q-Deformed quaternions and su(2) instantons

TL;DR

This paper introduces q-quaternions and constructs solutions to a deformed su(2) Yang-Mills instanton equations on a quantum Euclidean space, revealing a noncommutative moduli space and comparing different differential calculi.

Contribution

It develops the theory of q-quaternions, formulates instanton solutions in a quantum setting, and compares various differential calculi on related quantum groups.

Findings

Solutions depend on noncommuting parameters, suggesting a noncommutative moduli space.

Explicit comparison shows the differential calculi on R_q^4 and quantum groups essentially coincide.

The work links q-quaternions with quantum Euclidean space and instanton solutions.

Abstract

We have recently introduced the notion of a q-quaternion bialgebra and shown its strict link with the SO_q(4)-covariant quantum Euclidean space R_q^4. Adopting the available differential geometric tools on the latter and the quaternion language we have formulated and found solutions of the (anti)selfduality equation [instantons and multi-instantons] of a would-be deformed su(2) Yang-Mills theory on this quantum space. The solutions depend on some noncommuting parameters, indicating that the moduli space of a complete theory should be a noncommutative manifold. We summarize these results and add an explicit comparison between the two SO_q(4)-covariant differential calculi on R_q^4 and the two 4-dimensional bicovariant differential calculi on the bi- (resp. Hopf) algebras M_q(2),GL_q(2),SU_q(2), showing that they essentially coincide.