Partial actions of weak Hopf algebras: Smash product, globalization and Morita theory

• Felipe Castro ^[1] ; Antonio Paques ^[1] ; Glauber Quadros ^[1] ; Alveri Sant'Ana ^[1]
  1. [1] Universidade Federal do Rio Grande do Sul

     Universidade Federal do Rio Grande do Sul

     Brasil

     
• Localización: Journal of pure and applied algebra, ISSN 0022-4049, Vol. 219, Nº 12 (December 2015), 2015, págs. 5511-5538
• Idioma: inglés
• DOI: 10.1016/j.jpaa.2015.05.031
• Enlaces
  • Texto completo
• Resumen

  • In this paper we introduce the notion of partial action of a weak Hopf algebra on algebras, unifying the notions of partial group action [11], partial Hopf action [2], [3] and [9] and partial groupoid action [4]. We construct the fundamental tools to develop this new subject, namely, the partial smash product and the globalization of a partial action, as well as we establish a connection between partial and global smash products via the construction of a surjective Morita context. In particular, in the case that the globalization is unital, these smash products are Morita equivalent. We show that there is a bijective correspondence between globalizable partial groupoid actions and symmetric partial groupoid algebra actions, extending similar result for group actions [9]. Moreover, as an application we give a complete description of all partial actions of a weak Hopf algebra on its ground field, which suggests a method to construct more general examples.