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

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

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 $$m_i$$.