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I am not sure if you consider these creative but some typical examples of additive categories are the category $\mathcal{R}$-mod, of modules over a ring or a $k$-algebra $\mathcal{R}$, the category $ …

I was revising some old postgraduate notes of mine in homological algebra (written during a postgrad course on the topic, I had taken more than ten ;) years ago) and I came accross the following probl …

I am not sure whether the following point of view is what you are asking for, in the sense that it is not some further development of your observations, but a seemingly different categorical descripti …

The answer is yes, if we are talking about finite dimensional, Hopf algebras over a field: $\bullet$ $H$ being cosemisimple (as a coalgebra) is equivalent to the dual hopf algebra $H^*$ being semi …

I will try to provide an answer for a particular case of your last question: Let us consider (following my comment above) the case of $H=\mathbb{CZ}_2$ i.e. the group hopf algebra equipped with its no …

$\DeclareMathOperator\Hker{Hker}\DeclareMathOperator\Hcoker{Hcoker}\DeclareMathOperator\Im{Im}\DeclareMathOperator\coIm{coIm}\DeclareMathOperator\Id{Id}$The category $\mathcal{H}$ of finite dimension …

The term "Quantum groups" itself, implies that the development of the hopf algebra theory generalizes -in some categorical sense- usual group theory. There are various points that might support this v …

The category of the finite dimensional comodules of a hopf algebra over a field, is a rigid, monoidal category. (just as the category of the finite dimensional modules). If we take a fin dim, righ …

The following is not really an answer but a rather too-long comment, with respect to your second question: Does there exist deformations of the monoidal category of $U(\frak{g})$-modules which a …

I do not know much about the Tannaka-Krein duality itself. But regarding the last part of your question Also if somebody could cast some light on possible generalizations of this proposition (to the …