A hyperkahler structure on the cotangent bundle of a complex Lie group

TL;DR

This paper demonstrates that the cotangent bundle of a complexified compact Lie group possesses a hyperkahler structure, constructed via moduli space techniques related to Nahm's equations, expanding geometric understanding of Lie groups.

Contribution

It introduces a hyperkahler structure on the cotangent bundle of the complexification of a compact Lie group, realized through Nahm's equations moduli space.

Findings

The cotangent bundle admits a G-invariant hyperkahler structure.

The structure is constructed via a moduli space of solutions to Nahm's equations.

The hyperkahler structure is invariant under G's left and right translations.

Abstract

Let G be compact Lie group. It is shown that the cotangent bundle of the complexification of G admits a hyperkahler structure which is invariant under left and right translations by elements of G. The proof is to realize the cotangent bundle of the complex group as a moduli space of solutions to Nahm's equations on the closed interval.