Reduction of L_∞-Algebras of Observables on Multisymplectic Manifolds

We develop a reduction scheme for the L_∞-algebra of observables on a premultisymplectic manifold (M,ω) in the presence of a compatible Lie algebra action g M and subset N⊂ M. This reproduces in the symplectic setting the Poisson algebra of observables on the Marsden-Weinstein-Meyer symplectic reduced space, whenever the reduced space exists, but is otherwise distinct from the Dirac, Śniatycki-Weinstein, and Arms-Cushman-Gotay observable reduction schemes. We examine various examples, including multicotangent bundles and multiphase spaces, and we conclude with a discussion of applications to classical field theories and quantization.