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Let $G$ be a Hopf algebra, considered as a right $G$-comodule in the obvious way. The axioms of Hopf algebras imply that $$ G^{\operatorname{coinv}(G)} == \{g \in G : \Delta(g) = g \otimes 1\} = \mathbb{C}1. $$ Consider now that tensor product of comodules with $k$-factors $$ G\otimes \dotsb \otimes G. $$ Is there an explicit description of $$ (G\otimes \dotsb \otimes G)^{\operatorname{coinv}(G)}? $$

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Let $\Bbbk$ be a field and $H=\Bbbk G$ its group algebra. Then the regular coaction makes $H$ a right comodule over itself and we use the monoidal structure of right comodules to view $H^{\otimes n}$ as a right $H$-comodule.

Using the fact that $\{g\otimes h\otimes k\;|\;g,h,l\in G\}$ is a basis for $H^{\otimes 3}$ one can proof that $$(H\otimes H)^{coinv}=\Bbbk^{\oplus |G|}.$$ A basis for the vector space $(H\otimes H)^{coinv}$ is given by the elements $g\otimes g^{-1}$, for $g\in G$.

More generally, $$(H^{\otimes n})^{coinv}=Span_\Bbbk(\{g_1\otimes \ldots \otimes g_n \;|\; g_1\cdot\ldots\cdot g_n=1\}).$$ The dimension of this vector space is $$\dim (H^{\otimes n})^{coinv}=|G|^{n-1}.$$

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If $H$ is Hopf, then $H\otimes H$ with diagonal action is free, of the same dimension of $H$. The map is very explicit, you can surely can find it in any book of Hopf algebras (e.g. S. Montgomery). There is also the version for coaction. The statement is that it is isomorphic to $H\otimes V$ where V is a vector space, the $H$ coaction comes only from the left factor, and V=H as vector space. You can iterate fue procedure (or the explicit map) and get a formula for the coinvariants

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According to my understanding, you regard $G$ as a comodule on itself. So what you write is also Ok. Now I offer a way to construct a coinvariants on tensor product of comodules with k-factors. Without loss of generality, we can assume that $k=2$. Firstly, we should define a comodule structure on $G\otimes G$. This was done by in Susan Montgomery's book ``Hopf algebras and their actions on rings'' on page 14 (Definition 1.8.2). I posted it here. $$\Delta_{G\otimes G}:=(id \otimes m)(id \otimes \tau \otimes id)(\Delta \otimes \Delta),$$ where $m$ is the multiplication, $\tau$ is the switch map. So we can define

$$(G\otimes G)^{coinv G} :=\{g_1 \otimes g_2\in G\otimes G\mid \Delta_{G\otimes G} (g_1\otimes g_2):=g_1\otimes g_2 \otimes 1 \}.$$

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  • $\begingroup$ Yes, this is comodule structure on $G \otimes G$ that I am asking about. But I am looking for an explicit description. $\endgroup$ – Todd Claymore May 16 '20 at 16:38

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