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.
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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 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[G] and the space of function kG := Hom(G,k) are dual: given an element of the group algebra
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
Symmetries of group actions
Let G be a group and X a topological space. Any action
for
This may be described by a linear mapping
where
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
whenever
Let H be a Hopf algebra and A a left Hopf H-module algebra. The smash product algebra
and we write
In our case, A = F(X) and H = kG, and we have
In this case the smash product algebra
The cyclic homology of Hopf smash products has been computed. However, there the smash product is called a crossed product and denoted