Gröbner-Shirshov bases for Sklyanin algebras

In this thesis we study the theory of Gröbner-Shirshov bases for three- dimensional and four-dimensional Sklyanin algebras. First, we present a brief construction of free algebras, and then describe the theory of Gröbner-Shirshov bases of these algebras. In addition, we present examples on the computation of the bases, and in particular, we consider some relations with PBW algebras. Next, we address the origin and review some of the properties of three-dimensional Sklyanin algebras, especially the PBW property. With this, we classify the three-dimensional Sklyanin algebras that are or not PBW algebras into at least eight families, and we compute their Gröbner-Shirshov bases, obtaining in some cases finite bases and in others, apparently infinite ones. In the same way, we study four-dimensional Sklyanin algebras, reviewing some of their algebraic properties, their classification into six families of degenerate algebras, and we compute their Gröbner-Shirshov bases obtaining only for one family, a finite basis. Finally, we use a code developed in MATLAB to review the hand-made computations of the Gröbner-Shirshov bases in the different families of the three-dimensional Sklyanin algebras, and at the same time test the correctness of the code. Once verified, we use it to perform the calculations for four-dimensional Sklyanin algebras

• Autores: Herrera Cano, Karol Stefany
• Fecha de publicación: 2024
• Institución: Universidad Nacional de Colombia
• Repositorio: Universidad Nacional de Colombia
• Idioma: eng
• OAI Identifier: oai:repositorio.unal.edu.co:unal/86074
• Acceso en línea: https://repositorio.unal.edu.co/handle/unal/86074; https://repositorio.unal.edu.co/
• Palabra clave: Base de Gröbner-Shirshov; Algoritmo de Shirshov; Lema del diamante; Álgebra PBW; Álgebra de Sklyanin; Gröbner-Shirshov basis; Shirshov's algorithm; Diamond lemma; PBW algebra; Sklyanin algebra
• Rights: openAccess
• License: Reconocimiento 4.0 Internacional