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.

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.

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":

:$Delta(x)\; =\; x\; otimes\; x;$:$epsilon(x)\; =\; 1\_\{k\};$:$S(x)\; =\; x^\{-1\}.$

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".

ymmetries of group actions

Let "G" be a group and "X" a topological space. Any action $alphacolon\; G\; imes\; X\; o\; X$ of "G" on "X" gives a homomorphism $phi\_alphacolon\; G\; o\; mathrm\{Aut\}(F(X))$, where "F(X)" is an appropriate algebra of "k"-valued functions, such as the Gelfand-Naimark algebra $C\_0(X)$ of continuous functions vanishing at infinity. $phi\_\{alpha\}$ 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

:$lambdacolon\; kG\; otimes\; F(X)\; o\; F(X)$

:$((c\_1\; g\_1\; +\; c\_2\; g\_2\; +\; cdots\; )\; otimes\; f)(x)\; =\; c\_1\; f(g\_1\; cdot\; x)\; +\; c\_2\; f(g\_2\; cdot\; x)\; +\; cdots$

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)".

$lambda$ 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 $h\; cdot\; 1\_A\; =\; epsilon(h)1\_A$ 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, $lambda$ 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 [R. Akbarpour and M. Khalkhali. 2002. [http://arxiv.org/abs/math.KT/0011248v6 "Hopf Algebra Equivariant Cyclic Homology and Cyclic Homology of Crossed Product Algebras"] . arXiv:math/0011248v6 [math.KT] ] . However, there the smash product is called a crossed product and denoted $A\; times\; H$- not to be confused with the crossed product derived from $C^\{*\}$-dynamical systems [Gracia-Bondia, J. "et al." "Elements of Noncommutative Geometry". Birkauser: Boston, 2001. ] .

References