Unipotent element

In algebra , the term unipotent element is a generalization of the unipotent matrices known from linear algebra , for example the upper triangular matrices with ones on the main diagonal .

definition

Let it be a ring with a single element . An element is called unipotent if it is nilpotent , that is, if ${\displaystyle R}$ ${\displaystyle 1}$ ${\displaystyle r\in R}$ ${\displaystyle r-1}$

${\displaystyle (r-1)^{n}=0}$

for one is. ${\displaystyle n\in \mathbb {N} }$

Unipotent matrices

→ Main article : Unipotent matrix

For a ring and the square dies also form a ring. In this matrix ring , the identity matrix is the one element. The unipotent elements in this ring are called unipotent matrices. For example, all of the upper triangular matrices that have only ones on the diagonal are unipotent because they satisfy ${\displaystyle R}$ ${\displaystyle m\in \mathbb {N} }$ ${\displaystyle Mat(m,R)}$ ${\displaystyle I_{m}}$ ${\displaystyle A}$

${\displaystyle (A-I_{m})^{m}=0}$.

Unipotent operators

An operator acting on a vector space or module is called unipotent if ${\displaystyle T}$

${\displaystyle (T-Id)^{n}=0}$

applies to one . It is called locally unipotent if its restriction to every -invariant finite-dimensional subspace is unipotent. ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle T}$

Every automorphism of a finite-dimensional vector space over an algebraically closed field has a unique multiplicative Jordan-Chevalley decomposition of the form ${\displaystyle S}$ ${\displaystyle V}$

${\displaystyle S=S_{d}\circ S_{u}=S_{u}\circ S_{d}}$,

where are a semi-simple ( diagonalizable ) and a unipotent automorphism. If a - is stable subspace of , then - and is also stable with the decomposition ${\displaystyle S_{d}}$ ${\displaystyle S_{u}}$ ${\displaystyle W}$ ${\displaystyle S}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle S_{d}}$ ${\displaystyle S_{u}}$

${\displaystyle S|_{W}=S_{d}|_{W}\circ S_{u}|_{W}=S_{u}|_{W}\circ S_{d}|_{W}}$.

Unipotent algebraic groups

An element of an algebraic group is called unipotent if the operator defined by right multiplication with on the coordinate ring is locally unipotent. ${\displaystyle g}$ ${\displaystyle G}$ ${\displaystyle g}$ ${\displaystyle A\left[G\right]}$

An algebraic group over a field is called unipotent if all of its elements are unipotent. In particular, it then holds for every representation that is a unipotent matrix. ${\displaystyle G}$ ${\displaystyle K}$ ${\displaystyle g\in G}$ ${\displaystyle \rho \colon G\to GL(n,R)}$ ${\displaystyle \rho (g)}$

An algebraic group is unipotent if and only if it is isomorphic to a closed subgroup of a group of upper triangular matrices with ones on the diagonals .

Unipotent algebraic groups are characterized by the following property: for each linear effect of on a finite-dimensional vector space there is a vector with ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle V}$ ${\displaystyle v\in V\setminus \left\{0\right\}}$

${\displaystyle gv=v\ \forall g\in G}$.

literature

• Armand Borel : Linear algebraic groups . Springer, 1991.
• Jean-Pierre Serre : Groupes algébrique et corps des classes . Hermann, 1959.
• James E. Humphreys : Linear algebraic groups . Springer, 1981.
• Tatsuji Kambayashi, Masayoshi Miyanishi, Mitsuhiro Takeuchi: Unipotent algebraic groups . Springer, 1974.
• Ina Kersten : Linear Algebraic Groups . Universitätsverlag Göttingen, 2007, ISBN 978-3-940344-05-2 .
• Robert Steinberg : Conjugacy classes in algebraic groups . Lecture Notes in Mathematics, 366, Springer, 1974.

Web links

• VL Popov: Unipotent Element . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
• VL Popov: Unipotent Group . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).