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I am studying a subcategory $\mathcal{C}$ of modules for an associative noncommutative algebra $A$ (which is in fact also a Hopf algebra).

It is clear from our definition of $\mathcal{C}$ that it is an abelian symmetric monoidal category, which is nice. But recently, just out of curiosity, I proved that $\mathcal{C}$ is also rigid.

I would like to know what this implies for the representations of $A$. I have skimmed through a lot of books on category theory, and most of them mention rigid categories (so I think it has its relevance), but I am unaware of the consequences of this fact for representation theory.

Any comment with examples of the consequences of the rigidness of the category with details/references is welcome.

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    $\begingroup$ Are you working over a field? If so look up Tannakian categories. $\endgroup$
    – Ben
    Commented Jun 18 at 15:04
  • $\begingroup$ What do you mean 'consequences [...] for representation theory'? Rigidity is itself a property of the category of representations, isn't it? What exactly are you looking for? $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jun 18 at 20:50
  • $\begingroup$ @R.vanDobbendeBruyn I am asking myself if some proof or result we have for our modules would be simplified by invoking the fact that the category is rigid $\endgroup$
    – jg1896
    Commented Jun 19 at 11:58

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