Unipotent element of linear algebraic group

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Definition

Finite-dimensional case

Suppose G is a linear algebraic group over a field k, i.e., G is identified as a closed subgroup of the general linear group GL(n,k). Denote the identity element of G by 1.

An element g \in G is termed unipotent if g - 1 (the subtraction being done with respect to the additive structure of matrices) is a nilpotent element in the matrix ring M(n,k). By degree considerations, this is equivalent to requiring that (g - 1)^n = 0, where the powering is done in the matrix ring.