Finite generation of rings of differential operators of semigroup algebras. (Q1879655)

Let \(A\) be an integer matrix of size \(d\times n\). Assume that the columns of \(A\) generate \(\mathbb{Z}^d\) as an additive Abelian group. For a column \(a_i\), \(1\leqslant i\leqslant n\), from \(A\) denote by \(t^{a_i}\) the monomial with the multi-index \(a_i\) in the complex Laurent polynomial algebra \(B\) with variables \(t_1,\dots,t_d\). Consider the subalgebra \(R_A\) in \(B\) generated by \(t^{a_i}\), \(1\leqslant i\leqslant n\). Then the algebra of differential operators \(D(R_A)\) is defined as the subalgebra of the Weyl algebra \(B\langle\partial_1,\dots,\partial_d\rangle\) mapping \(R_a\) into itself. The main result of the paper states that \(D(R_A)\) is a finitely generated algebra. There is a natural filtration on \(D(R_A)\). There is given a criterion of finite generation of the associated graded algebra.