fusion product of the Weyl modules W (π_I), I=1, . N.In our previous works, we showed that the crystal B (λ)_{cl} of Lakshmibai-Seshadri paths (modulo the null root δ of g) of shape λ is isomorphic as a crystal to the crystal basis of the quantum Weyl module W_{q} (λ). Moreover, we showed that the crystal B (λ)_{cl} is isomorphic as a crystal to a tensor product B of the crystals B (π_I), I=1, ., nIn our series of works from 2005 to 2007, we defined a certain (nonnegative) integer-valued function (which we call the degree function) on the crystal B (λ)_{cl} above, and proved that this degree function can be identified (through the isomorphism between B (λ)_{cl} and B) with the "energy function" on B, which arose from the study of solvable lattice models in statistical mechanics. In particular, by restricting ourselves to the case of affine Lie algebras of type A, we obtain a description of Kostka polynomials in terms of Lakshmibai-Seshadri paths.

项目成果

Crystal of Lakshmibai-Seshadri paths associated to an integral weight of level zero for an affine Lie algebra

与仿射李代数零级积分权重相关的 Lakshmibai-Seshadri 路径晶体

• 发表时间: 2005
• 期刊: Int. Math. Res. Not no. 14
• 作者: S. Naito;D. Sagaki
• 通讯作者: D. Sagaki

Crystal Base Elements of an Extremal Weight Module Fixed by a Diagram Automorphism

• DOI: 10.1007/s10468-005-0234-x
• 发表时间: 2005-12
• 期刊: Algebras and Representation Theory
• 影响因子: 0.6
• 作者: S. Naito;Daisuke Sagaki

Crystal base elements of an extremal weight module fixed by a diagram automorphism II : case of affine Lie algebras

由图自同构固定的极值权模块的晶体基元 II ：仿射李代数的情况

• 发表时间: 2010
• 期刊: Progress in Mathematics Volume 284
• 作者: Satoshi Naito;Daisuke Sagaki
• 通讯作者: Daisuke Sagaki

Path model for a level-zero extremal weight module over a quantum affine algebra II

量子仿射代数 II 上零级极值权模块的路径模型

• 发表时间: 2006
• 期刊: Adv.Math. 200・1
• 作者: S.Naito;D.Sagaki
• 通讯作者: D.Sagaki

Construction of perfect crystals conjecturally corresponding to Kirillov-Reshetikhin modules over twisted quantum affine algebras

构造完美晶体，推测对应于扭曲量子仿射代数上的基里洛夫-列谢蒂欣模块

• 发表时间: 2006
• 期刊: Comm. Math. Phys 263, no. 3
• 作者: S. Naito;D. Sagaki
• 通讯作者: D. Sagaki