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Quantum homogeneous spaces with faithfully flat module structures

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Abstract

LetA be a Hopf algebra with bijective antipode andB⊃A a right coideal subalgebra ofA. Formally, the inclusionB⊃A defines a quotient mapG→X whereG is a quantum group andX a right homogeneousG-space. From an algebraic point of view theG-spaceX only has good properties ifA is left (or right) faithfully flat as a module overB.

In the last few years many interesting examples of quantumG-spaces for concrete quantum groupsG have been constructured by Podleś, Noumi, Dijkhuizen and others (as analogs of classical compact symmetric spaces). In these examplesB consists of infinitesimal invariants of the function algebraA of the quantum group. As a consequence of a general theorem we show that in all these casesA as a left or rightB-module is faithfully flat. Moreover, the coalgebraA/AB + is cosemisimple.

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Authors and Affiliations

  1. Mathematisches Institut der Universität München, Theresienstraße 39, 80333, München, Germany

    E. F. Müller & H. -J. Schneider

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  1. E. F. Müller
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  2. H. -J. Schneider
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Correspondence to E. F. Müller.

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Müller, E.F., Schneider, H.J. Quantum homogeneous spaces with faithfully flat module structures. Isr. J. Math. 111, 157–190 (1999). https://doi.org/10.1007/BF02810683

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