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June 1, 2025
Tannaka Reconstruction and the Monoid of Matrices
Posted by John Baez
You can classify representations of simple Lie groups using Dynkin diagrams, but you can also classify representations of ‘classical’ Lie groups using Young diagrams. Hermann Weyl wrote a whole book on this, The Classical Groups.
This approach is often treated as a bit outdated, since it doesn’t apply to all the simple Lie groups: it leaves out the so-called ‘exceptional’ groups. But what makes a group ‘classical’?
There’s no precise definition, but a classical group always has an obvious representation, you can get other representations by doing obvious things to this obvious one, and it turns out you can get all the representations this way.
For a long time I’ve been hoping to bring these ideas up to date using category theory. I had a bunch of conjectures, but I wasn’t able to prove any of them. Now Todd Trimble and I have made progress:
- John Baez and Todd Trimble, Tannaka reconstruction and the monoid of matrices.
We tackle something even more classical than the classical groups: the monoid of matrices, with matrix multiplication as its monoid operation.