The strong homotopy structure of Poisson reduction

Abstract

In this paper, we propose a reduction scheme for multivector fields phrased in terms of $L_{\infty }$-morphisms. Using well-known geometric properties of the reduced manifolds, we perform a Taylor expansion of multivector fields, which allows us to build up a suitable deformation retract of differential graded Lie algebras (DGLAs). We first obtained an explicit formula for the $L_{\infty }$-projection and -inclusion of generic DGLA retracts. We then applied this formula to the deformation retract that we constructed in the case of multivector fields on reduced manifolds. This allows us to obtain the desired reduction $L_{\infty }$-morphism. Finally, we perform a comparison with other reduction procedures.