**Definition**

Let *G* be an arbitrary group and *k* a field. The *group Hopf algebra* of *G* over *k*, denoted *kG* (or *k*), is as a set (and 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 *k* and the space of function *k**G* := Hom(*G*,*k*) are dual: given an element of the group algebra and a function on the group these pair to give an element of *k* via which is a well-defined sum because it is finite.

Read more about this topic: Group Hopf Algebra

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