Left-invariant hypersymplectic metrics on Lie groups

ADRIÁN ANDRADA, ISABEL DOTTI

La Cumbre, Córdoba, Argentina

Workshop; Sixth Workshop on Lie Theory and Geometry; 2007

FaMAF, UNC

A hypersymplectic metric on a manifold M^{4n} is a pseudo-Riemannian metric of signature (2n; 2n) such that its holonomy group is contained in Sp(n;R). These metrics were introduced by N. Hitchin in the early ´90s, and since then they have appeared in different contexts, both in mathematics and physics, especially in string theory. Compact complex surfaces admitting hypersymplectics metrics were classiffied by H. Kamada in 2002, and in 2004 Fino, Pedersen, Poon and Sorensen exhibited hypersymplectic structures on a class of 2-step nilmanifolds in their search of neutral Calabi-Yau metrics. In this talk we show first our classification of 4-dimensional solvable Lie groups admitting left-invariant hypersymplectic metrics. Later, we exhibit a procedure to construct hypersymplectic structures on R^{4n} which are complete and invariant by a nilpotent Lie group acting simply and transitively on R^{4n}. The degree of nilpotency is related to the flatness of the metric, since we show that the metric is flat if and only if the group is at most 2-step nilpotent.