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

## Contents

## Statement

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

- The additive group of the field (the unipotent case)
- The multiplicative group of the field (which, as a set, comprises all the non-identity elements of the field) (the semisimple case)
- 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

## 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 . 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 roots of unity for any positive integer . This is a cyclic subgroup of order dividing (equality occurs if the field is algebraically closed of characteristic not dividing ), and is a closed normal subgroup. | The subgroup is the kernel of the multiplicative group endomorphism . 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 | the additive group of any finite subfield of order , a natural number, is a closed normal subgroup | The subgroup is the kernel of the additive group endomorphism (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. |