Working over an arbitrary field F of characteristic not 2, we consider linear algebraic
groups over F. We view these as functors, represented by finitely generated F-Hopf
algebras, from the category of commutative, associative, F-algebras Alg_F, to the
category of groups. Classical examples of these groups, such as the special linear
group SL_n are split, however there are also linear algebraic groups arising from central
simple F-algebras which are non-split. For example, associated to a non-split central
simple F-algebra A of degree n is a non-split special linear group SL(A). It is well
known that central simple algebras are twisted forms of matrix algebras. This means
that over the separable closure of F, denoted F_sep, we have A⊗_F F_sep ∼= M_n(F_sep) and that there is a twisted Gal(F_sep/F)-action on M_n(F_sep) whose fixed points are A. We
show that a similar method of twisted Galois descent can be used to obtain all non-split
semisimple linear algebraic groups associated to central simple algebras as fixed
points within their split counterparts. In particular, these techniques can be used
to construct the spin and half-spin groups Spin(A, τ ) and HSpin(A, τ ) associated
to a central simple F-algebra of degree 4n with orthogonal involution. Furthermore,
we develop a theory of twisted Galois descent for Hopf algebras and show how the
fixed points obtained this way are the representing Hopf algebras of our non-split
groups. Returning to the view of group schemes as functors, we discuss how the group
schemes we consider are sheaves on the étale site of Alg_F whose stalks are Chevalley
groups over local, strictly Henselian F-algebras. This allows us to use the generators
and relations presentation of Chevalley groups to explicitly describe group scheme
morphisms. After showing how the Kronecker tensor product of matrices induces
maps between simply connected groups, we give an explicit description of these maps
in terms of Chevalley generators. This allows us to compute the kernel of these new
maps composed with standard isogenies and thereby construct new tensor product
maps between non-simply connected split groups. These new maps are Gal(F_sep/F)-morphisms and so we apply our techniques of twisted Galois descent to also obtain
new tensor product morphisms between non-split groups schemes. Finally, we use
one of our new split tensor product maps to compute the degree three cohomological
invariants of HSpin_4n for all n.
| Identifer | oai:union.ndltd.org:uottawa.ca/oai:ruor.uottawa.ca:10393/43009 |
| Date | 10 December 2021 |
| Creators | Ruether, Cameron |
| Contributors | Zaynullin, Kirill |
| Publisher | Université d'Ottawa / University of Ottawa |
| Source Sets | Université d’Ottawa |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | application/pdf |
| Rights | Attribution 4.0 International, http://creativecommons.org/licenses/by/4.0/ |
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