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 \(E=\{\prod^{m}_{j=1}e_{j}^{i_{j}}\mid i_{j}=0\mbox{ or }1\}\) 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, \(\Delta\) 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?

#RepresentationTheory #LieGroups #CliffordAlgebra #Mathematics #Proof

#groups #RepresentationTheory #LieGroups

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?

#DifferentialGeometry #LieGroups #Manifolds

@xameer

Cf. #DifferentialLogic • Discussion 3
• https://inquiryintoinquiry.com/2020/06/17/differential-logic-discussion-3/

#Physics once had a #FrameProblem (#Complexity of #DynamicUpdating) long before #AI did but physics learned to reduce complexity through the use of #DifferentialEquations and #GroupSymmetries (combined in #LieGroups). One of the promising features of #MinimalNegationOperators is their relationship to #DifferentialOperators. So I’ve been looking into that. Here’s a link, a bit in medias res, but what I’ve got for now.

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 \( x'(t) = A(t) x(t) \) which I describe below the fold. (1/n, n≈7)

#MagnusExpansion #DifferentialEquations #MatrixExponential #QuantumMechanics #LieGroups #NumericalAnalysis #GeometricNumericalIntegration