We consider Lie groups endowed with a pair of anticommuting left-invariant abelian complex structures and a left-invariant, possibly indefinite, metric such that results to be a hyperkähler manifold. We study the algebraic structure and geometric properties of such Lie groups with an abelian hyperkähler structure. It results that such groups are always 3-step nilpotent and there is a correspondence between the associated hyperkähler Lie algebras and certain triples defined for a complex (associative) commutative algebra such that . This correspondence allows us to compute the Riemannian curvature of the pseudo-metric, describe the holonomy algebra and show that hyperkähler Lie groups with abelian complex structures are complete and locally symmetric. This clearly implies that every simply-connected Lie group endowed with an abelian hyperkähler structure is actually a symmetric space. In constrast to the definite case, there exist non-flat examples of abelian hyperkähler Lie groups; they cannot be 2-step nilpotent and their dimension is always equal to or greater than 16. Moreover, using the triple description, we classify up to Lie algebra isomorphism all Lie algebras admitting an abelian hyperkähler structure for . Some remarks on their classification up to triholomorphic symplectomorphism will also be mentioned. [BS_HK] I. Bajo, E. Sanmart'{i}n, « Indefinite hyperkähler metrics on Lie groups with abelian complex structures », 2019, to appear in Transformation Groups.
