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,…,n} be the standard set of generators corresponding to a basis of the root system. Consider the adjoint-orbit space M={Ad g (H 1), g∈G}, identified with the homogeneous space G/L where L={g∈G: Ad g (H 1)=H 1}. We prove that the coordinate functions f i (g):=λ i (Ad g (H 1)), i=1,…,n, where {λ 1,…,λ n } is basis of \({\mathcal G}^{\prime }\) 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 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.
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Acknowledgments
The author would like to thank Prof. Marc A Rieffel for inviting him to the Department of Mathematics, University of California at Berkeley, where most of the work was done. He is grateful to him for stimulating discussion, and also for explaining some of his work on coadjoint orbits, which gave the author the motivation to consider the problem of quantum actions on homogeneous spaces. He is also grateful to late Prof. S C Bagchi for some useful discussion on Lie groups. The author would like to thank the anonymous referee for pointing out mistakes in the older versions and constructive criticism which helped improve the paper. The author gratefully acknowledges support from IUSSTF for Indo-US Fellowship, Department of Science and Technology Goverment of India for the Swarnajayanti Fellowship and project, and also the Indian National Science Academy.
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Communicating Editor: B V Rajarama Bhat
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GOSWAMI, D. Quadratic independence of coordinate functions of certain homogeneous spaces and action of compact quantum groups. Proc Math Sci 125, 127–138 (2015). https://doi.org/10.1007/s12044-015-0211-1
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DOI: https://doi.org/10.1007/s12044-015-0211-1