Modified Makar-Limanov and Derksen invariants

Abstract

We investigate modified Makar-Limanov and Derksen invariants of an affine algebraic variety. The modified Makar-Limanov invariant is the intersection of kernels of all locally nilpotent derivations with slices and the modified Derksen invariant is the subalgebra generated by these kernels. We prove that the modified Makar-Limanov invariant coincides with the Makar-Limanov invariant if there exists a locally nilpotent derivation with a slice. Also we construct an example of a variety admitting a locally nilpotent derivation with a slice such that the modified Derksen invariant does not coincide with the Derksen invariant.