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

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?

• There is an operadic version of Milnor-Moore's theorem which characterizes enveloping algebras in arbitrary symmetric monoidal categories (whenever this notion makes sense). This is theorem 6.1 in arxiv.org/abs/math/0306212. Jun 7, 2020 at 20:34

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 $$g\in G$$, $$x\in P$$.