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1. FGC for Hopf algebras

Questions related to FGC conjecture for Hopf algebras

    1. Exact sequences of Hopf algebras


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      Problem 1.1.

      [Nicolas Andruskiewitsch] Consider an exact sequence of Hopf algebras $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$, where $B$ is finite-dimensional. If $A$ and $C$ satisfy the FGC conjecture, does $B$ satisfy it as well?
      1. Consider a supergroup scheme $G$. Does the Drinfeld double satisfy the FGC conjecture? (It has been done for group schemes by Negron.)
      2. (Pablo Sanchez Ocal) Consider twisted tensor products.
      3. (Van Nguyen) Consider the case when $C$ is a group algebra.
          For recent progress, see https://arxiv.org/abs/2407.05881.


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          Problem 1.2.

          [Hector Pena Pollastri] Let $B$ be a finite-dimensional bialgebra over a field $\mathbf{k}$. What are examples of $B$ for which the FGC conjecture is not true?


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              Problem 1.3.

              [Nicolas Andruskiewitsch] Let $H$ be a finite-dimensional Hopf algebra. Let $\mathcal{C}$ denote the semisimplification of its representation category. Does $\mathcal{C}$ have nice subcategories $\mathcal{D}$? Consider Hopf algebras of tame representation type and their semisimplifications.


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                  Problem 1.4.

                  [Karin Erdmann] Does there exist a scheme parameterizing types of Hopf algebras? (Nichols algebras would be one of the types.)


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                      Problem 1.5.

                      [Antoine Touze] Let $H$ be a finite-dimensional Hopf algebra over a field $\mathbf{k}$. Suppose $H$ acts on a finitely generated algebra $A$ over $\mathbf{k}$. When is $H^0(H, A)$ (i.e. the invariants) finitely generated?
                        1. Remark. This could help prove that FGC for $H^*(\mathcal{C})$ implies FGC for $\text{Ext}(X, X)$ over $H^*(\mathcal{C})$.


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                              Problem 1.6.

                              [Nicolas Andruskiewitsch] There is a list of a few finite-dimensional Nichols algebras $B(X)$ for $X$ an irreducible Yetter-Drinfeld module over a finite non-abelian group $G$. For these, determine when the FGC conjecture holds.

                                  Cite this as: AimPL: Finite tensor categories: their cohomology and geometry, available at http://aimpl.org/finitetensor.