Proud, Richard (1997) Unipotent subgroups of reductive algebraic groups. PhD thesis, University of Warwick.

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Abstract

Let G be
a connected reductive algebraic group
defined
over an algebraically closed
field
of
good characteristic p>0.
Suppose
uEG
has
order p.
In [T2] it is
shown
that
u
lies in
a
closed reductive subgroup of
G
of type Al. This is the best
possible group
theoretic
analogue
of
the Jacobson-Morozov
theorem
for
simple
Lie
algebras.
Testerman's key
result
is
a
type
of
`exponentiation
process'.
For
our given element u,
this
process constructs a
1-dimensional
connected abelian unipotent subgroup of
G, hence isomorphic to Ga,
containing u.
This in
turn yields the
required
Al
overgroup of u.

Now let 1#uEG be
an arbitrary unipotent element.
Such
an element
has
order pt,
for
some tEN. In this thesis
we extend
the
above result, and show
that
u
lies in
a
t-dimensional
closed connected abelian unipotent subgroup of
G,
provided p>
29
when
G'
contains a simple
component of
type E8,
and
that
p
is
good
for the remaining components.
The
structure of
the
resulting unipotent overgroup
is
also explicitly given.
This is the best
possible result,
in
terms
of
`minimal dimension',
which we could
hope for.
In Chapter 1
we
discuss the theory
of
Witt
vectors, associated with a commutative ring
with
identity. They
are closely related
to the study of connected abelian unipotent algebraic
groups.
The
unipotent overgroups are constructed using a variation of the usual exponen-
tiation process.
The
necessary material on
formal
power series rings
is
given
in 1.3. The
Artin-Hasse
exponentials of
1.4
play a crucial role
in this
construction.
The
connection
between Witt
groups and
Artin-Hasse
exponentials
is discussed in 1.5.

In Chapter 2
we apply the techniques of
Chapter 1 to the
various simple algebraic groups.
For
each type, a particular
isogeny
class
is
chosen and
the
required overgroup
is
constructed
for the
regular
(and
subregular) classes.
In 2.9
we pass
to the
adjoint case.
In Chapter 3
we extend
the
results of
Chapter 2 to include
all unipotent classes
in
all
reductive algebraic groups
(under
certain restrictions).
In 3.1 the Cayley Transform for the
classical groups
is
combined with
the ideas
of
Chapter 1 to
give an explicit construction of
the
unipotent overgroups
for
every unipotent class.
In 3.2
we
discuss
semiregular unipotent
elements.
Finally, in 3.3,
we prove the main
theorem of
this thesis.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Algebra, Vector analysis
Official Date:	September 1997
Dates:	Date Event September 1997 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Testerman, D. M.
Sponsors:	Engineering and Physical Sciences Research Council (EPSRC) ; University of Warwick. Institute of Mathematics
Extent:	vi, 138 leaves : illustrations
Language:	enk

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