Dolbeault-Dirac operators, quantum Clifford algebras and the Parthasarathy formula

TL;DR

This paper investigates Dolbeault-Dirac operators on quantized flag manifolds, revealing they generally do not follow Parthasarathy-type formulas due to quadratic relations in quantum root vectors and Clifford algebras.

Contribution

It demonstrates the failure of Parthasarathy formulas in the quantum setting and analyzes the quadratic nature of relations in quantum root vectors and Clifford algebras.

Findings

Dolbeault-Dirac operators do not satisfy Parthasarathy formulas in general.

Quantum root vectors exhibit quadratic commutation relations.

Some quantum Clifford algebras have non-quadratic-constant relations.

Abstract

We consider Dolbeault-Dirac operators on quantized irreducible flag manifolds as defined by Kr\"ahmer 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: we always have quadratic commutation relations for the relevant quantum root vectors, up to terms in the quantized Levi factor; there are examples of quantum Clifford algebras where the commutation relations are not of quadratic-constant type.