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.

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 are the functions with compact support.

However, the group algebra ${\displaystyle k[G]}$ and ${\displaystyle k^{G}}$ – the commutative algebra of functions of G into k – are dual: given an element of the group algebra ${\displaystyle x=\sum _{g\in G}a_{g}g}$ and a function on the group ${\displaystyle f\colon G\to k,}$ these pair to give an element of k via ${\displaystyle (x,f)=\sum _{g\in G}a_{g}f(g),}$ which is a well-defined sum because it is finite.

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 ${\displaystyle {\mathcal {G}}(kG)}$, the set of group-like elements of kG (i.e. elements ${\displaystyle a\in kG}$ such that ${\displaystyle \Delta (a)=a\otimes a}$ and ${\displaystyle \epsilon (a)=1}$), is precisely G.

Let G be a group and X a topological space. Any action ${\displaystyle \alpha \colon G\times X\to X}$ of G on X gives a homomorphism ${\displaystyle \phi _{\alpha }\colon G\to \mathrm {Aut} (F(X))}$, where F(X) is an appropriate algebra of k-valued functions, such as the Gelfand–Naimark algebra ${\displaystyle C_{0}(X)}$ of continuous functions vanishing at infinity. The homomorphism ${\displaystyle \phi _{\alpha }}$ is defined by ${\displaystyle \phi _{\alpha }(g)=\alpha _{g}^{*}}$, with the adjoint ${\displaystyle \alpha _{g}^{*}}$ defined by

for ${\displaystyle g\in G,f\in F(X)}$, and ${\displaystyle x\in X}$.