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Summary: LINEAR ALGEBRAIC GROUPS WITHOUT THE
NORMALIZER THEOREM
DANIEL ALLCOCK
Abstract. One can develop the basic structure theory of linear
algebraic groups (the root system, Bruhat decomposition, etc.) in a
way that bypasses several major steps of the standard development,
including the self-normalizing property of Borel subgroups.
An awkwardness of the theory of linear algebraic groups is that one
must develop a lot of material about general linear algebraic groups
before one can really get started. Our goal here is to show how to
develop the root system, etc., using only the completeness of the flag
variety and some facts about solvable groups. In particular, one can
skip over the usual analysis of Cartan subgroups, the fact that G is the
union of its Borel subgroups, the connectedness of torus centralizers,
and the normalizer theorem (i.e., a Borel subgroup is self-normalizing).
The main idea is a new approach to the structure of rank 1 groups; the
key step is lemma 5.
All algebraic geometry is over a fixed algebraically closed field. G
always denotes a connected linear algebraic group with Lie algebra g, T
a maximal torus, and B a Borel subgroup containing it. We assume the
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