Given a -bialgebra , and a left Hopf action of on a -algebra , one defines the crossed product algebra (in Hopf algebra literature also called the smash product algebra or Hopf smash product; distinguish from the rather different smash product in topology) as the -algebra whose underlying vector space is and the product is given by
The idea is that if the bialgebra is in fact a Hopf algebra embedded as – whatever the product in the latter is (but assumed to satisfy ) – and if the action is inner within , i.e. , then we have
and hence the formula for the product above is a tautology: .
Similarly, given a right Hopf action of on , one defines the crossed product algebra whose underlying space is . The left and right versions are isomorphic if has an invertible antipode; this extends the correspondence between the left and right actions obtained by composing with the antipode map.
Every smash product algebra of the form is naturally equipped with a monomorphism of algebras and with a right -coaction making into a right -comodule algebra. Map , is then a map of right -comodule algebra (where the coaction on is ), and is the subalgebra of -coinvariants.
If is a Hopf algebra, then the homomorphism is a convolution invertible linear map with convolution inverse defined by for , where is the antipode of . Conversely,
Proposition Let be a Hopf algebra, a right -comodule algebra, and a map of right -comodule algebra. Clearly acts on by for and , where the product on the right-hand side is in . Conclusion: where the smash product is with respect to that action.
There is also a more general cocycled crossed product. For a bialgebra and an algebra , if we consider the category of extensions which are compatibly left -modules and right -comodules, and where , then the crossed product algebras are the canonical representatives of cleft Hopf-Galois extensions which are a more invariant concept.
Let be an algebra, a Hopf algebra, a measuring, i.e. a -linear map satisfying for all , , and which we assume unital, i.e. for all . We do not assume that is an action.
Let further a (convolution) invertible -linear map be given.
We say that is a 2-cocycle (relative to the measuring ) if the following two cocycle identities hold
These identities clearly generalize the classical factor system?s in group theory (linearly extended to the case of group algebras, for the finite groups at least). Therefore it is an example of a nonabelian cocycle in Hopf algebra theory. However, its role in the general theory is less well understood than the group case.
Define the cocycled crossed product on by
for all , . The cocycled crossed product is an associative algebra iff is a cocycle.
If so, we call cocycled crossed product algebra. Map is a right -coaction, making into a right -comodule algebra, which is cleft extensions]] are always isomorphic (as -extensions) to the cocycled crossed product algebras.
If then we say that is a trivial cocycle and then the compatibility conditions above reduce to demanding that the measuring is an action. The cocycled crossed product then reduces to the usual smash product algebra.
Theorem. Suppose we are given two measurings with cocycles respectively. Then there exists an isomorphism of -extensions of , (i.e. an isomorphism of -algebras, left -modules and right -comodules) iff there is an invertible element such that for all ,
The isomorphism is then given by
Y. Doi, M. Takeuchi, Cleft comodule algebras for a bialgebra, Comm. Alg. 14 (1986) 801–818
S. Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics 82, AMS 1993.
S. Majid, Foundations of quantum group theory, Cambridge University Press 1995.