Let be a Hopf algebra, a right -comodule algebra is an -extension of a subalgebra if is precisely the subalgebra of -coinvariants. The -extension is cleft if there exist a convolution-invertible -comodule map .
If is a cleft -extension, then the cleavage can always be chosen normalized in the sense that ; because if it is not normalized we can rescale to form a normalized cleavage (indeed, is group-like, hence is invertible with inverse ).
It is easy to show that the rule
defines a measuring i.e. and if is chosen normalized, then . Define a convolution invertible map by
Then the pair defines the data for the cocycled crossed product algebra which is canonically isomorphic to as an -extension of , and i.e. as a right -comodule algebra with the isomorphism fixing as given.
Conversely, every cocycled product is cleft via and the cocycle built out of is the same one, which helped build the cocycled crossed product.
Every cleft extension is a particular case of a Hopf-Galois extension.
Y. Doi, M. Takeuchi, Cleft comodule algebras for a bialgebra, Comm. Alg. 14 (1986) 801–818
S. Majid, Foundations of quantum group theory, Cambridge University Press 1995, 2000.
S. Montgomery, Hopf algebras and their actions on rings, CBMS 82, AMS 1993.
There are generalizations for Hopf algebroids:
There are some globalizations of cleft extensions. For the smash product case of the globalization some details are written in