Generalization of the concept of homogeneous element

Question

Let $G$ be a finite group. Recall that $kG$ is a Hopf algebra and, since $dim(kG)<\infty$, $()^∗$ is also a Hopf algebra with its structure dual to that of $$. As it is well known, an associative algebra $A$ is $G$-graded (i.e $A=\oplus_{\sigma\in G} A_\sigma , A_\sigma A_\tau\subseteq A_{\sigma\tau}$) iff it is a $(kG)^*$-module algebra with $(kG)^*$-action given by $p_\sigma\cdot a=a_\sigma$ for all $a\in A$ ($p_\sigma\in (kG)^*$ is such that $p_\sigma(\tau)=\delta_{\sigma,\tau}$).

Now, suppose more generaly that $A$ is an $H$-module algebra, where $H$ is a finite dimensional Hopf algebra.

Question: Who plays the role of homogeneous elements in this more general concept? In other words, what is the natural generalization for a homogeneous element?

Answer 1

I'll first make things more clear by directly writing down the Hopf algebra $kG^*$:

• As a $k$-vector space, it has a basis given by the elements $\{p_g \mid g \in G\}$.
• The unit is $\sum_{g \in G} p_g$.
• The multiplication is $p_g^2 = p_g$ and $p_g p_h = 0$ for $g \neq h$.
• The counit is $\varepsilon(p_e) = 1$ and $\varepsilon(p_g) = 0$ for $g \neq e$.
• The comultiplication is $p_g \mapsto \sum_{xy = g} p_x \otimes p_y$
• The antipode is $p_g \mapsto p_{g^{-1}}$.

We then have the rigid monoidal category $\operatorname{\mathsf{Rep}} kG^*$ of left modules over $kG^*$, which is the category of $G$-graded vector spaces:

• A left module $V$ decomposes into its homogeneous parts $V = \bigoplus_{g \in G} V_g$ by applying the projectors $p_g$.
• For homogeneous $V_g$ and $W_h$ we have $V_g \otimes_k W_h$ is homogeneous in $gh$.
• The duality makes $(V_g)^*$ homogeneous in $g^{-1}$.

An algebra object, or monoid object, internal to the category $\operatorname{\mathsf{Rep}} kG^*$ is what we call a $G$-graded algebra. I think that this construction is the one we want to generalise. So for an arbitrary Hopf algebra $H$, we would be looking for algebra objects internal to $\operatorname{\mathsf{Rep}} H$.

However, in an arbitrary Hopf algebra $H$ you may not have projectors like $p_g$ which can "pick out" homogeneous subspaces. For example, taking $H = kG$ the usual group Hopf algebra, then $\operatorname{\mathsf{Rep}} kG$ is just $G$-representations, and its unclear what "homogeneous" should mean. The most canonical thing you can do here is to take the central idempotents which act as projectors to isotypic components of representations (which does actually make sense), but figuring out how these isotypic components multiply together is usually much harder than in the graded vector space case. For example, when $G$ is the symmetric group, it is an unsolved problem to describe the coefficients appearing in this multiplication in a combinatorial way.