Δ-locally nilpotent algebras, their ideal structure and simplicity criteria

Abstract

The class of Δ-locally nilpotent algebras introduced in the paper is a wide generalization of the algebras of differential operators on commutative algebras. Examples include all the rings $D\left(A\right)$ of differential operators on commutative algebras in arbitrary characteristic, all subalgebras of $D\left(A\right)$ that contain the algebra A, the universal enveloping algebras of nilpotent, solvable and semi-simple Lie algebras, the Poisson universal enveloping algebra of an arbitrary Poisson algebra, iterated Ore extensions $A[x_1,\dots ,x_n;{\delta }_1,\dots ,{\delta }_n]$, certain generalized Weyl algebras, and others.

In [8], simplicity criteria are given for the algebras differential operators on commutative algebras. To find the simplicity criterion was a long standing problem from 60'th. The aim of the paper is to describe the ideal structure of Δ-locally nilpotent algebras and as a corollary to give simplicity criteria for them. These results are generalizations of the results of [8]. Examples are considered.