Oberseminar Deformationsquantisierung und Geometrie: Ana María Chaparro Castañeda (Rio)

Let k be a field of characteristic zero and A = k[x1, ..., xn]/I with I = (f1, ..., fk) be an affine algebra.
We study Nambu-Poisson brackets on A of arity m ≥ 2, focusing on the case when m is even. We
construct an L∞-algebroid on the cotangent complex LA|k, generalizing previous work on the case when
A is a Poisson algebra. This structure is referred to as the higher form brackets. The main tool is a
P∞-structure on a resolvent R of A. These P∞- and L∞-structures are merely Z2-graded for m ̸= 2. We
discuss several examples and propose a method to obtain new ones that we call the outer tensor product.
We compare our higher form brackets with the form bracket of Vaisman. We introduce the notion of a
Lie-Rinehart m-algebra, the form bracket of a Nambu-Poisson bracket of even arity being an example.
We find a flat Nambu connection on the conormal module.