Geometry of bundle-valued multisymplectic structures with Lie algebroids

Abstract

We study multisymplectic structures taking values in vector bundles with connections from the viewpoint of the Hamiltonian symmetry. We introduce the notion of bundle-valued $n$-plectic structures and exhibit some properties of them. In addition, we define bundle-valued homotopy momentum sections for bundle-valued $n$-plectic manifolds with Lie algebroids to discuss momentum map theories in both cases of quaternionic K\"{a}hler manifolds and hyper-K\"{a}hler manifolds. Furthermore, we generalize the Marsden-Weinstein-Meyer reduction theorem for symplectic manifolds and construct two kinds of reductions of vector-valued 1-plectic manifolds.