Group Hopf algebra
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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 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 functions kG := 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.
Hopf algebra structure
The required Hopf algebra compatibility axioms are easily checked. Notice that , the set of group-like elements of kG (i.e. elements such that and ), is precisely G.
Symmetries of group actions
Let G be a group and X a topological space. Any action of G on X gives a homomorphism , where F(X) is an appropriate algebra of k-valued functions, such as the Gelfand-Naimark algebra of continuous functions vanishing at infinity. is defined by with the adjoint defined by
for , and .
This may be described by a linear mapping
where , are the elements of G, and , which has the property that group-like elements in kG give rise to automorphisms of F(X).
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 and
whenever , and in sumless Sweedler notation. When has been defined as in the previous section, this turns 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 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.
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