Second cohomology group for trivial group action of additive group of two-dimensional vector space on additive group of a field

Suppose $$K$$ is a field. The goal of this page is to describe the algebraic second cohomology group for trivial group action:

$$H^2_{\mbox{alg}}(G_a \times G_a;G_a)$$

where $$G_a$$ is the additive group of $$K$$, and also to discuss the corresponding group extensions.

Case of characteristic zero
In this case, we obtain:

$$H^2_{\mbox{alg}}(G_a \times G_a;G_a) \cong G_a$$


 * All the non-identity elements are in one orbit under the automorphism action, and this orbit gives the unitriangular matrix group of degree three over $$K$$.
 * The identity element corresponds to $$G_a \times G_a \times G_a$$, i.e., the additive group of the three-dimensional vector space over $$G_a$$.

Case of prime characteristic not equal to two
In this case, we obtain:

$$H^2_{\mbox{alg}}(G_a \times G_a;G_a) \cong G_a \times G_a \times G_a$$

Case of characteristic two
In this case, we obtain (?):

$$H^2_{\mbox{alg}}(G_a \times G_a;G_a) \cong G_a \times G_a \times G_a$$

Related facts

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

Related cohomology groups

 * Second cohomology group for trivial group action of elementary abelian group of prime-square order on group of prime order