Pseudo-parabolic category over quaternionic projective plane

TL;DR

This paper studies the quantization of the quaternionic projective plane, showing that the associated module category is semi-simple and equivalent to quantized equivariant vector bundles, advancing understanding of quantum geometry of symmetric spaces.

Contribution

It demonstrates the semi-simplicity and categorical equivalence of the quantized module category over the quaternionic projective plane, a novel result in quantum geometry.

Findings

The module category _t( P^2) is semi-simple.

The category is equivalent to quantized equivariant vector bundles.

Provides new insights into quantum geometry of quaternionic projective spaces.

Abstract

Quaternionic projective plane $\mathbb{H} P^2$ is the next simplest conjugacy class of the symplectic group $SP(6)$ with pseudo-Levi stabilizer subgroup after the sphere $\mathbb{S}^4\simeq \mathbb{H} P^1$. Its quantization gives rise to a module category $\mathcal{O}_t\bigl(\mathbb{H} P^2\bigr)$ over finite-dimensional representations of $U_q\bigl(\mathfrak{s}\mathfrak{p}(6)\bigr)$, a full subcategory in the category $\mathcal{O}$. We prove that $\mathcal{O}_t\bigl(\mathbb{H} P^2\bigr)$ is semi-simple and equaivalent to the category of quantized equivariant vector bundles on $\mathbb{H} P^2$.