• Quadratic independence of coordinate functions of certain homogeneous spaces and action of compact quantum groups

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/125/01/0127-0138

• # Keywords

Quantum isometry; compact quantum group; homogeneous spaces; simple Lie groups.

• # Abstract

Let 𝐺 be one of the classical compact, simple, centre-less, connected Lie groups of rank 𝑛 with a maximal torus 𝑇, the Lie algebra $\mathcal{G}$ and let $\{E_{i},F_{i},H_{i},i=1,\ldots,n\}$ be tha standard set of generators corresponding to a basis of the root system. Consider the adjoint-orbit space $M=\{\text{Ad}_{g}(H_{1}), g\in G\}$, identified with the homogeneous space $G/L$ where $L=\{g\in G : \text{Ad}_{g}(H_{1})=H_{1}\}$. We prove that the coordinate functions $f_{i}(g):=\gamma_{i}(\text{Ad}_{g}(H_{1}))$, $i=1,\ldots,n$, where $\{\gamma_{1},\ldots,\gamma_{n}\}$ is basis of $\mathcal{G}'$ are quadratically independent' in the sense that they do not satisfy any nontrivial homogeneous quadratic relations among them. Using this, it is proved that there is no genuine compact quantum group which can act faithfully on $C(M)$ such that the action leaves invariant the linear span of the above coordinate functions. As a corollary, it is also shown that any compact quantum group having a faithful action on the noncommutative manifold obtained by Rieffel deformation of 𝑀 satisfying a similar linearity' condition must be a Rieffel-Wang type deformation of some compact group.

• # Author Affiliations

1. Indian Statistical Institute, 203, B. T. Road, Kolkata 700 108, India

• # Proceedings – Mathematical Sciences

Volume 130, 2020
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019