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The quaternions and Lie algebras I
Someone who has just read the previous post on how exponentiating quaternions gives a nice parameterization of might object as follows: “that’s nice and all, but there has to be a general version of...
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The quaternions and Lie algebras II
We now know what a Lie algebra is and we know they are abstractions of infinitesimal symmetries, which are given by derivations. Today we will see what we can say about associating infinitesimal...
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The representation theory of SU(2)
Today we will give four proofs of the classification of the (finite-dimensional complex continuous) irreducible representations of (which you’ll recall we assumed way back in this previous post). As a...
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The Jacobson radical
The Artin-Wedderburn theorem shows that the definition of a semisimple ring is enormously restrictive. Even fails to be semisimple! A less restrictive notion, but one that still captures the notion of...
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Noncommutative probability and group theory
There are, roughly speaking, two kinds of algebras that can be functorially constructed from a group . The kind which is covariantly functorial is some variation on the group algebra , which is the...
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The double commutant theorem
Let be an abelian group and be a collection of endomorphisms of . The commutant of is the set of all endomorphisms of commuting with every element of ; symbolically, . The commutant of is equal to the...
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Four flavors of Schur-Weyl duality
If is a finite-dimensional complex vector space, then the symmetric group naturally acts on the tensor power by permuting the factors. This action of commutes with the action of , so all permutations...
View ArticleA transcript of my qualifying exam
I passed my qualifying exam last Friday. Here is a copy of the syllabus and a transcript. Although I’m sure there are more, I’m only aware of two other students at Berkeley who’ve posted transcripts of...
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Hecke operators are also relative positions
Continuing yesterday’s story about relative positions, let be a finite group and let and be finite -sets. Yesterday we showed that -orbits on can be thought of as “atomic relative positions” of...
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The representation theory of the additive group scheme
In this post we’ll describe the representation theory of the additive group scheme over a field . The answer turns out to depend dramatically on whether or not has characteristic zero. Preliminaries...
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