Classification of connected unipotent abelian algebraic groups over an algebraically closed field

Statement

In characteristic zero: connected case

Suppose $K$ is an algebraically closed field of characteristic zero and $G$ is a finite-dimensional connected unipotent abelian algebraic group over $K$. Then, $G$ is isomorphic, as an algebraic group, to the direct product of finitely many copies of the additive group of $K$. Further, the number of copies used equals the dimension of $G$.

In particular, this means that for every fixed dimension, there is a unique isomorphism class of connected unipotent abelian algebraic group of that dimension.

In characteristic zero: general case

For an algebraically closed field of characteristic zero, all unipotent algebraic groups are connected, so the above classification for the connected case is also a classification of the general case.

In prime characteristic: connected case

Suppose $K$ is an algebraically closed field of characteristic a prime number $p$ and $G$ is a finite-dimensional connected unipotent abelian algebraic group over $K$. Then, $G$ is isomorphic, as an algebraic group, to the direct product of finitely many algebraic groups, each of which is the additive group of a truncated ring of Witt vectors over $K$. Further, the sum of the lengths of each of the truncated rings used in the direct product equals the dimension of $G$ as an algebraic group over $K$.

In particular, this means that for dimension $k$, the number of isomorphism classes of connected unipotent abelian algebraic groups equals the number of unordered integer partitions of $k$.

In prime characteristic: general case

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-- roughly, must be isogenous to a connected group of the same dimension, so we can use the classification for the connected case.

Related facts

Similar classifications for algebraic groups

• Classification of connected one-dimensional algebraic groups over an algebraically closed field: There are three types of such groups: the additive group, multiplicative group, and elliptic curve groups. If we are interested in the unipotent case, then the additive group is the only possibility.
• Classification of connected two-dimensional unipotent algebraic groups over an algebraically closed field
• Classification of connected three-dimensional unipotent algebraic groups over an algebraically closed field

Similar classifications for finite groups

• Classification of finite abelian groups: For the case of order $p^{k}$, the finite abelian groups of that order are parametrized by the set of unordered integer partitions of $k$. For any unordered integer partition $k=m_{1}+m_{2}+\dots +m_{r}$, the corresponding abelian group is the direct product of the cyclic groups $\mathbb {Z} /p^{m_{i}}\mathbb {Z}$. This is related as follows: an abelian group of prime power order $p^{k}$ can be thought of as the $\mathbb {F} _{p}$-points of a $k$-dimensional connected unipotent abelian algebraic group over the algebraic closure of $\mathbb {F} _{p}$. Each cyclic group $\mathbb {Z} /p^{m_{i}}\mathbb {Z}$ comprises the $\mathbb {F} _{p}$-points of the truncated ring of Witt vectors of length $m_{i}$.