Let $H$ be a cosemisimple Hopf algebra (or just a bialgebra). Considering $H$ as a left $H$comodule (i.e. take $\Delta_H$ to be the coaction), can there exist a (nonidentity) algebra map $\sigma:H \to H$ which is also a left $H$comodule map. It seems to be that such things cannot exist, but I can see a good way to prove it . . . For example if we assume that the subcoalgebras of $H$ are nonisomorphic and that $H$ is generated by the elements of a single subcoalgebra, then the result seems to work . . . but it is not clear to me that these conditions are necessary.
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2$\begingroup$ For a nonidentity example, maybe one could take the group algebra of a finite group and the Hopf algebra automorphism induced by an automorphism of this group? I don't understand the notion of cosemisimplicity well enough to know if this can be made to work. $\endgroup$– Mark WildonApr 6, 2021 at 11:14

$\begingroup$ Another thought: how about the coordinate ring of the algebraic group $\mathrm{GL}_2(\mathbb{C})$, as generated by the matrix coefficient functions $f_{11},f_{12},f_{21},f_{22}$ and the inverse of the determinant. It is graded by degree and the dual of the degree $r$ component is the Schur algebra $S(2,r)$, which is semisimple over $\mathbb{C}$. Does this make it cosemisimple? If so, there should be an example using the automorphism of $\mathrm{GL}_2(\mathbb{C})$ sending a matrix to the inverse of its transpose; for instance this induces $f_{11} \mapsto f_{22} d^{1}$. $\endgroup$– Mark WildonApr 6, 2021 at 16:47

$\begingroup$ For the "just a bialgebra" case, consider $H = K[x]/(x^2  x)$ with $x$ grouplike and $\sigma(x) = 0$. $\endgroup$– lambdaApr 6, 2021 at 17:38

$\begingroup$ @Mark: the coordinate ring of $\mathrm{GL}_2(\mathbb{C})$ is indeed cosemisimple since $\mathrm{GL}_2(\mathbb{C})$ is reductive . . . I need to think about the automorphism though . . . $\endgroup$– Tim CrombyApr 7, 2021 at 13:38
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