Closed normal subgroup for Lie structure need not be closed normal subgroup of algebraic group

Statement
It is possible to have an algebraic group $$G$$ over a field admitting analytic structure (such as the field of real numbers or field of complex numbers), and a normal subgroup $$H$$ of $$G$$ such that $$H$$ is a closed normal subgroup of $$G$$ under the topology for the induced Lie group structure, but is not a closed normal subgroup under the Zariski topology for the algebraic group structure.

Related facts

 * Quotient map of Lie group structures for algebraic groups need not be quotient map of algebraic groups: Uses essentially the same examples, gives a new algebraic group structure to the quotient group.

Over the real numbers
Let $$G = \R$$ and $$H = \mathbb{Z}$$ be the subgroup of integers. Then:


 * $$H$$ is a discrete closed normal subgroup of $$G$$ with the Lie group topology, which is the usual Euclidean topology.
 * $$H$$ is not closed in $$G$$ under the Zariski topology because the only proper closed subsets of a one-dimensional space are the finite subsets.

Note that the quotient group $$G/H$$ is a Lie group isomorphic to the circle group. In fact, the circle group can also be given the structure of an algebraic group in a natural way by using coordinates. However, the quotient mapping from the real numbers to the circle group (realized as $$t \mapsto (\cos(2\pi t),\sin(2\pi t))$$ is not a regular morphism, i.e., it is not a mapping of algebraic varieties. For more, see quotient map of Lie group structures for algebraic groups need not be quotient map of algebraic groups

Over the complex numbers
Let $$G = \mathbb{C}$$ and $$H = \mathbb{Z}[i]$$ be the subgroup of Gaussian integers, i.e., numbers of the form $$\{a + bi \mid a,b \in \mathbb{Z} \}$$. Then:


 * $$H$$ is a discrete closed normal subgroup of $$G$$ with the Lie group topology, which is the usual Euclidean topology.
 * $$H$$ is not closed in $$G$$ under the Zariski topology because the only proper closed subsets of a one-dimensional space are the finite subsets.

Note that the quotient group $$G/H$$ can be realized as an elliptic curve group in multiple ways.