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Differential geometry of quantum homogeneous spaces


The research group of A. Joseph at the Weidman Institute is internationally recognized for its fundamental contributions to the theory of Lie algebras and quantum groups. The researcher I.Heckenberger is a member of a group in Leipzig. He has been working on the field of quantum groups with focus on no commutative differential geometry for ten years. In the framework of quantum groups one investigates deformations of universal enveloping algebras of semi simple Lie algebras and quintile coordinate rings of semi simple algebraic groups. One approach to no commutative differential geometry on quantum groups is the notion of covariant differential calculus introduced by S. L. Woronowicz. A powerful method to obtain classification results within the scope of this theory is to determine explicitly the locally finite part of the dual Hop algebra of the quantum group. A. Joseph and G. Letter have answered this question independently of differential calculi. Recently Heckenberger has been investigating differential calculi on quintile irreducible flag manifolds. For classical irreducible flag manifolds the de Ram complex is known to be the dual of the Bernstein--Gland--Gland resolution. The first aim of the research project is to obtain an analogous result in the quantum setting. For this a detailed understanding of higher order differential calculi, generalized Vera modules for quintile universal enveloping algebras and the duality between them shall be established. A second related aim is to evaluate the quantum KPRV determinants pertaining to generalize Vera modules introduced by A. Joseph and D. To Doric in order to describe those exceptional generalized Vera modules who’s locally finite endomorphism do not come all from the enveloping algebra. Apart from the Boreal case, so far only the degrees of these determinants are known.

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