Structure preserving transformations in hyperkähler Euclidean spaces

Abstract

The definition and structure of hyperkähler structure preserving transformations (invariance group) for quaternionic structures have been recently studied and some preliminary results on the Euclidean case discussed. In this work we present the whole structure of the invariance Lie algebra in the Euclidean case for any dimension.

Symmetry and quaternionic integrable systems

“…Physical applications of hyperhamiltonian dynamics, and in particular of quaternionic and Dirac oscillators, have been considered in previous papers [22,23]. We have then characterized the symmetry properties of quaternionic oscillators; the required computations do actually to some extent reproduce those needed to study invariance properties of the quaternionic structure behind quaternionic oscillators, and have hence been only partially detailed here, referring to other works [24,25] for details. As symmetries are invariant under diffeomorphisms, they are also properties of any quaternionic integrable system, and can be used to detect such systems.…”

“…The maps preserving the quaternionic structure (i.e. carrying a given hyperkahler structure into an equivalent one) are considered as the canonical maps for the quaternionic structure; see [24] for their characterization, and [25] for a fully explicit discussion in Euclidean R 4n spaces. Note canonical maps induce necessarily a map It may be worth providing a description in local coordinates, also to fix notation to be widely used in the following.…”

“…They realize the Hopf fibration of S 3 [29]. We stress that -albeit we have not written this explicitly to avoid a heavy notation -in (25) and (26) ν is a function of ρ; thus it is a constant of motion, but takes in general different values on different spheres.…”

Canonical transformations for hyperkahler structures and hyperhamiltonian dynamics

“…H ⊆ hSp. We can expect that, unless the hyperkahler structure has some special (invariance) property, the two will just coincide (in [24] we will find this is the case in Euclidean spaces; this fact should be seen as a check that our notion of hyperkahler maps is an appropriate one. ⊙…”

“…what are the appropriate definitions of hyperkahler and canonical transformations in general (see Sect.III); we will also characterize them by providing equations to be satisfied by the transformations. In a companion paper [24] we will obtain a full characterization (that is, we solve the characterizing equations) of hyperkahler maps in the Euclidean case; this is related to the general case via the result presented here in Sect.IV. Albeit such a full…”