Return to Answer

The case of irreducible, cocommutative Hopf algebras, over a field with $char(k)> 0$, is discussed in Sweedler's textbook on Hopf algebras, Ch.$XIII$, sect. $13.2$. (See prop. $13.2.2$, $13.2.3$).

For the case of cocommutative Hopf algebras, over an algebraically closed field $k$, with $char(k)\geq 3$ you can have a look at:

Modules of solvable infinitesimal groups and the structure of representation-finite cocommutative Hopf algebras, R.Farnsteiner, D.Voigt, Math. Proc. Cambridge Philos. Soc., v.127, p.441-459, 1999

and the references therein (among them, there is also an interesting older paper by the same authors discussing the case of cocommutative hopf algebras of finite representation type, over an algebraically closed field of $char(k)>0$).

Edit (July 2018): Since user's Qiaochu Yuan answer, provides a form of counter-example to the theorem, for non-algebraically closed fields, i think it would be of some added value to indicate a case in which the Cartier-Kostant-Milnor-Moore theorem is also valid for non-algebraically closed fields: this happens if we take as an assumption that we are dealing with a pointed hopf algebra. So, the (somewhat more general) statement of the theorem goes like:

If $H$ is a pointed, cocommutative hopf algebra over a field $k$ of characteristic $0$, we have that $$ H\cong U\big(P(H)\big)\ltimes kG(H) $$ where $P(H)$ is the lie algebra of the primitives, $U(.)$ its UEA and $G(H)$ the group of the group-likes.

The algebraically closed field case (mentioned in the OP), may be considered a special case of the above, taking into account that cocommutative hopf algebras over algebraically closed fields, can be easily seen to be pointed.

The case of irreducible, cocommutative Hopf algebras, over a field with $char(k)> 0$, is discussed in Sweedler's textbook on Hopf algebras, Ch.$XIII$, sect. $13.2$. (See prop. $13.2.2$, $13.2.3$).

For the case of cocommutative Hopf algebras, over an algebraically closed field $k$, with $char(k)\geq 3$ you can have a look at:

Modules of solvable infinitesimal groups and the structure of representation-finite cocommutative Hopf algebras, R.Farnsteiner, D.Voigt, Math. Proc. Cambridge Philos. Soc., v.127, p.441-459, 1999

and the references therein (among them, there is also an interesting older paper by the same authors discussing the case of cocommutative hopf algebras of finite representation type, over an algebraically closed field of $char(k)>0$).

Edit (July 2018): Since user's Qiaochu Yuan answer, provides a form of counter-example to the theorem, for non-algebraically closed fields, i think it would be of some added value to indicate cases in which the Cartier-Kostant-Milnor-Moore theorem remains valid for non-algebraically closed fields: this happens if we take as an assumption that we are dealing with pointed hopf algebras. So, the (somewhat more general) statement of the theorem goes like:

If $H$ is a pointed, cocommutative hopf algebra over a field $k$ of characteristic $0$, we have that $$ H\cong U\big(P(H)\big)\ltimes kG(H) $$ where $P(H)$ is the lie algebra of the primitives, $U(.)$ its UEA and $G(H)$ the group of the group-likes.

The algebraically closed field case (mentioned in the OP), may be considered a special case of the above, taking into account that cocommutative hopf algebras over algebraically closed fields, can be easily seen to be pointed.

The case of irreducible, cocommutative Hopf algebras, over a field with $char(k)> 0$, is discussed in Sweedler's textbook on Hopf algebras, Ch.$XIII$, sect. $13.2$. (See prop. $13.2.2$, $13.2.3$).

For the case of cocommutative Hopf algebras, over an algebraically closed field $k$, with $char(k)\geq 3$ you can have a look at:

Modules of solvable infinitesimal groups and the structure of representation-finite cocommutative Hopf algebras, R.Farnsteiner, D.Voigt, Math. Proc. Cambridge Philos. Soc., v.127, p.441-459, 1999

and the references therein (among them, there is also an interesting older paper by the same authors discussing the case of cocommutative hopf algebras of finite representation type, over an algebraically closed field of $char(k)>0$).

Edit (July 2018): Since user's Qiaochu Yuan answer, provides a form of counter-example to the theorem, for non-algebraically closed fields, i think it would be of some added value to indicate a case in which the Cartier-Kostant-Milnor-Moore theorem is also valid for non-algebraically closed fields: this happens if we take as an assumption that we are dealing with a pointed hopf algebra. So, the (somewhat more general) statement of the theorem goes like:

If $H$ is a pointed, cocommutative hopf algebra over a field $k$ of characteristic $0$, we have that $$ H\cong U\big(P(H)\big)\sharp kG(H) $$ where $P(H)$ is the lie algebra of the primitives, $U(.)$ its UEA and $G(H)$ the group of the group-likes.

The algebraically closed field case (mentioned in the OP), may be considered a special case of the above, taking into account that cocommutative hopf algebras over algebraically closed fields, can be easily seen to be pointed.

The case of irreducible, cocommutative Hopf algebras, over a field with $char(k)> 0$, is discussed in Sweedler's textbook on Hopf algebras, Ch.$XIII$, sect. $13.2$. (See prop. $13.2.2$, $13.2.3$).

For the case of cocommutative Hopf algebras, over an algebraically closed field $k$, with $char(k)\geq 3$ you can have a look at:

Modules of solvable infinitesimal groups and the structure of representation-finite cocommutative Hopf algebras, R.Farnsteiner, D.Voigt, Math. Proc. Cambridge Philos. Soc., v.127, p.441-459, 1999

and the references therein (among them, there is also an interesting older paper by the same authors discussing the case of cocommutative hopf algebras of finite representation type, over an algebraically closed field of $char(k)>0$).

Edit (July 2018): Since user's Qiaochu Yuan answer, provides a form of counter-example to the theorem, for non-algebraically closed fields, i think it would be of some added value to indicate a case in which the Cartier-Kostant-Milnor-Moore theorem is also valid for non-algebraically closed fields: this happens if we take as an assumption that we are dealing with a pointed hopf algebra. So, the (somewhat more general) statement of the theorem goes like:

If $H$ is a pointed, cocommutative hopf algebra over a field $k$ of characteristic $0$, we have that $$ H\cong U\big(P(H)\big)\ltimes kG(H) $$ where $P(H)$ is the lie algebra of the primitives, $U(.)$ its UEA and $G(H)$ the group of the group-likes.

The algebraically closed field case (mentioned in the OP), may be considered a special case of the above, taking into account that cocommutative hopf algebras over algebraically closed fields, can be easily seen to be pointed.

The case of irreducible, cocommutative Hopf algebras, over a field with $char(k)> 0$, is discussed in Sweedler's textbook on Hopf algebras, Ch.$XIII$, sect. $13.2$. (See prop. $13.2.2$, $13.2.3$).

For the case of cocommutative Hopf algebras, over an algebraically closed field $k$, with $char(k)\geq 3$ you can have a look at:

Modules of solvable infinitesimal groups and the structure of representation-finite cocommutative Hopf algebras, R.Farnsteiner, D.Voigt, Math. Proc. Cambridge Philos. Soc., v.127, p.441-459, 1999

and the references therein (among them, there is also an interesting older paper by the same authors discussing the case of cocommutative hopf algebras of finite representation type, over an algebraically closed field of $char(k)>0$).

The case of irreducible, cocommutative Hopf algebras, over a field with $char(k)> 0$, is discussed in Sweedler's textbook on Hopf algebras, Ch.$XIII$, sect. $13.2$. (See prop. $13.2.2$, $13.2.3$).

For the case of cocommutative Hopf algebras, over an algebraically closed field $k$, with $char(k)\geq 3$ you can have a look at:

Modules of solvable infinitesimal groups and the structure of representation-finite cocommutative Hopf algebras, R.Farnsteiner, D.Voigt, Math. Proc. Cambridge Philos. Soc., v.127, p.441-459, 1999

and the references therein (among them, there is also an interesting older paper by the same authors discussing the case of cocommutative hopf algebras of finite representation type, over an algebraically closed field of $char(k)>0$).

Edit (July 2018): Since user's Qiaochu Yuan answer, provides a form of counter-example to the theorem, for non-algebraically closed fields, i think it would be of some added value to indicate a case in which the Cartier-Kostant-Milnor-Moore theorem is also valid for non-algebraically closed fields: this happens if we take as an assumption that we are dealing with a pointed hopf algebra. So, the (somewhat more general) statement of the theorem goes like:

If $H$ is a pointed, cocommutative hopf algebra over a field $k$ of characteristic $0$, we have that $$ H\cong U\big(P(H)\big)\sharp kG(H) $$ where $P(H)$ is the lie algebra of the primitives, $U(.)$ its UEA and $G(H)$ the group of the group-likes.

The algebraically closed field case (mentioned in the OP), may be considered a special case of the above, taking into account that cocommutative hopf algebras over algebraically closed fields, can be easily seen to be pointed.