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. 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 of upper-triangular matrices with 's along the diagonal, so they are the group of matrices Then, a unipotent group can be defined as a subgroup of some . Using scheme theory the group can be defined as the group scheme 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, rx, 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 GLn(k)). For example, the standard representation of on with standard basis has the fixed vector . 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. In particular, this implies there are no non-trivial semisimple representations. Examples Un Of course, the group of matrices is unipotent. Using the lower central series where and there are