The Parthasarathy formula and a spectral triple for the quantum Lagrangian Grassmannian of rank two

TL;DR

This paper proves that the Dolbeault--Dirac operator on the quantum Lagrangian Grassmannian of rank two satisfies a Parthasarathy-type formula, confirming the spectral triple structure for this quantum space.

Contribution

It establishes a Parthasarathy formula for the quantum Lagrangian Grassmannian of rank two and verifies the spectral triple constructed by Kr"ahmer.

Findings

Dolbeault--Dirac operator satisfies Parthasarathy formula

Spectral triple for quantum Lagrangian Grassmannian confirmed

Advances understanding of quantum flag manifolds

Abstract

We show that the Dolbeault--Dirac operator on the quantum Lagrangian Grassmannian of rank two, an example of a quantum irreducible flag manifold, satisfies an appropriate version of the Parthasarathy formula. We use this result to complete the proof that the candidate spectral triple for this space, as defined by Kr\"ahmer, is a spectral triple.