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

Abstract

Let $G$ be one of the classical compact, simple, centre-less, connected Lie groups or rank $n$ with a maximal torus $T$, the Lie algebra $\clg$ and let $\{ E_i, F_i, H_i, i=1, \ldots, n \}$ be the standard set of generators corresponding to a basis of the root system. Consider the adjoint-orbit space $M=\{ {\rm Ad}_g(H_1),~g \in G \}$, identified with the homogeneous space $G/L$ where $L=\{ g \in G:~{\rm Ad}_g(H_1)=H_1\}$. We prove that the `coordinate functions' $\{ f_i, i=1, \ldots, n \}$, (where $f_i(g):=\lambda_i({\rm Ad}_g(H_1))$, $\{ \lambda_1, \ldots, \lambda_n\}$ is basis of $\clg^\prime$) 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 faithtully on $C(M)$ such that the action leaves invariant the linear span of the above cordinate 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.