I am working my way through Adams's "Lectures on Exceptional Lie Groups", and I am not satisfied with the proof for his proposition 4.2 (which states the even subalgebra for the Clifford Algebra $Cl(V{)}_0$ has 1 irreducible representation when $dim(V)=m=2n+1$ and 2 irreducible representations when $dim(V)=m=2n$ with certain specific weights).

The argument seems to be to relate representations of an Abelian subgroup $E=\{\prod _{j=1}^m{e}_j^{i_j}\mid i_j=0\text{ or }1\}$ and $E_0=E\cap Cl(V{)}_0$ [where $e_j$ form the canonical basis for $V$] to representations of $\mathbb{R}[E]/(\nu +1)\cong Cl(V)$, the quotient of the group algebra $\mathbb{R}[E]$ modulo the identification of the square of the generators $e_j^2=\nu$ with -1.

I'm with Adams until he picks a complex 1-dimensional representation $W$ of $F$, because he starts working with *COMPLEX* representations. But Adams triumphantly announces "We thus get a representation, $\mathrm{\Delta }$ of $E_0$..." then shows it is irreducible. I'm fine with it being irreducible from the character relations, that's fine.

Even supposing this is an irreducible representation for $\mathbb{C}[E_0]$, I don't quite see how to obtain an irrep for $Cl(V{)}_0$; I am guessing just extend it "in the obvious way"? Does this preserve irreducibility?

I've worked out that the injectivity radius under the Euclidean metric for the #unitary group U(n) is π and for real and special subgroups O(n), SO(n), and SU(n) is π√2.

This seems like a pretty basic property, but I can't find a single reference that gives the injectivity radii for any of these groups. Anyone know of one?

Quantum mechanics anyone? Dozens have been disappointed by UCLA’s administration ineptly standing in the way of Dr. Mike Miller being able to offer his perennial Winter UCLA math class (Ring Theory this quarter), so a few friends and I are putting our informal math and physics group back together.

We’re mounting a study group on quantum mechanics based on Peter Woit‘s Introduction to Quantum Mechanics course from 2022. We’ll be using his textbook Quantum Theory, Groups and Representations:An Introduction (free, downloadable .pdf) and his lectures from YouTube.

Shortly, we’ll arrange a schedule and some zoom video calls to discuss the material. If you’d like to join us, send me your email or leave a comment so we can arrange meetings (likely via Zoom or similar video conferencing).

Our goal is to be informal, have some fun, but learn something along the way. The suggested mathematical background is some multi-variable calculus and linear algebra. Many of us already have some background in Lie groups, algebras, and representation theory and can hopefully provide some help for those who are interested in expanding their math and physics backgrounds.

Everyone is welcome! 

#group-theory #lie-groups #peter-woit #physics #quantum-mechanics #representation-theory

https://boffosocko.com/2023/01/26/quantum-mechanics-study-group-for-peter-woit/

I really enjoyed the paper

Oteo & Ros, Why Magnus expansion?, URL: https://doi.org/10.1080/00207160.2021.1938011 (paywall)

and not just because it cites a paper of mine (though it does help!)

It's a historical/personal reflection on the Magnus expansion, a series solution to the differential equation $x^{\prime }(t)=A(t)x(t)$ which I describe below the fold. (1/n, n≈7)

Hi guys! Is there someone here doing research in #PDE, #LieGroups, #ComplexAnalysis?