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. 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

U

n

{\displaystyle \mathbb {U} _{n}}

of upper-triangular matrices with

1

{\displaystyle 1}

's along the diagonal, so they are the group of matrices

U

n

=

{

[

1

∗

⋯

∗

∗

0

1

⋯

∗

∗

⋮

⋮

⋮

⋮

0

0

⋯

1

∗

0

0

⋯

0

1

]

}

.

{\displaystyle \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\}.}

Then, a unipotent group can be defined as a subgroup of some

U

n

{\displaystyle \mathbb {U} _{n}}

. Using scheme theory the group

U

n

{\displaystyle \mathbb {U} _{n}}

can be defined as the group scheme

Spec

(

C

[

x

11

,

x

12

,

…

,

x

n

n

,

1

det

]

(

x

i

i

=

1

,

x

i

>

j

=

0

)

)

{\displaystyle {\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)}

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].