LND-Filtrations and Semi-Rigid Domains

TL;DR

This paper explores the structure of semi-rigid k-domains using LND-filtrations, introduces new classes of such domains with rich invariants, and provides an algorithm for isomorphism detection, offering counterexamples to the cancellation problem.

Contribution

It introduces the LND-filtration as a key tool for studying semi-rigid k-domains and constructs new examples with unique invariant properties.

Findings

LND-filtration effectively characterizes semi-rigid k-domains.

New classes of semi-rigid k-domains with large invariant sub-algebras are constructed.

An algorithm for explicit isomorphism between cylinders over these domains is developed.

Abstract

We investigate the filtration corresponding to the degree function induced by a non-zero locally nilpotent derivation and its associated graded algebra. We show that this kind of filtration, referred to as the LND-filtration, is the ideal candidate to study the structure of semi-rigid k-domains, that is, k-domains for which every non-zero locally nilpotent derivation gives rise to the same filtration. Indeed, the LND-filtration gives a very precise understanding of these structure, it is impeccable for the computation of the Makar-Limanov invariant, and it is an efficient tool to determine their isomorphism types and automorphism groups. Then, we construct a new interesting class of semi-rigid k-domains in which we elaborate the fundamental requirement of LND-filtrations. The importance of these new examples is due to the fact that they possess a relatively big set of invariant sub-algebras, which can not be recoverd by known invariants such as the Makar-Limanov and the Derksen invariants. Also, we define a new family of invariant sub-algebras as a generalization of the Derksen invariant. Finally, we introduce an algorithm to establish explicit isomorphisms between cylinders over non-isomorphic members of the new class, providing by that new counter-examples to the cancellation problem.