arXiv subjects

Bachar Alhajjar

Publications and source records attributed to Bachar Alhajjar.

3 recordsLinked to original sources

arXiv2015

Methods to compute ring invariants and applications: a new class of exotic threefolds

We develop some methods to compute the Makar-Limanov and Derksen invariants, isomorphism classes and automorphism groups for k-domains B, which are constructed from certain Russell k-domains. We propose tools and techniques to distinguish between k-domains with the same Makar-Limanov and Derksen invariants. In particular, we introduce the exponential chain associated to certain modifications. We extract C-domains from the class B that have smooth contractible factorial Spec(B), which are diffeomorphic to R 6 but not isomorphic to C 3, that is, exotic C 3. We examine associated exponential chains to prove that exotic threefolds Spec(B) are not isomorphic to Spec(R), for any Russell C-domain R.

math.AC↗

arXiv2015

LND-Filtrations and Semi-Rigid Domains

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.

math.AC↗

arXiv2013

The Filtration Induced by a Locally Nilpotent Derivation

We investigate the filtration corresponding to the degree function induced by a non-zero locally nilpotent derivations and its associated graded algebra. As an application we provide an efficient method to recover the Makar-Limanov invariants, isomorphism classes and automorphism groups of classically known algebras. We also present a new class of examples which can be fully described with this method.

math.AC↗