Dolbeault–Dirac Operators, Quantum Clifford Algebras and the Parthasarathy Formula

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. The final version of this research has been published in Advances in Applied Clifford Algebras. © 2016 Springer Verlag