Unipotent Algebraic Groups
Abstract: Let Ga be the additive algebraic group over an algebraically closed
field k of odd characteristic. This thesis considers group extensions H of Ga by Ga. By
[Ros56], every unipotent group has a closed subgroup isomorphic to Ga and a quotient which
is isomorphic to Ga. By studying extensions of Ga by Ga, we study every connected two
dimensional unipotent group. In Chapter 4, we provide... read morethe description of an arbitrary H
with cohomological data as in [Jan03] and the description of Lazard's exponential as in
[BD06]. From the orbit method, we give the representations of the finite group H = H(Fq) in
terms of the cohomological data of the extension H. In Chapter 5 we find conditions for an
algebraic torus T = Gm(Fq) to act on H by automorphisms. Given G = H ⋊ T , we provide a
description of the representations of G both over a characteristic 0 field and over a field
of characteristic l which divides the order of T using the representations found in 4.
Other "terminal" results appear where appropriate. We show that every perfect group scheme
is a subgroup of a product of perfectized Witt vectors in Chapter 3 by modifying an
argument of Serre in [Ser88] and give a bound on the dimension of the cohomology of an
arbitrary reductive group from first principles in 6. Matrix embeddings for arbitrary two
dimensional H can be found in Chapter 4.
Thesis (Ph.D.)--Tufts University, 2017.
Submitted to the Dept. of Mathematics.
Advisor: George McNinch.
Committee: Paul Sobaje, Richard Weiss, and Robert Lemke Oliver.
Keyword: Mathematics.read less