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Algunas interacciones entre extensiones de Hopf Galois y anillos no
conmutativos

Resumen: En este artículo, nuestros objetos de interés son las extensiones de
Hopf Galois (p. ej., álgebras de Hopf, extensiones de Galois de cuerpos
algebraicos, álgebras fuertemente graduadas, productos cruzados, fibrados
principales, etc.) y familias de anillos no conmutativos (p. ej., anillos
polinomiales torcidos, extensiones PBW y extensiones PBW torcidas, etc.).
Recopilamos y sistematizamos preguntas, problemas, propiedades y avances
recientes en ambas teorías desarrollando explícitamente ejemplos y haciendo
cálculos que generalmente se omiten en la literatura. En particular, para las
extensiones de Hopf Galois consideramos enfoques desde el punto de vista de
torsores cuánticos (también conocidos como "heaps" cuánticos) y sistemas de
Hopf Galois, mientras que para algunas familias de anillos no conmutativos
presentamos avances en la caracterización de propiedades homológicas y
teóricas de anillos. Cada tema desarrollado se ejemplifica con abundantes
referencias a obras clásicas y actuales, por lo que este trabajo sirve de
referencia para los interesados en cualquiera de las dos teorías. A lo largo
de este trabajo, se presentan las interacciones entre ambos.

Palabras Clave: álgebra de Hopf; extensión de Hopf Galois; anillo no
conmutativo; extensión de Ore; extensión PBW torcida.

Algumas interações entre extensões de Hopf Galois e anéis não comutativos

Resumo: Neste artigo, nossos objetos de interesse são extensões de Hopf Galois
(por exemplo, álgebras de Hopf, extensões de Galois de cuerpos algebraicos,
álgebras fortemente graduadas, produtos cruzados, fibrados principais, etc.),
e famílias de anéis não comutativos (por exemplo, anéis polinomiais torcidos,
extensões PBW e extensões PBW torcidas, etc.). Coletamos e sistematizamos
questões, problemas, propriedades e avanços recentes em ambas as teorias,
desenvolvendo explicitamente exemplos e fazendo cálculos que geralmente são
omitidos na literatura. Em particular, para extensões de Hopf Galois
consideramos abordagens do ponto de vista de torsores quânticos (também
conhecidos como heaps quânticos) e sistemas Hopf Galois, enquanto para algumas
famílias de anéis não comutativos apresentamos avanços na caracterização de
propriedades homológicas e teóricas de anéis. Cada tema desenvolvido é
exemplificado com abundantes referências a obras clássicas e atuais, por isso
este trabalho serve como referência para os interessados em qualquer uma das
duas teorias. Ao longo desde trabalho, as interações entre os dois são
apresentadas.

Palavras-chave: álgebra de Hopf; extensão de Hopf Galois; anel não comutativo;
extensão de Ore; extensão PBW torcida.

Fabio Calderón

Fabio Calderón is a PhD(c) at Universidad Nacional de Colombia (Bogotá,
Colombia). In 2021-2022 he did a research stay at Rice University (Houston,
TX, USA) funded by the Fulbright Visiting Student Researcher program. His
topics of research are noncommutative algebra and (generalizations of) Hopf
algebras.

ORCID: 0000-0003-1777-0805

Armando Reyes

Armando Reyes is Associate Professor (Exclusive Dedication) at the Department
of Mathematics, Universidad Nacional de Colombia - Sede Bogotá. He is the
leader of the research group LEMAGNOC (Métodos algebraicos, analíticos y
geométricos en geometría no conmutativa). He is Investigador Senior by
Minciencias.

ORCID: 0000-0002-5774-0822