**Hopf Module Algebras and The Hopf Smash Product**

Let *H* be a Hopf algebra. A (left) *Hopf* H*-module algebra* *A* is an algebra which is a (left) module over the algebra *H* such that and

whenever, and in sumless Sweedler notation. Obviously, as defined in the previous section turns into a left Hopf *kG*-module algebra, and hence allows us to consider the following construction.

Let *H* be a Hopf algebra and *A* a left Hopf *H*-module algebra. The *smash product* algebra is the vector space with the product

- ,

and we write for in this context.

In our case, *A = F(X)* and *H = kG*, and we have

- .

In this case the smash product algebra is also denoted by .

The cyclic homology of Hopf smash products has been computed. However, there the smash product is called a crossed product and denoted - not to be confused with the crossed product derived from -dynamical systems.

Read more about this topic: Group Hopf Algebra

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