Unipotent linear algebraic group

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Definition

A linear algebraic group G over a field k (which therefore comes equipped with an embedding as a closed subgroup of the general linear group GL(n,k)) is termed a unipotent linear algebraic group if it satisfies the following equivalent conditions:

  1. Every element in G is a unipotent element, i.e., subtracting 1 from it gives a nilpotent element in the matrix ring.
  2. G is conjugate in GL(n,k) 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 n - 1, where n 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]] k, which is the full group GL(1,k), is abelian and hence nilpotent, but is not unipotent.

A related fact in the context of Lie algebras is Engel's theorem.