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In this paper we introduce the notion of partial action of a weak Hopfalgebra on algebras, unifying the notions of partial group action [11], partialHopf action ([2],[3],[9]) and partial groupoid action [4]. We construct thefundamental tools to develop this new subject, namely, the partial smashproduct and the globalization of a partial action, as well as, we establish aconnection between partial and global smash products via the construction of asurjective Morita context. In particular, in the case that the globalization isunital, these smash products are Morita equivalent. We show that there is abijective correspondence between globalizable partial groupoid actions andsymmetric partial groupoid algebra actions, extending similar result for groupactions [9]. Moreover, as an application we give a complete description of allpartial actions of a weak Hopf algebra on its ground field, which suggests amethod to construct more general examples.

Partial actions of weak Hopf algebras: smash products, globalization and Morita theory