Abstract
We consider Dolbeault–Dirac operators on quantized irreducible flag manifolds as defined by Krähmer and Tucker-Simmons. We show that, in general, these operators do not satisfy a formula of Parthasarathy–type. This is a consequence of two results that we prove here: first that we always have quadratic commutation relations for the relevant quantum root vectors, up to terms in the quantized Levi factor; second that there are examples of quantum Clifford algebras where the commutation relations are not of quadratic-constant type, unlike the classical case.
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Matassa, M. Dolbeault–Dirac Operators, Quantum Clifford Algebras and the Parthasarathy Formula. Adv. Appl. Clifford Algebras 27, 1581–1609 (2017). https://doi.org/10.1007/s00006-016-0730-6
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DOI: https://doi.org/10.1007/s00006-016-0730-6