Equivariant Zariski Structures

TL;DR

This paper introduces a new class of noncommutative algebras linked to quantum groups, assigns a first-order theory to their geometric models, and proves model-theoretic properties including categoricity and quantifier elimination, establishing them as Zariski structures.

Contribution

It defines a novel class of noncommutative algebras with associated geometric theories and proves their model-theoretic properties, connecting algebraic and geometric frameworks.

Findings

Establishment of uncountable categoricity for the structures.

Quantifier elimination to existential formulas.

Existence of an appropriate dimension theory.

Abstract

A new class of noncommutative $k$-algebras (for $k$ an algebraically closed field) is defined and shown to contain some important examples of quantum groups. To each such algebra, a first order theory is assigned describing models of a suitable corresponding geometric space. Model-theoretic results for these geometric structures are established (uncountable categoricity, quantifier elimination to the level of existential formulas) and that an appropriate dimension theory exists, making them Zariski structures.