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Partial (co)actions of Hopf algebras and partial Hopf-Galois theory. (English) Zbl 1168.16021
Partial group actions were first considered by R. Exel, [Proc. Lond. Math. Soc., III. Ser. 74, No. 2, 417-443 (1997; Zbl 0874.46041)], in the context of operator algebras. Several subsequent studies have developed a purely algebraic theory; in particular, a partial group action Galois theory was developed by M. Dokuchaev, M. Ferrero and A. Pacques [J. Pure Appl. Algebra 208, No. 1, 77-87 (2007; Zbl 1142.13005)]. A coring point of view was considered by S. Caenepeel and E. De Groot [in Proc. int. conf. math. appl., ICMA 2004, Kuwait: Kuwait Foundation for the Advancement of Sciences (KFAS). 117-134 (2005; Zbl 1085.16029)].
The paper under review is devoted to partial (co)actions of Hopf algebras on algebras. Several duality results are presented, and a Galois theory for partial comodule algebras is developed. Necessary for this analysis are ‘lax corings’, generalizing notions introduced and studied by R. Wisbauer [J. Algebra 245, No. 1, 123-160 (2001; Zbl 1002.16035)].

MSC:
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16S40 Smash products of general Hopf actions
13B05 Galois theory and commutative ring extensions
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