Unipotent

Unipotent

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

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.

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

Read more about Unipotent:  Unipotent Algebraic Groups, Unipotent Radical, Jordan Decomposition

Other articles related to "unipotent":

Unipotent - Jordan Decomposition
... over a perfect field can be written uniquely as the product g = gugs of commuting unipotent and semisimple elements gu and gs ... linear algebraic group over a perfect field is the product of a unipotent group and a semisimple group ...
Springer Correspondence - Example
... For the special linear group SLn, the unipotent conjugacy classes are parametrized by partitions of n if u is a unipotent element, the corresponding partition is given by the ... of the Weyl group corresponds to the regular unipotent class, and the sign representation corresponds to the identity element of G) ...
Reductive Group
... group is an algebraic group G over an algebraically closed field such that the unipotent radical of G is trivial (i.e ... the group of unipotent elements of the radical of G) ... algebraically closed, a reductive group is a smooth affine algebraic group such that the unipotent radical of G over the algebraic closure is trivial ...
Ratner's Theorems
... are a group of major theorems in ergodic theory concerning unipotent flows on homogeneous spaces proved by Marina Ratner around 1990 ... The study of the dynamics of unipotent flows played a decisive role in the proof of the Oppenheim conjecture by Grigory Margulis ... theorems have guided key advances in the understanding of the dynamics of unipotent flows ...
Deligne–Lusztig Theory - Lusztig's Classification of Irreducible Characters - Jordan Decomposition
... An irreducible character is called unipotent if it occurs in some R1T, and is called semisimple if its average value on regular unipotent elements is non-zero (in which case the average value is 1 or ... character (corresponding to some semisimple element s of the dual group), and a unipotent representation of the centralizer of s ... of the irreducible character is the product of the dimensions of its semisimple and unipotent components ...