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 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 an unipotent affine algebraic group, all elements are unipotent (see below for the definition of an element being unipotent in such a group). An element, x, of an affine algebraic group is unipotent when its associated right translation operator, rx, on the affine coordinate ring A of G is locally unipotent as an element of the ring of linear endomorphism of A. (Locally unipotent means that its restriction to any finite-dimensional stable subspace of A is unipotent in the usual ring 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)).