On sums of homogeneous locally nilpotent derivations

TL;DR

This paper provides a criterion to determine when the sum of two homogeneous locally nilpotent derivations on a graded algebra is locally nilpotent, based on their degrees, and also addresses their commutators.

Contribution

It introduces a new criterion for local nilpotency of sums of homogeneous LNDs and analyzes their commutators within graded algebras.

Findings

Criterion for local nilpotency of sum based on degrees

Solution for commutators of homogeneous LNDs

Applicable to integrally closed graded algebras

Abstract

Let A be a commutative associative integrally closed k-algebra without zero divisors effectively graded by a lattice. We obtain a criterion of local nilpotency of the sum of two homogeneous locally nilpotent derivations (LNDs) of fiber type on A in terms of their degrees. The same problem is solved for commutators of two homogeneous LNDs.