Hopf gradings on complex commutative group rings

Question

Let $G$ be a finite abelian group. The complex group ring $\mathbb{C}G$ admits a structure of Hopf algebra when the multiplication is the usual multiplication in a group ring and the co-multiplication is defined by $\Delta (g)=g\otimes g$ and the anti-pod is $S(g)=g^{-1}$ for any $g\in G$.

Now a Hopf-grading on a Hopf algebra is a grading of the algebra which "respects" the multiplication and the co-multiplication. However, it seems that this is not enough and there is some extra compatibility condition which I do not understand.

I tried reading about it in Montgomery book "Hopf algebras and their actions on rings", but I still can't fully understand it.

So for now, I want to leave the general definition and to concentrate only on the Hopf algebra $\mathbb{C}G$ as above.

I hope someone can tell me what are all the Hopf-gradings on $\mathbb{C}G$, and if this is too much work, I would like to know just what conditions should I check and perhaps a small example (for example $\mathbb{C}C_4)$.

Thanks in advance for any help.

Answer 1

There is two notion of gradation - one is based on the filtration on vector space, but the second is related to group action.

If $H$ is a Hopf algebra over field $k$, then it is $\mathbb{N}$-filtered $k$-Hopf algebra iff there exist filtration $\{H_i\}_{i\in \mathbb{N}}$ on $H$ such that

• $H_iH_j\subset H_{i+j}$
• $\Delta H_n \subset \sum\limits_{k=0}^{n} H_{k}\otimes H_{n-k}$
• $S(H_i)\subset H_i$

If $H\cong \bigoplus\limits_{i\in\mathbb{N}}H_{(i)}$ for some linear supspaces $H_{(i)}$ of $H$, and the above conditions hold if we replace $H_i$ by $H_{(i)}$ then we have $\mathbb{N}$-graded $k$-Hopf algebra. It turns out that if $H$ is connected ($H_{(0)}\cong k$) then the condition for $S$ is unnecessary and $S$ can be uniquely determined from other components of bialgebra.

The above definition can be generalize for directed sets instead of $\mathbb{N}$. Such sets should be also monoid.

The second notion of gradation is a gradation by group (Montgomery Hopf algebras and their actions on rings, 10.5.1). Suppose we have finite abelian group $G$. Then it is know that $\hat G \cong G$, so there exist bicharacter $\lambda$ on $G$. Hopf algebra $H$ (in some braided monoidal category) is called $G$-graded $\lambda$-Hopf algebra (or $G$-graded Hopf algebra with respect to given bicharacter $\lambda$) if it has gradation (as a vector space) $H\cong \bigoplus\limits_{g\in G} H_g$ and

• $H$ is a $G$-graded algebra $\left(\right.H_gH_h\subset H_{g\cdot h}$, where $\cdot$ is a multiplication in $G\left.\right)$
• $H$ is a $G$-graded coalgebra $\left( \Delta(H_g)\subset \sum\limits_{p\cdot q =g}H_p\otimes H_q \right)$
• $\Delta$ and $\varepsilon$ are algebra maps with $m_{H\otimes H}$ given by $$m_{H\otimes H}\left(a_{g_1}\otimes b_{h_{1}},a_{g_2}\otimes b_{h_{2}}\right)=\lambda(h_1,g_2)a_{g_1}a_{g_{2}}\otimes b_{h_{1}}b_{h_{2}},$$ where $A\cong \bigoplus\limits_{g\in G} a_{g}$ and $B\cong \bigoplus\limits_{h\in G}b_h$.
• $S\circ m=m\circ \Psi \circ (S\otimes S)$, where $\Psi$ is a braiding in our braided monoidal category.

Often in this context is it assumed that $G$-graded Hopf algebra is connected, so we also need $\varepsilon(H_g)=0$ unless $g=e$ ($e$-neutral element in $G$).

As an example let us take $kG$ as a Hopf algebra. In the natural way it forms $kG$-comodule algebra ($H$-comodule algebra $A$ is a $k$-algebra which is simultaneously $H$-comodule with a coaction $\delta$ which is an algebra homomorphism). Indeed, take $\delta=\Delta$.

One can easily check that any $kG$-comodule $A$ is a $G$-graded $k$-module. We have coaction $\tilde{\delta}(a)=\sum\limits_{g\in G}a_g\otimes g$ on $A$, where $a_g\in A_g$ and $A\cong \bigoplus\limits_{g\in G}A_g$. In this case we have $1_A\in A_e$ and for any $a\in A_g,\ b\in A_h$ we have an equality $$\tilde{\delta}(ab)=ab\otimes gh \in A_{gh}$$ Hence $A$ is a $G$-graded $k$-algebra. Converse statement is also true.

If you are interested in the second notion you can find a lot of informations in this article and in section 3 here. Some aspects of gradation on $\mathbb{C}G$ you can find also in this article.