Uma introdução ao plano quântico e aos números binomiais quânticos

Abstract

This work presents an introductory study of the quantum plane, understood as a particular case of automorphism-type skew polynomial rings. To this end, we begin by examining Ore extensions, which provide the algebraic framework necessary to understand the noncommutative multiplication that characterizes the quantum plane. Next, we analyze the quantum plane, highlighting its fundamental noncommutation rule and the structural properties that follow from it. The investigation then proceeds to the study of power expansions in the quantum plane and the construction of quantum binomial numbers, emphasizing their role in generalizing classical combinatorics and in forming the quantum Pascal triangle. The work is developed through bibliographical research, using as main references the classical texts of Ore on noncommutative extensions, modern expositions on skew polynomial rings, introductory works on the quantum plane, and specific references on quantum binomial numbers and combinatorics. The organization is structured around the study of Ore extensions, followed by the analysis of the quantum plane and, finally, the quantum binomial numbers, with illustrative examples aimed at making the topic more accessible.