Cotangent bundles of 4-dimensional hypercomplex Lie groups
This paper deals with Lie groups \(\text{Aff}(\mathbb{C})\) (the group of affine motions of \(\mathbb{C}\)), \(S_{\mathbb{R}}\) (the solvable Lie group that acts simply transitively and isometrically on the 4-dimensional and hyperbolic space) and \(\mathbb{H}^*\) (the multiplicative group of non-zero quaternions). The author proves the existence of invariant conformally hyper-Kähler metrics on the cotangent bundles \(T^*\text{Aff}(\mathbb{C})\) and \(T^* S_{\mathbb{R}}\) endowed with a Lie group structure determined by the Obata connection on \(\text{Aff}(\mathbb{C})\) and \(S_{\mathbb{R}}\). Furthermore, she proves that \(T^*\text{Aff}(\mathbb{C})\) and \(T^* \mathbb{H}^*\) admits a hyper-Kähler with torsion structure which is invariant with respect to the Lie group structure determined by the affine structure associated to the left invariant symplectic form on \(\text{Aff}(\mathbb{C})\) and Obata connection on \(\mathbb{H}^*\).
scientific article; zbMATH DE number 1889173