Unipotent

In mathematics, a unipotent element, r, of a ring, R, is one such that r − 1 is a nilpotent element; in other words, (r − 1)ⁿ is zero for some n.

In particular, a square matrix, M, is a unipotent matrix, if and only if its characteristic polynomial, P(t), is a power of t − 1. Equivalently, M is unipotent if all its eigenvalues are 1.

The term quasi-unipotent means that some power is unipotent, for example for a diagonalizable matrix with eigenvalues that are all roots of unity.

In an unipotent affine algebraic group all elements are unipotent (see below for the definition of an element being unipotent in such a group).

Unipotent algebraic groups

An element, x, of an affine algebraic group is unipotent when its associated right translation operator, r_x, on the affine coordinate ring A[G] of G is locally unipotent as an element of the ring of linear endomorphism of A[G]. (Locally unipotent means that its restriction to any finite-dimensional stable subspace of A[G] is unipotent in the usual ring sense.)