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Journal of Lie Theory 28 (2018), No. 1, 211--244
Copyright Heldermann Verlag 2018

On the Dolbeault-Dirac Operators on Quantum Projective Spaces

Marco Matassa
Laboratoire de Mathématiques, Université Clermont Auvergne, 63000 Clermont-Ferrand, France

We consider Dolbeault-Dirac operators on quantum projective spaces, following Krähmer and Tucker-Simmons. The main result is an explicit formula for their squares, up to terms in the quantized Levi factor, which can be expressed in terms of some central elements. This computation is completely algebraic. These operators can also be made to act on appropriate Hilbert spaces. Using the formula mentioned above, we easily find that they have compact resolvent, thus obtaining a result similar to that of D'Andrea and Dabrowski.

Keywords: Dirac operators, quantum projective spaces, quantum groups, noncommutative geometry.

MSC: 58B32, 17B37, 46L87

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