Algebraic sets, ideals of points and the Hilbert's Nullstellensatz theorem for skew PBW extensions

TL;DR

This paper extends concepts of algebraic geometry, including algebraic sets and Nullstellensatz, to skew PBW extensions, a class of non-commutative algebras relevant in physics and algebraic geometry.

Contribution

It introduces algebraic sets and ideals of points for bijective skew PBW extensions and proves a version of Nullstellensatz for certain quasi-commutative cases.

Findings

Defined algebraic sets and ideals of points for skew PBW extensions.

Constructed a Zariski topology for these algebraic sets.

Proved an adapted Nullstellensatz theorem for specific skew PBW extensions.

Abstract

In this paper we define the algebraic sets and the ideal of points for bijective skew PBW extensions with coefficients in left Noetherian domains. Some properties of affine algebraic sets of commutative algebraic geometry will be extended, in particular, a Zariski topology will be constructed. Assuming additionally that the extension is quasi-commutative with polynomial center and the ring of coefficients is an algebraically closed field, we will prove an adapted version of the Hilbert's Nullstellensatz theorem that covers the classical one. The Gr\"obner bases of skew PBW extensions will be used for defining the algebraic sets and for proving the main theorem. Many key algebras and rings coming from mathematical physics and non-commutative algebraic geometry are skew PBW extensions.