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)<sup>n</sup> 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. Thus all the eigenvalues of a unipotent matrix 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 the theory of algebraic groups, a group element is unipotent if it acts unipotently in a certain natural group representation. A unipotent affine algebraic group is then a group with all elements unipotent.
Definition
Definition with matrices
Consider the group <math>\mathbb{U}_n</math> of upper-triangular matrices with <math>1</math>'s along the diagonal, so they are the group of matrices
:<math>\mathbb{U}_n = \left\{
\begin{bmatrix}
1 & * & \cdots & * & * \\
0 & 1 & \cdots & * & * \\
\vdots & \vdots & &\vdots & \vdots \\
0& 0& \cdots & 1 &* \\
0 & 0 & \cdots & 0 & 1
\end{bmatrix}
\right\}.</math>
Then, a unipotent group can be defined as a subgroup of some <math>\mathbb{U}_n</math>. Using scheme theory the group <math>\mathbb{U}_n</math> can be defined as the group scheme
:<math>\text{Spec}\left(
\frac{\mathbb{C}\!\left[x_{11},x_{12},\ldots, x_{nn}, \frac{1}{\text{det\right]}{
(x_{ii} = 1, x_{i > j} = 0)
}
\right)</math>
and an affine group scheme is unipotent if it is a closed group scheme of this scheme.
Definition with ring theory
An element x of an affine algebraic group is unipotent when its associated right translation operator, r<sub>x</sub>, 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-theoretic sense.)
An affine algebraic group is called unipotent if all its elements are unipotent. Any unipotent algebraic group is isomorphic to a closed subgroup of the group of upper triangular matrices with diagonal entries 1, and conversely any such subgroup is unipotent. In particular any unipotent group is a nilpotent group, though the converse is not true (counterexample: the diagonal matrices of GL<sub>n</sub>(k)).
For example, the standard representation of <math>\mathbb{U}_n</math> on <math>k^n</math> with standard basis <math>e_i</math> has the fixed vector <math>e_1</math>.
Definition with representation theory
If a unipotent group acts on an affine variety, all its orbits are closed, and if it acts linearly on a finite-dimensional vector space then it has a non-zero fixed vector. In fact, the latter property characterizes unipotent groups.<sup>page 8</sup>
:<math>0 \to M\times U \to G \to A \to 0</math>
where <math>A</math> is an abelian variety, <math>M</math> is of multiplicative type (meaning, <math>M</math> is, geometrically, a product of tori and algebraic groups of the form <math>\mu_n</math>) and <math>U</math> is a unipotent group.
Characteristic p
When the characteristic of the base field is p there is an analogous statement