Hyper-K\"ahler quotients of solvable Lie groups

TL;DR

This paper explores hyper-K"ahler quotients of solvable Lie groups, leading to new complete hyper-K"ahler metrics on Euclidean spaces by applying quotient constructions to specific Lie groups with invariant structures.

Contribution

It introduces a novel application of hyper-K"ahler quotient construction to solvable Lie groups, resulting in new explicit hyper-K"ahler metrics on Euclidean spaces.

Findings

Recovered known hyper-K"ahler metrics via Lie group structures

Constructed new complete hyper-K"ahler metrics on Euclidean spaces

Provided local expressions for these metrics

Abstract

In this paper we apply the hyper-K\"ahler quotient construction to Lie groups with a left invariant hyper-K\"ahler structure under the action of a closed abelian subgroup by left multiplication. This is motivated by the fact that some known hyper-K\"ahler metrics can be recovered in this way by considering different Lie group structures on $\H^p \times \H^q$ ($\H$: the quaternions). We obtain new complete hyper-K\"ahler metrics on Euclidean spaces and give their local expressions.