A survey on twists: from twisted group algebras to Hopf algebras

• Ano de defesa: 2024
• Autor(a) principal: Giacomin, Lucas Nacif
• Tipo de documento: Dissertação
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Biblioteca Digitais de Teses e Dissertações da USP
• Palavras-chave em Português: Cociclos; Cocycles; Twisted algebras; Twists de álgebras; Zhang twist; Zhang twists
• Link de acesso: https://www.teses.usp.br/teses/disponiveis/45/45131/tde-14112024-100428/

Twisting in mathematics involves the deformation of an existing structure, often to explore new algebraic properties. This dissertation examines the concept of twisting, focusing on the deformation of products of algebras. The historical development of twisted group algebras, first studied by Schur in 1904, acts as the stepping stone to the discussion, particularly exploring the relation between projective representations and group cohomology. The study extends to Zhangs formalization of twistings on graded algebras, offering insights into the construction of twisted algebras and their properties. Additionally, the dissertation explores the implications of twisting in the realm of Hopf algebras, emphasizing the significance of cocycles in quantum group theory and its relation to the Quantum Yang-Baxter Equation. By extending the concept of twisting systems to Hopf algebras, the research provides a comprehensive overview of the subject, highlighting recent advancements and theoretical applications. The work aims to be an introductory guide to the theory of twists, with a structured exposition of key concepts, and their significance in the field.