Q-Deformed quaternions and su(2) instantons

Abstract

We have recently introduced the notion of a q-quaternion bialgebra and shown its strict link with the SOq(4)-covariant quantum Euclidean space ℝ4q. 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 SOq(4)-covariant differential calculi on ℝ4q and the two 4-dimensional bicovariant differential calculi on the bi- (resp. Hopf) algebras Mq(2), GLq(2), SUq(2), showing that they essentially coincide.