Tue 9:00 from Mon 30th Sep 2024 Wed 10:00 Fri 10:00 all lectures in B3.02Manifesto
Week 2
(a) Noetherian and Artinian conditions and Zorn's Lemma
(b) Free groups F(X) and finitely presented groups < X | R>.
(a) Algebra uses induction arguments well beyond the finite induction of
Peano's
axioms. The ascending and descending chain conditions are logical
tools that
allow us pass from one object to another (e.g. to a bigger or smaller
submodule)
with a guarantee that the process eventually
terminates. Zorn's lemma is the
algebraist's preferred form of the Axiom of Choice. It is an axiom
comparable
in importance to the least upper bound statement of the
completeness of the
reals in first year analysis.
(b) The final part of this week's lectures dealt with the construction of the
free group
F(X) on a set of generators X = {x_al} (usually finite), and the notion
of a group
< X | R > defined by generators and relations.
Some extra snippets on groups
Extracts from my lecture notes MA3E1
Groups and repns (better
written than anything I can hope to do this term).
FP groups
Two challenge questions
Binary groups
Week 3
I started with a brief preview of noncommutative rings highlighting
the difficulties of the subject. The main topic was Clifford algebras and
the
construction of the spin groups Spin(2n) that double cover SO(2n).
Later, I made
a start on Lie algebras, giving the definition, and
explaining what they
mean in terms of the tangent
space to a Lie group.
Worksheet on Week 3
Week 4
The main topic is the Wedderburn-Artin theorem characterising
semisimple rings R as finite direct sums of full matrix rings over division
algebras. The argument works with the category of modules over
R, with the
semisimple assumption that every short exact sequence splits. This turns
out to be an extremely strong assumption.
