Lie Algebra bundles on s-Kahler manifolds, with applications to Abelian varieties

We prove that one can obtain natural bundles of Lie algebras on rank two s-K¨ahler manifolds, whose fibres are isomorphic respectively to so(s+1, s+1), su(s+1, s+1) and sl(2s+2,R). These bundles have natural flat connections, whose flat global sections generalize the Lefschetz operators of K¨ahler geometry and act naturally on cohomology. As a first application, we build an irreducible representation of a rational form of su(s + 1, s + 1) on (rational) Hodge classes of Abelian varieties with rational period matrix.