Further developments of Cartier–Gabriel–Kostant–Milnor–Moore Structure Theorem for cocommutative Hopf algebras

Question

A very well-known theorem in Hopf algebra theory (see, for example, Lorenz - A tour of representation theory or the EGNO book (Etingof, Gelaki, Nikshych, and Ostrik - Tensor categories)) states that if $H$ is a cocommutative Hopf algebra over an algebraically closed field $\Bbbk$ of characteristic zero then $$H \cong U(P(H)) \mathbin\# \Bbbk G(H)$$ as Hopf algebras, where $P(H)$ is the space of primitive elements of $H$ and $G(H)$ the group of group-like elements. As far as I know, this theorem is attributed differently to many people, but I believe the list in the title should be comprehensive of all of them. For example, I know that the Milnor–Moore part of the theorem is the one stating that an irreducible cocommutative Hopf algebra $H'$ over a field of characteristic zero satisfies $H' \cong U(P(H'))$.

I know that Nichols proved, in The Kostant structure theorem for $K/k$ Hopf algebras, an analogue of the full CGKMM theorem for Hopf algebroids of the form $(K,H)$, where $K$ is a field extension of $\Bbbk$, that Moerdijk and Mrčun proved, in On the universal enveloping algebra of a Lie algebroid, an analogue of Milnor–Moore for bialgebroids and that, later, Kališnik and Mrčun proved, in A Cartier–Gabriel–Kostant structure theorem for Hopf algebroids, an analogue of the full Cartier–Gabriel–Kostant–Milnor–Moore for Hopf algebroids over the algebra $\mathcal{C}^{\infty}_c(\mathcal{M})$ of smooth functions with compact support over a smooth real manifold $\mathcal{M}$.

Is anybody aware of some further developments/extensions of this result to other classes of Hopf algebras (for example, weak Hopf algebras, general Hopf algebroids, (co)quasi-Hopf algebras, etc.)?

Otherwise, is there anybody aware of counter-examples that suggest this could not be generalized further in some direction?

Edit: Just to add some further motivation to this quest. Let $G$ be a linear algebraic group over an algebraically closed field $\Bbbk$ of characteristic zero. The coordinate algebra $\mathscr{O}(G)$ is a commutative Hopf algebra. If we consider the subspace $\mathscr{O}(G)^\circ$ of $\mathsf{Hom}_\Bbbk\left(\mathscr{O}(G),\Bbbk\right)$ formed by all those linear functionals which vanish on a finite-codimensional ideal, this is a cocommutative Hopf algebra. By the CGKMM Theorem we know that $$\mathscr{O}(G)^\circ \cong U\left(P\left(\mathscr{O}(G)^\circ\right)\right) ~\#~ \Bbbk G\left(\mathscr{O}(G)^\circ\right).$$ It happens that $P\left(\mathscr{O}(G)^\circ\right) \cong \mathfrak{g}$, the tangent Lie algebra to $G$ at the neutral element, and $G\left(\mathscr{O}(G)^\circ\right) \cong \mathsf{Alg}_\Bbbk\left(\mathscr{O}(G),\Bbbk\right) \cong G$, so that $$\mathscr{O}(G)^\circ \cong U\left(\mathfrak{g}\right) ~\#~ \Bbbk G,$$ where on the right-hand side $G$ is considered as a discrete group and $U(\mathfrak{g}) \cong \mathsf{Dist}_G$, the hyperalgebra of distributions on $G$.

Is there anybody aware of some further results like this one for groupoids?

Answer 1

The super version of the theorem, refers to "hopf superalgebras", or "$\mathbb{Z}_2$-graded hopf algebras" or "hopf algebras in the braided monoidal category of $\mathbb{CZ}_2$-modules":
Let $\mathcal{H}$ be a super-cocommutative hopf superalgebra over an algebraically closed field $k$ of char zero. Then we have the hopf superalgebra isomorphism: $$ \mathcal{H}\cong k[G(\mathcal{H})]\ltimes_{\pi} U\big(P(\mathcal{H})\big) $$ where, $k[G(\mathcal{H})]$ is the group algebra of the the group $G(\mathcal{H})$ of the grouplikes of $\mathcal{H}$, $U\big(P(\mathcal{H})\big)$ is the universal enveloping algebra of the lie superalgebra $P(\mathcal{H})$ of the primitive elements of $\mathcal{H}$ and the smash product $\ltimes_{\pi}$ is with respect to the representation of $G(\mathcal{H})$ on $P(\mathcal{H})$ determined by: $\pi:G\to Aut(P)$, $\pi(g)x=gxg^{-1}$, for all $g\in G$, $x\in P$.

This has been an old result, first shown by Kostant, at:
B.Kostant, "Graded manifolds, graded Lie theory and prequantization", Differential Geometrical Methods in Mathematical Physics. Lecture Notes in Mathematics, vol 570, p. 177-306, (1977))

P.S.: What i do not know, is whether there is a corresponding version of the theorem for Hopf algebras in braided monoidal categories. I am not aware of some reference in that direction (although, i think the generalization should not be that difficult to prove). The only related reference i know of is Braided bialgebras of Hecke-type, A. Ardizzoni, C.Menini, D. Stefan, Journal of Algebra 321 (2009) 847–865, where a result -analogous to the Milnor-Moore part of the theorem- is proved for connected, braided bialgebras which are infinitesimally cocommutative (see theorem 5.5 and the resulting discussion).

Edit: another possible line of generalization of the theorem , has to do with the class of quasitriangular hopf algebras. These, generalise the cocommutative hopf algebras, so it is natural to consider the possibility of a "quasitriangular version" of the theorem. See for example: Classification of quasitriangular Hopf algebras