Difference between revisions of "Unipotent group"
(Importing text file) |
Ulf Rehmann (talk | contribs) m (MR/ZBL numbers added) |
||
Line 14: | Line 14: | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Springer (1991) {{MR|1102012}} {{ZBL|0726.20030}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann (1959) {{MR|0103191}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J.E. Humphreys, "Linear algebraic groups" , Springer (1981) {{MR|0610979}} {{MR|0396773}} {{ZBL|0471.20029}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> T. Kambayachi, M. Miyanishi, M. Takeuchi, "Unipotent algebraic groups" , Springer (1974) {{MR|376696}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> R. Steinberg, "Conjugacy classes in algebraic groups" , ''Lect. notes in math.'' , '''366''' , Springer (1974) {{MR|0352279}} {{ZBL|0281.20037}} </TD></TR></table> |
Revision as of 14:52, 24 March 2012
A subgroup of a linear algebraic group
consisting of unipotent elements (cf. Unipotent element). If
is identified with its image under an isomorphic imbedding in a group
of automorphisms of a suitable finite-dimensional vector space
, then a unipotent group is a subgroup contained in the set
![]() |
of all unipotent automorphisms of . Fixing a basis in
, one may identify
with the general linear group
, where
is an algebraically closed ground field; the linear group
is then also called a unipotent group. An example of a unipotent group is the group
of all upper-triangular matrices in
with 1's on the main diagonal. If
is a subfield of
and
is a unipotent subgroup in
, then
is conjugate over
to some subgroup of
. In particular, all elements of
have in
a common non-zero fixed vector, and
is a nilpotent group. This theorem shows that the unipotent algebraic groups are precisely the Zariski-closed subgroups of
for varying
.
In any linear algebraic group there is a unique connected normal unipotent subgroup
(the unipotent radical) with reductive quotient group
(cf. Reductive group). To some extent this reduces the study of the structure of arbitrary groups to a study of the structure of reductive and unipotent groups. In contrast to the reductive case, the classification of unipotent algebraic groups is at present (1992) unknown.
Every subgroup and quotient group of a unipotent algebraic group is again unipotent. If , then
is always connected; moreover, the exponential mapping
(where
is the Lie algebra of
) is an isomorphism of algebraic varieties; if
, then there exist non-connected unipotent algebraic groups: e.g. the additive group
of the ground field (which may be identified with
) is a
-group and so contains a finite unipotent group. In a connected unipotent group
there is a sequence of normal subgroups
such that all quotients
are one-dimensional. Every connected one-dimensional unipotent algebraic group is isomorphic to
. This reduces the study of connected unipotent algebraic groups to a description of iterated extensions of groups of type
.
Much more is known about commutative unipotent algebraic groups (cf. [4]) than in the general case. If , then they are precisely the algebraic groups isomorphic to
; here, the isomorphism
is given by the exponential mapping. If
, then the connected commutative unipotent algebraic groups
are precisely the connected commutative algebraic
-groups. Now
need not be isomorphic to
: for this it is necessary and sufficient that
for all
. In the general case
is isogenous (cf. Isogeny) to a product of certain special groups (so-called Witt groups, cf. [2]).
If and
are connected unipotent algebraic groups and
, then the variety
is isomorphic to an affine space. Any orbit of a unipotent algebraic group of automorphisms of an affine algebraic variety
is closed in
[5].
References
[1] | A. Borel, "Linear algebraic groups" , Springer (1991) MR1102012 Zbl 0726.20030 |
[2] | J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann (1959) MR0103191 |
[3] | J.E. Humphreys, "Linear algebraic groups" , Springer (1981) MR0610979 MR0396773 Zbl 0471.20029 |
[4] | T. Kambayachi, M. Miyanishi, M. Takeuchi, "Unipotent algebraic groups" , Springer (1974) MR376696 |
[5] | R. Steinberg, "Conjugacy classes in algebraic groups" , Lect. notes in math. , 366 , Springer (1974) MR0352279 Zbl 0281.20037 |
Unipotent group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unipotent_group&oldid=17020