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If you consider the category of $G$-graded vector spaces, the homogenous elements can be written as $U_g$ for $g \in G$.

The category of $G$-graded vector spaces is isomorphic to the category of $k^G-$modules (where $k^G$ is the dual Hopf algebra to the group algebra $k[G]$).

I have a representation over $G$: $$\rho_0: G \rightarrow \mathrm{End}(V_0)$$ and want to define the following map for $V, V_0$ vector spaces and $U_g$ a graded vector spaces: $$ \lambda_{g, V}: V_0 \otimes (V \otimes U_g) \rightarrow (V_0 \otimes V) \otimes U_g, v_0 \otimes v \otimes u \mapsto \rho_0(g)(v_0) \otimes v \otimes u$$

As one sees, the $g$ which I use in the representation is "taken from" the grading of the $G$-graded homogenous vector space.

Now consider the more general case: Let $H$ be a finite dimensional Hopf algebra and look at the modules over the dual:

$$H^*-\mathrm{mod}$$

I want to adapt a proof - in which I need the above map $\lambda_{g, V}$ - to this new, more general situation.

Here $U_g$ would be replaced by a module $N \in H^*-\mathrm{mod}$, I still have a representation $\rho_0: H \rightarrow \mathrm{End}(V_0)$, but now I don't know which $h$ I have to use. I guess what I am asking is:

Are there any components in $H^*-\mathrm{mod}$ which can be identified with an element in $H$? I guess, homogenous doesn't make sense as I don't have a grading, but kind of "basic" elements.

We have that $H^* \subset H^*-\mathrm{mod}$, but the elements in $H^*$ are maps from $H$ to $k$ - still now indexing with an $h \in H$ possible.

I'm not so good at writing questions, so I you have any questions or comments or if I have missed something, please comment!

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A $G$-graded vector space is a comodule over the Hopf algebra $\Bbbk[G]$. Graded components are, broadly speaking, suitable subcomodules of your $G$-graded vector space: those for which the coaction $\delta:U\to U\otimes \Bbbk[G]$ behaves as follows: $$\delta: U_g\to U_g\otimes \Bbbk[G]: u\mapsto u\otimes g.$$ Therefore, if you want to generalize your construction you may perform something similar to the following. Let $H$ be any Hopf algebra. Take $V_0$ to be a left $H$-module, $V$ a vector space, $U$ a right $H$-comodule and define $$ \lambda: V_0\otimes (V\otimes U) \to (V_0\otimes V)\otimes U: v_0\otimes v\otimes u\mapsto \sum_uu_{[1]}v_0 \otimes v\otimes u_{[0]}, $$ where $\delta(u)=\sum_u u_{[0]} \otimes u_{[1]}$ by resorting to Sweedler's Sigma Notation.

However, if you may provide some more context concerning your question (such as why you put the parenthesis in that particular position, why you would like to define such a $\lambda$, etc.) maybe we can be more helpful.

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  • $\begingroup$ Thanks for this approach! A little bit about my context: Suppose $(M, \rho)$, is a left $H$-module (so $M$ is a vector space), then we can define a functor $\Psi_{(M, \rho)}: \mathrm{vect}\rightarrow \mathrm{vect}, V \mapsto M \otimes V, f \mapsto \mathrm{id_M} \otimes f$. I want this to be a module functor over $H$-comod (or equivalently $H^*$-mod). In my example I wanted this to be a module functor over the $G$-graded vector space and used the $\lambda_{g, V}$ as above to get the "functorial morphism" which is needed for the module functor structure. $\endgroup$ – P. Schulze Jun 11 at 8:27
  • $\begingroup$ With module functor structure I mean that $\Psi_{(M, \rho)}(V \otimes N) \rightarrow \Psi_{(M, \rho)}(V) \otimes N$ where $N$ is a $H$-comod. $\endgroup$ – P. Schulze Jun 11 at 8:30
  • $\begingroup$ So I'm not quite sure that I really understand your suggestion for $\lambda$. To show that the module structure holds, I used that $\rho(g_1g_2) = \rho(g_1) \circ \rho(g_2)$. Aditionally I later want to turn $\Psi: H-\mathrm{mod} \rightarrow \mathrm{End}_{G-\mathrm{vect}}(\mathrm{vect}), (M, \rho) \mapsto \Psi_{(M, \rho)} $ into a functor and there I need for an intertwiner $f: (M_0, \rho_0) \rightarrow (M_1, \rho_1)$ that $f \circ \rho_0(g) = \rho_1(g) \circ f$. I hope this was understandable so far. This doesn't seem to work if my $\lambda$ doesn't contain the $\rho$. $\endgroup$ – P. Schulze Jun 11 at 8:35
  • $\begingroup$ I feel confused. For me a $\mathcal{M}$-module category is a category $\mathcal{C}$ with a suitable action of a monoidal category $\mathcal{M}$ on it, and a module functor between module categories $\mathcal{C}$ and $\mathcal{D}$ is a functor $\mathcal{F}:\mathcal{C}\to\mathcal{D}$ which is $\mathcal{M}$-linear. Is this what you are looking at? If yes, which are the categories involved in your construction? If not, where does the $V$ and the morphism $\Psi_{(M,\rho)}(V\otimes N)\to \Psi_{(M,\rho)}(V)\otimes N$ live? $\endgroup$ – Ender Wiggins Jun 11 at 9:15
  • $\begingroup$ Oh I should have mentioned that, my fault! I consider the finite dimensional vector spaces over a field $k$ $\mathrm{vect}$ as a right module category over $H$-mod. Per definition a $H$-module is a $k$-vector space + a $k$-linear map such that I can give the action by the forgetful functor and the tensor product of vector spaces. Hence can $\Psi_{(M, \rho)}: \mathrm{vect} \rightarrow \mathrm{vect}, V \rightarrow M \otimes V$ be considered as a module functor. I hope I gave you all the important details now. $\endgroup$ – P. Schulze Jun 11 at 9:23

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