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

Statement
The only connected  algebraic groups of dimension equal to 1 over an  algebraically closed field are the following:


 * 1) The additive group of the field (the unipotent case)
 * 2) The multiplicative group of the field (which, as a set, comprises all the non-identity elements of the field) (the semisimple case)
 * 3) An elliptic curve group over the field (the complete variety/abelian variety case)

Note that for the case of a linear algebraic group (or equivalently, an affine algebraic group), case (3) does not apply and thus cases (1) and (2) are the only possibilities.

Related classifications

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

Not all quotient maps are algebraic
Quotient map of Lie group structures for algebraic groups need not be quotient map of algebraic groups: The significance in this context is as follows. Consider the field of complex numbers $$\mathbb{C}$$. The additive group is a connected one-dimensional linear algebraic group. We can quotient this out by a lattice and get a complex torus. This quotient map is a quotient map of Lie group structures; however, it is not a quotient map of algebraic groups. Thus, the quotient does not immediately get an algebraic group structure. In fact, the quotient can be given an algebraic group structure more indirectly as an elliptic curve group. However, the quotient map itself is not a regular morphism and in fact its explicit description involves transcendental functions.

The one-dimensional algebraic groups are algebraic quotients of themselves
The connected one-dimensonal algebraic groups are not simple, i.e., they do have proper nontrivial closed normal subgroups. However, the quotients by these normal subgroups are in many cases isomorphic to the groups themselves. In other words, the groups are not Hopfian groups, even when restricted to algebraic maps.