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

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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

Caveats

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.

Case Possibilities for proper closed normal subgroup (equivalent to possibilities for finite subgroup) How we can see that the quotient by any such subgroup is isomorphic to the original group if the field is algebraically closed
multiplicative group of a field (see also subgroup structure of multiplicative group of a field) subgroup of n^{th} roots of unity for any positive integer n. This is a cyclic subgroup of order dividing n (equality occurs if the field is algebraically closed of characteristic not dividing n), and is a closed normal subgroup. The subgroup is the kernel of the multiplicative group endomorphism x \mapsto x^n. This is a surjective endomorphism because the field is algebraically closed, hence, by the first isomorphism theorem, the multiplicative group is isomorphic to its quotient by the kernel.
additive group of a field of characteristic p the additive group of any finite subfield of order p^n, n a natural number, is a closed normal subgroup The subgroup is the kernel of the additive group endomorphism x \mapsto x^{p^n} - x (this is an endomorphism because it uses the Frobenius endomorpism). This is a surjective endomorphism because the field is algebraically closed, hence, by the first isomorphism theorem, the additive group is isomorphic to its quotient by the kernel.