Group Hopf algebra

In mathematics, the group Hopf algebra of a given group is a certain construct related to the symmetries of group actions. Deformations of group Hopf algebras are foundational in the theory of quantum groups.
Definition.
Let "G" be a group and "k" a field. The "group Hopf algebra" of "G" over "k", denoted "kG" (or "k"["G"]), is as a set (and a vector space) the free vector space on "G" over "k". As an algebra, its product is defined by linear extension of the group composition in "G", with multiplicative unit the identity in "G"; this product is also known as convolution.
Note that while the group algebra of a "finite" group can be identified with the space of functions on the group, for an infinite group these are different. The group algebra, consisting of "finite" sums, corresponds to functions on the group that vanish for cofinitely many points; topologically (using the discrete topology), these correspond to functions with compact support.
However, the group algebra formula_1 and the space of functions formula_2 are dual: given an element of the group algebra formula_3 and a function on the group formula_4 these pair to give an element of "k" via formula_5 which is a well-defined sum because it is finite.
Hopf algebra structure.
We give "kG" the structure of a cocommutative Hopf algebra by defining the coproduct, counit, and antipode to be the linear extensions of the following maps defined on "G":
The required Hopf algebra compatibility axioms are easily checked. Notice that formula_9, the set of group-like elements of "kG" (i.e. elements formula_10 such that formula_11 and formula_12), is precisely "G".
Symmetries of group actions.
Let "G" be a group and "X" a topological space. Any action formula_13 of "G" on "X" gives a homomorphism formula_14, where "F"("X") is an appropriate algebra of "k"-valued functions, such as the Gelfand-Naimark algebra formula_15 of continuous functions vanishing at infinity. The homomorphism formula_16 is defined by formula_17, with the adjoint formula_18 defined by
for formula_20, and formula_21.
This may be described by a linear mapping
where formula_24, formula_25 are the elements of "G", and formula_26, which has the property that group-like elements in formula_27 give rise to automorphisms of "F"("X").
formula_28 endows "F"("X") with an important extra structure, described below.
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 formula_29 and
whenever formula_31, formula_32 and formula_33 in sumless Sweedler notation. When formula_28 has been defined as in the previous section, this turns "F"("X") into a left Hopf "kG"-module algebra, which allows the following construction.
Let "H" be a Hopf algebra and "A" a left Hopf "H"-module algebra. The "smash product" algebra formula_35 is the vector space formula_36 with the product
and we write formula_38 for formula_39 in this context.
In our case, formula_40 and formula_41, and we have
In this case the smash product algebra formula_43 is also denoted by formula_44.
The cyclic homology of Hopf smash products has been computed. However, there the smash product is called a crossed product and denoted formula_45- not to be confused with the crossed product derived from formula_46-dynamical systems.