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.

1 Definition
2 Hopf algebra structure
3 Symmetries of group actions
4 Hopf module algebras and the Hopf smash product
5 References

[edit] Definition

Let G be an arbitrary 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.

[edit] 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:

$\Delta(x) = x \otimes x;$

ε(x) = 1 k;

S (x) = x - 1 .

The required Hopf algebra compatibility axioms are easily checked. Notice that $\mathcal{G}(kG)$ , the set of group-like elements of kG (i.e. elements $a \in kG$ such that $\Delta(a) = a \otimes a$ ), is precisely G.

[edit] Symmetries of group actions

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

$\alpha^*_g(f)x = f(\alpha(g,x))$

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

This may be described by a linear mapping

$\lambda\colon kG \otimes F(X) \to F(X)$

$((c_1 g_1 + c_2 g_2 + \cdots ) \otimes f)(x) = c_1 f(g_1 \cdot x) + c_2 f(g_2 \cdot x) + \cdots$

where $c_1,c_2,\ldots \in k$ , $g_1, g_2,\ldots$ are the elements of G, and $g_i \cdot x := \alpha(g_i,x)$ , which has the property that group-like elements in kG give rise to automorphisms of F(X).

$λ$ endows F(X) with an important extra structure, described below.

[edit] 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 $h \cdot 1_A = \epsilon(h)1_A$ and

$h \cdot (ab) = (h_{(1)} \cdot a)(h_{(2)} \cdot b)$

whenever $a,b \in A$ , $h \in H$ and $\Delta(h) = h_{(1)} \otimes h_{(2)}$ in sumless Sweedler notation. Obviously, $λ$ as defined in the previous section turns $F (X)$ into a left Hopf kG-module algebra, and hence allows us to consider the following construction.

Let H be a Hopf algebra and A a left Hopf H-module algebra. The smash product algebra $A \# H$ is the vector space $A \otimes H$ with the product

$(a \otimes h)(b \otimes k) := a(h_{(1)} \cdot b) \otimes h_{(2)}k$ ,

and we write $a \# h$ for $a \otimes h$ in this context.

In our case, A = F(X) and H = kG, and we have

$(a \# g_1)(b \# g_2) = a(g_1 \cdot b) \# g_1 g_2$ .

The cyclic homology of Hopf smash products has been computed^[1]. However, there the smash product is called a crossed product and denoted $A \rtimes H$ - not to be confused with the crossed product derived from $C *$ -dynamical systems^[2].