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

Proceedings Mathematical Sciences

Debashish Goswami¹ 

[1] Indian Statistical Institute, 0, B. T. Road, Kolkata 00 0, India$$

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

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

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 $M={ext{Ad}_{g}(H_{1}), gin G}$, identified with the homogeneous space 𝐺/𝐿 where $L={gin G : ext{Ad}_{g}(H_{1})=H_{1}}$. We prove that the coordinate functions $f_{i}(g):=𝛾_{i}(ext{Ad}_{g}(H_{1}))$, $i=1,ldots,n$, where ${𝛾_{1},ldots,𝛾_{n}}$ 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.

【 授权许可】

Unknown