Corona algebras, and quantum groups

Corona 代数和量子群

• 批准号: 228065-2010
• 负责人: Kucerovsky, Daniel
• 金额: $ 1.75万
• 依托单位: University of New Brunswick
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=570350
• 关键词: Corona; algebras; quantum; groups

A C*-algebra is a noncommutative generalization of a function algebra over a topological space, thus is sometimes called a noncommutative topological space. The methods used for studying the topology of C*-algebras, also known as geometric functional analysis, are very different from those of classical topology, but the rather ambitious program of classifying all C*-algebras is like the classical program of classifying topological spaces by means of their topological invariants. The recent plethoria of counterexamples indicate that new approaches may be needed. However, one of the high points of the existing theory is Kirchberg's classification of purely infinite simple C*-algebras, and by generalizing one of Kirchberg's key theorems, we obtain the corona factorization property, which characterizes a noncommutative topology property in analytic terms and has deep consequences for the behaviour of a C*-algebra. Groups are mathematical objects that encode symmetry properties of a structure, and can be used to construct group algebras. Quantum groups are thought of as generalizations of groups, much in the same sense as C*-algebras are thought of as noncommutative topological spaces. It is possible for a C*-algebra to also be a quantum group. We apply the powerful machinery of the C*-algebraic classification program to a certain class of quantum group, obtaining interesting results, possibly eventually leading us to new examples of quantum groups. This is a highly novel approach to the challenging problem of classification of quantum groups.

C*-代数是拓扑空间上函数代数的非交换推广，因此有时被称为非交换拓扑空间。用于研究C*-代数的拓扑的方法，也称为几何泛函分析，与经典拓扑学的方法有很大的不同，但对所有C*-代数进行分类的雄心勃勃的程序类似于利用其拓扑不变量来对拓扑空间进行分类的经典程序。最近过多的反例表明，可能需要新的方法。然而，现有理论的一个亮点是Kirchberg对纯无限单C~*-代数的分类，通过推广Kirchberg的一个关键定理，我们得到了冠因式分解性质，它在解析意义下刻画了非对易拓扑性，并对C~*-代数的行为有深刻的影响。 群是对结构的对称性进行编码的数学对象，可用于构造群代数。量子群被认为是群的推广，就像C*-代数被认为是非交换拓扑空间一样。C*-代数也可能是量子群。我们将C*-代数分类程序的强大机制应用于一类量子群，得到了有趣的结果，最终可能会引导我们找到量子群的新例子。对于量子群分类这一具有挑战性的问题，这是一种非常新颖的方法。