"What are the properties of the product of two non-reductive algebraic groups? More specifically, is the product of two non-reductive groups always non-reductive? If not, what conditions determine the reductivity or non-reductivity of the product? How does the unipotent radical of the product behave in relation to the unipotent radicals of the original groups?"
Properties of the product of two non-reductive algebraic groups:
Non-reductivity: The product of two non-reductive groups isn't always non-reductive; this outcome relies on certain specific conditions.
Conditions that determine whether a group is reductive or non-reductive include:
- The structure of the individual groups
- The manner in which these groups interact within the product
- The characteristics of their respective unipotent radicals
Unipotent radical behavior:
The unipotent radical of the product is connected to the unipotent radicals of the original groups through their structural interactions.
However, this relationship is not straightforward; the behavior can become quite complex based on how the groups are combined.
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