Unipotent linear algebraic group
A linear algebraic group over a field (which therefore comes equipped with an embedding as a closed subgroup of the general linear group ) is termed a unipotent linear algebraic group if it satisfies the following equivalent conditions:
- Every element in is a unipotent element, i.e., subtracting from it gives a nilpotent element in the matrix ring.
- is conjugate in to a subgroup of the upper-triangular unipotent matrix group.
Relation with being nilpotent
Any unipotent linear algebraic group is nilpotent, with its nilpotency class at most , where is the degree of the general linear group it is embedded in. However, the converse is not true: it is possible to have a linear algebraic group that is nilpotent as an abstract group but is not unipotent. For instance, the [[[multiplicative group of a field]] , which is the full group , is abelian and hence nilpotent, but is not unipotent.
A related fact in the context of Lie algebras is Engel's theorem.