The notion of quasi-elliptic rings appeared as a result of an attempt to
classify a wide class of commutative rings of operators found in the theory of
integrable systems, such as rings of commuting differential, difference,
differential-difference, etc. operators. They are contained in a certain
non-commutative “universal” ring – a purely algebraic analogue of the ring of
pseudodifferential operators on a manifold, and admit (under certain mild
restrictions) a convenient algebraic-geometric description. An important
algebraic part of this description is the Schur-Sato theory – a generalisation
of the well known theory for ordinary differential operators. Some parts of
this theory were developed earlier in a series of papers, mostly for dimension
two.
In this paper we present this theory in arbitrary dimension. We apply this
theory to prove two classification theorems of quasi-elliptic rings in terms of
certain pairs of subspaces (Schur pairs). They are necessary for the
algebraic-geometric description of quasi-elliptic rings mentioned above.
The theory is effective and has several other applications, among them is a
new proof of the Abhyankar inversion formula.
