Classification of connected unipotent three-dimensional algebraic groups over an algebraically closed field

In characteristic zero
Suppose $$K$$ is an algebraically closed field of characteristic zero and $$G$$ is a connected unipotent two-dimensional algebraic group over $$K$$. Then, there are two possibilities for $$G$$:


 * 1) $$G$$ is an abelian algebraic group, and is isomorphic to a direct product of three copies of the additive group of $$K$$. In other words, it is the additive group of a three-dimensional vector space over $$K$$.
 * 2) $$G$$ is an algebraic group of nilpotency class two, and is isomorphic to the unitriangular matrix group of degree three over $$K$$.

In prime characteristic not equal to two
Suppose $$K$$ is an algebraically closed field of characteristic equal to a prime number $$p$$ not equal to 2 and $$G$$ is a connected unipotent two-dimensional algebraic group over $$K$$. Then, there are five possibilities of $$G$$.

The three possibilities for abelian $$G$$ are given by the classification of connected unipotent abelian algebraic groups over an algebraically closed field, and correspond to the set of unordered integer partitions of 3. These are:

There are two possibilities for non-abelian $$G$$, both of nilpotency class two, namely:


 * The upper-triangular unipotent matrix group of degree three over $$K$$.
 * semidirect product of length two Witt ring and additive group: A semidirect product of the truncated ring of Witt vectors of length two by the additive group, where the action is suitably defined.

Related classifications

 * Classification of groups of prime-cube order is a finite group classification that mimics the prime characteristic case. In fact, for $$p > 2$$, the groups of order $$p^3$$ can be naturally viewed as the $$\mathbb{F}_p$$-points of connected unipotent three-dimensional algebraic groups over $$\mathbb{F}_p$$.