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?

  • 2
    $\begingroup$ 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. $\endgroup$
    – Adrien
    Jun 7, 2020 at 20:34

1 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

  • $\begingroup$ You have my sincere gratitude for all your inputs. Alas I cannot +1 you more than once. I am leaving the question unanswered to see if somebody else shows up with other references $\endgroup$ Jun 10, 2020 at 6:25
  • 2
    $\begingroup$ Thank you. I have just now realized that you have been a coauthor of A. Ardizzoni and D. Stefan, so maybe, you were already aware of their work mentioned in my PS above :) $\endgroup$ Jun 10, 2020 at 11:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.