Equivariant Spectral Triples for Homogeneous Spaces of the Compact Quantum Group \(U_q(2)\)

IF 0.9 3区数学 Q3 MATHEMATICS, APPLIED

Mathematical Physics, Analysis and Geometry Pub Date : 2022-08-08 DOI:10.1007/s11040-022-09432-7

Satyajit Guin, Bipul Saurabh

引用次数: 2

Abstract

In this article, we study homogeneous spaces \(U_q(2)/_\phi \mathbb {T}\) and \(U_q(2)/_\psi \mathbb {T}\) of the compact quantum group \(U_q(2),\,q\in {\mathbb {C}}\setminus \{0\}\). The homogeneous space \(U_q(2)/_\phi \mathbb {T}\) is shown to be the braided quantum group \(SU_q(2)\). The homogeneous space \(U_q(2)/_\psi \mathbb {T}\) is established as a universal \(C^*\)-algebra given by a finite set of generators and relations. Its \({\mathcal {K}}\)-groups are computed and two families of finitely summable odd spectral triples, one is \(U_q(2)\)-equivariant and the other is \(\mathbb {T}^2\)-equivariant, are constructed. Using the index pairing, it is shown that the induced Fredholm modules for these families of spectral triples give each element in the \({\mathcal {K}}\)-homology group \(K^1(C(U_q(2)/_\psi \mathbb {T}))\).

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紧量子群齐次空间的等变谱三元组 \(U_q(2)\)

本文研究紧量子群\(U_q(2),\,q\in {\mathbb {C}}\setminus \{0\}\)的齐次空间\(U_q(2)/_\phi \mathbb {T}\)和\(U_q(2)/_\psi \mathbb {T}\)。齐次空间\(U_q(2)/_\phi \mathbb {T}\)显示为编织量子群\(SU_q(2)\)。将齐次空间\(U_q(2)/_\psi \mathbb {T}\)建立为一个由有限生成器和关系集给出的普遍\(C^*\) -代数。计算了其\({\mathcal {K}}\) -群，构造了两个有限可和奇谱三元组，一个是\(U_q(2)\) -等变组，另一个是\(\mathbb {T}^2\) -等变组。利用索引配对，证明了这些谱三元组族的诱导Fredholm模给出了\({\mathcal {K}}\) -同源群\(K^1(C(U_q(2)/_\psi \mathbb {T}))\)中的每个元素。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematical Physics, Analysis and Geometry 数学-物理：数学物理

CiteScore

2.10

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： MPAG is a peer-reviewed journal organized in sections. Each section is editorially independent and provides a high forum for research articles in the respective areas. The entire editorial board commits itself to combine the requirements of an accurate and fast refereeing process. The section on Probability and Statistical Physics focuses on probabilistic models and spatial stochastic processes arising in statistical physics. Examples include: interacting particle systems, non-equilibrium statistical mechanics, integrable probability, random graphs and percolation, critical phenomena and conformal theories. Applications of probability theory and statistical physics to other areas of mathematics, such as analysis (stochastic pde''s), random geometry, combinatorial aspects are also addressed. The section on Quantum Theory publishes research papers on developments in geometry, probability and analysis that are relevant to quantum theory. Topics that are covered in this section include: classical and algebraic quantum field theories, deformation and geometric quantisation, index theory, Lie algebras and Hopf algebras, non-commutative geometry, spectral theory for quantum systems, disordered quantum systems (Anderson localization, quantum diffusion), many-body quantum physics with applications to condensed matter theory, partial differential equations emerging from quantum theory, quantum lattice systems, topological phases of matter, equilibrium and non-equilibrium quantum statistical mechanics, multiscale analysis, rigorous renormalisation group.