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Total nonnegativity, quantum algebras and growth of algebras

总非负性、量子代数和代数增长

基本信息

批准号：
EP/K035827/1
负责人：
Tom Lenagan
金额：
$ 2.69万
依托单位：
University of Edinburgh
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2013
资助国家：
英国
起止时间：
2013 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FK035827%2F1
关键词：
Total nonnegativity quantum algebras growth

项目摘要

This is wide ranging project that involves the three areas of noncommutativealgebra, Poisson algebraic geometry and linear algebra. Also, the solutionsoften involve representation theory and combinatorics. In addition, theproject will consider problems concerning growth of algebras. The development of the theory of quantum algebras was motivated by problems inPhysics from the 1980s onwards. Totally nonnegative matrices have beeninvolved in problems in such diverse areas as mechanical systems, birth anddeath processes, planar resistor networks, computer aided geometric design,juggling, etc. Results concerning growth of algebras have been obtained fromthe 1960s onwards, but the subject was in a quiescent state until the 2000swhen significant advances have been made. In the past five years, surprising links between the three areas mentioned inthe first paragraph have been discovered and investigated. A partialunderstanding of these connections has been gained, especially in the particularcase of coordinate algebras of matrices. The present project aims to furtherthis understanding by deepening the knowledge of the matrix case and byexpanding the scope of the knowledge to include algebras such asgrassmannians, partial flag varieties and De Concini-Kac-Procesi algebras. The growth part of the project will concentrate on two specific types ofgrowth: quadratic growth/Gelfand-Kirillov dimension two, and intermediategrowth (super polynomial, but subexponential).

这是一个涉及面很广的项目，涉及非对易代数、泊松代数几何和线性代数三个领域。此外，解决方案还涉及到表示论和组合学。此外，该项目还将考虑有关代数增长的问题。从20世纪80年代开始，量子代数理论的发展就受到物理学问题的推动。完全非负矩阵涉及到机械系统、生灭过程、平面电阻网络、计算机辅助几何设计、杂耍等领域的问题，从20世纪60年代开始，关于代数增长的研究已经取得了一些成果，但直到2000年才有了显著的进展。在过去的五年里，发现和调查了第一段中提到的三个领域之间令人惊讶的联系。对这些联系有了部分的了解，特别是在矩阵的坐标代数的特殊情况下。本项目旨在通过加深对矩阵情况的了解并将知识范围扩大到包括诸如Grassmannians、部分旗簇和De Concini-Kac-Procesi代数等代数来促进这一理解。该项目的增长部分将集中于两种特定类型的增长：二次增长/Gelfand-Kirillov二维增长和中间增长(超多项式，但次指数)。