Group Hopf Algebra - Hopf Module Algebras and The Hopf Smash Product

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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