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

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      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

       

      Debashish Goswami1

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

       
  • Proceedings – Mathematical Sciences | News

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