Classification of twisted generalized Weyl algebras over polynomial rings (Q2285288)

The main work of this paper is solving the graded isomorphism problem for a wide class of twisted generalized Weyl algebras, examples of which include infinite dimensional primitive quotients of the universal enveloping algebras of certain important classes of Lie algebras. Explicitly, let \(R\) be a polynomial ring in \(m\) variables over a field of characteristic zero, the authors classify all rank \(n\) twisted generalized Weyl algebras over \(R\), up to \(\mathbb{Z}^n\)-graded isomorphisms. To achieve this classification, the authors convert the problem to the task of classifying solutions to the binary and ternary consistency equations, then reduce it to the special case \(n = 2\), and finally solve this special case by applying methods from previous work of the authors. As a consequence, when \(R\) is a polynomial ring and the automorphisms are given by additive shifts, the binary consistency relation actually implies the ternary consistency relation.