The Schouten-Nijenhuis bracket in infinite dimensions
Peter W. Michor
TL;DR
This work extends the Schouten-Nijenhuis bracket to infinite-dimensional smooth convenient manifolds by first constructing a direct bracket for summable skew multivector fields using the completed bornological tensor product, thereby obtaining a graded Lie algebra compatible with wedge products. It then leverages the duality with differential forms to derive alternative formulas and naturality properties, and finally generalizes the bracket to all multivector fields via Tulczyjew’s approach and the Lie differential operator, ensuring bilinearity, skew-symmetry, and the Jacobi identity in the infinite-dimensional setting. The framework supports Poisson structures and their deformations in infinite dimensions and situates the results among established infinite-dimensional Poisson theories, highlighting potential extensions and applications in dual subbundles and queer Poisson structures. The paper integrates the Kriegl–Michor calculus on convenient manifolds with Tulczyjew’s finite-dimensional insights to provide a robust, natural, and extensible bracket for a broad class of geometric objects on infinite-dimensional manifolds.
Abstract
The Schouten-Nijenhuis bracket on smooth infinite-dimensional manifolds $M$ is developed in two steps: For summable multivector fields whose pointwise dual are all differential form, and in an extended form for multivector fields which are sections of $L^{\bullet}_{\text{skew}}(T^*M,\mathbb R)$. We need to either assume that $C^{\infty}(M)$ separates points on $TM$, or consider sheaves of local sections.