Automorphisms for skew PBW extensions and skew quantum polynomial rings

TL;DR

This paper investigates the structure of automorphisms in skew PBW extensions and skew quantum polynomial rings, extending known results and applying localization techniques to characterize their automorphisms over Ore domains.

Contribution

It generalizes automorphism results for skew PBW extensions and skew quantum polynomials using filtered-graded methods and localization, building on Artamonov's work.

Findings

Automorphisms act as scalar multiplications on generators under certain conditions.

Provides characterizations of automorphisms over Ore domains.

Extends known automorphism results to more general skew PBW extensions.

Abstract

In this work we study the automorphisms of skew $PBW$ extensions and skew quantum polynomials. We use Artamonov's works as reference for getting the principal results about automorphisms for generic skew $PBW$ extensions and generic skew quantum polynomials. In general, if we have an endomorphism on a generic skew $PBW$ extension and there are some $x_i,x_j,x_u$ such that the endomorphism is not zero on this elements and the principal coefficients are invertible, then endomorphism act over $x_i$ as $a_ix_i$ for some $a_i$ in the ring of coefficients. Of course, this is valid for quantum polynomial rings, with $r=0$, as such Artamonov shows in his work. We use this result for giving some more general results for skew $PBW$ extensions, using filtred-graded techniques. Finally, we use localization for characterize some class the endomorphisms and automorphisms for skew $PBW$ extensions and skew quantum polynomials over Ore domains.