Commutativity and conjugates | Group theory episode 5

#monoids #complexnumbers #grouptheory

Binary operations can be commutative or not. We "fix" commutativity with the very important concept of conjugates: when two group elements are conjugate to each other, they behave in a very similar way. You can use this to change your mathematical perspective, to make difficult problems easier to solve or understand. And we also discuss how commutators are used in Lie groups and in quantum physics.

[AA1] An important note at 5:04 is that you have to read matrix multiplication from right to left. You first apply the matrix on the right to a vector, and then you apply the one on the left to the result. So if you want to work out these examples yourself, make sure you do it in the correct order.

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Don't forget to follow these super interesting links:

[M3M] https://www.math3ma.com/blog/a-group-...
This intuitive blog post on Math3ma talks about the center of a group. This is the subgroup containing all elements that commute. Its cosets turn out to be related to conjugates, which measure how non-commutative you are!

[SOC]    • Cyclic Groups  (Abstract Algebra)  
Cyclic groups, well explained.

[WP] https://en.wikipedia.org/wiki/Commutator
Two different definitions of commutators: One for groups, and one for rings. The one I show in the video is actually only available in rings.

[STE] https://math.stackexchange.com/questi...
In a group, we only have a single binary operation. In rings, we don't have division. This explains the different definitions of commutators in groups & rings.

[WB] https://en.wikibooks.org/wiki/Quantum...
An entire chapter about operators and commutators in quantum physics.

[PG]    • Ever heard of Quantum Operators and C...  
Commutators in quantum physics, explained very elegantly.

0:00 Introduction
0:37 Commutativity
2:02 Cyclic groups are always abelian
3:51 Commutators
5:11 Commutators in quantum physics
7:53 Lie groups & Lie algebras
14:28 Conjugacy
17:18 The sandwich product
19:31 Different perspectives
21:47 Conjugacy classes

This video is published under a CC Attribution license
( https://creativecommons.org/licenses/... )