Monoidal Triangular Categories: Representation Theory, Cohomology, and Geometry

幺半群三角范畴:表示论、上同调和几何

基本信息

项目摘要

Representation theory has emerged as a central area of modern mathematics with connections to combinatorics, algebraic geometry, topology and number theory. In addition, it has many applications to chemistry and physics. Representations are mappings of complicated algebraic objects (such as groups, rings, Lie algebras, and Lie superalgebras) to an array of numbers (matrices). These realizations by matrices encode important data that can yield deep insights into these complicated algebraic objects. In recent years, a useful approach has been to understand the entire collection of representations of an object. Representations for a certain algebraic object often form a tensor triangulated category. Techniques from homological algebra can be used to build bridges between tensor triangulated categories and geometric objects. Uncovering this hidden geometry often leads to new insights about the algebraic object and its representations. This project includes research and training opportunities for graduate students and postdoctoral fellows in algebra and representation theory. In this project the PI will develop new methods to understand tensor structures in monoidal tensor categories. This will entail the development of monoidal triangular geometry and explicit computations of Balmer spectra. The PI will also introduce new geometric and topological methods to provide concrete calculations of cohomology for algebraic/finite groups, Lie superalgebras, quantum groups, and Frobenius kernels. The development of the cohomology theory was an important tool to resolving 30 year old problems that deal with tilting modules in connection with filtrations of representations of algebraic groups. The PI will study representations of Lie superalgebras via the application of super geometry to construct representations and to compute higher sheaf cohomology. The PI will develop a new Lie theory that entails the use of detecting and parabolic subalgebras, in addition to, a nilpotent cone to formulate a geometric setting in order to compute characters for irreducible representations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
表示理论已经成为现代数学的一个中心领域,与组合学、代数几何、拓扑学和数论有联系。此外,它在化学和物理方面也有许多应用。表示是复杂代数对象(如群、环、李代数和李超代数)到数字数组(矩阵)的映射。这些矩阵的实现编码了重要的数据,可以对这些复杂的代数对象产生深刻的见解。近年来,一种有用的方法是理解对象表示的整个集合。对某一代数对象的表示通常形成一个张量三角化范畴。同调代数的技术可以用来建立张量三角分类和几何对象之间的桥梁。揭示这种隐藏的几何图形通常会导致对代数对象及其表示的新见解。该项目为代数和表示理论的研究生和博士后提供研究和培训机会。在这个项目中,PI将开发新的方法来理解单轴张量类别中的张量结构。这将需要发展单面三角形几何和巴尔默谱的显式计算。PI还将引入新的几何和拓扑方法,为代数/有限群、李超代数、量子群和Frobenius核提供上同调的具体计算。上同调理论的发展是解决30年来处理与代数群表示的过滤有关的倾斜模的问题的重要工具。PI将研究李超代数的表示,通过应用超几何来构造表示和计算高轴上同调。PI将发展一个新的李理论,该理论需要使用检测和抛物子代数,以及一个幂零锥来制定一个几何设置,以便计算不可约表示的字符。该奖项反映了美国国家科学基金会的法定使命,并通过使用基金会的知识价值和更广泛的影响审查标准进行评估,被认为值得支持。

项目成果

期刊论文数量(6)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
On Donkin's Tilting Module Conjecture I: Lowering the Prime
论唐金倾斜模块猜想一:降低素数
  • DOI:
    10.1090/ert/608
  • 发表时间:
    2022
  • 期刊:
  • 影响因子:
    0.6
  • 作者:
    Bendel, Christopher P.;Nakano, Daniel K.;Pillen, Cornelius;Sobaje, Paul
  • 通讯作者:
    Sobaje, Paul
q-SCHUR ALGEBRAS CORRESPONDING TO HECKE ALGEBRAS OF TYPE B
  • DOI:
    10.1007/s00031-020-09628-7
  • 发表时间:
    2020-10
  • 期刊:
  • 影响因子:
    0.7
  • 作者:
    Chun-Ju Lai;Daniel K. Nakano;Ziqing Xiang
  • 通讯作者:
    Chun-Ju Lai;Daniel K. Nakano;Ziqing Xiang
The Nilpotent Cone for Classical Lie Superalgebras
经典李超代数的幂零锥
Noncommutative tensor triangular geometry
非交换张量三角几何
  • DOI:
  • 发表时间:
    2022
  • 期刊:
  • 影响因子:
    1.7
  • 作者:
    Nakano, Daniel;Vashaw, Kent;Yakimov, Milen.
  • 通讯作者:
    Yakimov, Milen.
Noncommutative Tensor Triangular Geometry and the Tensor Product Property for Support Maps
非交换张量三角形几何和支持图的张量积性质
  • DOI:
    10.1093/imrn/rnab221/6354855
  • 发表时间:
    2021
  • 期刊:
  • 影响因子:
    1
  • 作者:
    Nakano, Daniel K.;Vashaw, Kent B.;Yakimov, Milen T.
  • 通讯作者:
    Yakimov, Milen T.
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Daniel Nakano其他文献

On the realization of orbit closures as support varieties
论轨道闭合作为支撑品种的实现

Daniel Nakano的其他文献

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{{ truncateString('Daniel Nakano', 18)}}的其他基金

Representation Theory and Geometry in Monoidal Categories
幺半群范畴中的表示论和几何
  • 批准号:
    2401184
  • 财政年份:
    2024
  • 资助金额:
    $ 23.24万
  • 项目类别:
    Continuing Grant
Representations, Cohomology, and Geometry in Tensor Triangulated Categories
张量三角范畴中的表示、上同调和几何
  • 批准号:
    1701768
  • 财政年份:
    2017
  • 资助金额:
    $ 23.24万
  • 项目类别:
    Continuing Grant
Representation Theory, Geometry, and Cohomology in Tensor Triangulated Categories
张量三角范畴中的表示论、几何和上同调
  • 批准号:
    1402271
  • 财政年份:
    2014
  • 资助金额:
    $ 23.24万
  • 项目类别:
    Standard Grant
Cohomology, Geometry and Representation Theory: Algebraic Groups, Quantum Groups and Lie Superalgebras
上同调、几何和表示论:代数群、量子群和李超代数
  • 批准号:
    1002135
  • 财政年份:
    2010
  • 资助金额:
    $ 23.24万
  • 项目类别:
    Standard Grant
Vertical Integration of Research and Education in Mathematics at the University of Georgia
佐治亚大学数学研究与教育的垂直整合
  • 批准号:
    0738586
  • 财政年份:
    2008
  • 资助金额:
    $ 23.24万
  • 项目类别:
    Continuing Grant
Cohomological Methods in the Representation Theory of Algebraic Groups, Quantum Groups and Superalgebras
代数群、量子群和超代数表示论中的上同调方法
  • 批准号:
    0654169
  • 财政年份:
    2007
  • 资助金额:
    $ 23.24万
  • 项目类别:
    Continuing Grant
Cohomology and Representation Theory
上同调和表示论
  • 批准号:
    0400548
  • 财政年份:
    2004
  • 资助金额:
    $ 23.24万
  • 项目类别:
    Standard Grant
Cohomology and Representation Theory: Reductive Algebraic Groups and Related Structures
上同调和表示论:还原代数群及相关结构
  • 批准号:
    0136082
  • 财政年份:
    2001
  • 资助金额:
    $ 23.24万
  • 项目类别:
    Standard Grant
Cohomology and Representation Theory: Algebraic Groups, Finite Groups and Lie Algebras
上同调和表示论:代数群、有限群和李代数
  • 批准号:
    9800960
  • 财政年份:
    1998
  • 资助金额:
    $ 23.24万
  • 项目类别:
    Standard Grant
Mathematical Sciences: Cohomology and Representation Theory of Algebraic Groups and Lie Algebras
数学科学:代数群和李代数的上同调和表示论
  • 批准号:
    9500715
  • 财政年份:
    1995
  • 资助金额:
    $ 23.24万
  • 项目类别:
    Standard Grant

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Potassium Atoms in 2D Triangular Superlattice
二维三角形超晶格中的钾原子
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