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. 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), 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 functions kG := Hom(G,k) are dual: given an element of the group algebra x = ∑ g ∈ G a g g {displaystyle x=sum _{gin G}a_{g}g} and a function on the group f : G → k , {displaystyle fcolon G o k,} these pair to give an element of k via ( x , f ) = ∑ g ∈ G a g f ( g ) , {displaystyle (x,f)=sum _{gin 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 G ( k G ) {displaystyle {mathcal {G}}(kG)} , the set of group-like elements of kG (i.e. elements a ∈ k G {displaystyle ain kG} such that Δ ( a ) = a ⊗ a {displaystyle Delta (a)=aotimes a} and ϵ ( a ) = 1 {displaystyle epsilon (a)=1} ), is precisely G. Let G be a group and X a topological space. Any action α : G × X → X {displaystyle alpha colon G imes X o X} of G on X gives a homomorphism ϕ α : G → A u t ( F ( X ) ) {displaystyle phi _{alpha }colon G o mathrm {Aut} (F(X))} , where F(X) is an appropriate algebra of k-valued functions, such as the Gelfand-Naimark algebra C 0 ( X ) {displaystyle C_{0}(X)} of continuous functions vanishing at infinity. ϕ α {displaystyle phi _{alpha }} is defined by ϕ α ( g ) = α g ∗ {displaystyle phi _{alpha }(g)=alpha _{g}^{*}} with the adjoint α g ∗ {displaystyle alpha _{g}^{*}} defined by for g ∈ G , f ∈ F ( X ) {displaystyle gin G,fin F(X)} , and x ∈ X {displaystyle xin X} .