A formal Frobenius theorem and argument shift (Q736199)

The authors proved a formal Frobenius theorem, which is the formal counterpart of the classical theorem on the integrability of smooth distributions. First, they define formal counterparts of smooth geometric objects: formal vector fields, formal distributions, formal integrals and formal integrability of a distribution. For example, a formal vector field on an affine space \(\mathbb K^n\) over a field \(\mathbb K\) is a vector \(v=(v^1(x),\dots,v^n(x))\), whose components are formal power series. The formal Frobenius theorem is then applied to construct a commutative set of polynomials in the Poisson algebra associated with a finite-dimensional Lie algebra over any field \(\mathbb K\) of characteristic zero. Finally, a completeness criterion for this set is proved.