On homotopy nilpotency

We review established and recent results on the homotopy nilpotence of spaces. In particular, the homotopy nilpotency of the loop spaces \(\Omega(G/K)\) of homogenous spaces \(G/K\) for a compact Lie group \(G\) and its closed homotopy nilpotent subgroup \(K \lt G\) is discussed.

Fourier multipliers for Triebel–Lizorkin spaces on compact Lie groups

We investigate the boundedness of Fourier multipliers on a compact Lie group when acting on Triebel-Lizorkin spaces. Criteria are given in terms of the Hörmander-Mihlin-Marcinkiewicz condition. In our analysis, we use the difference structure of the unitary dual of a compact Lie group. Our results cover the sharp Hörmander-Mihlin theorem on Lebesgue spaces and also other historical results on the subject.

HIGHER GENERATION BY ABELIAN SUBGROUPS IN LIE GROUPS

To a compact Lie group G one can associate a space E(2;G) akin to the poset of cosets of abelian subgroups of a discrete group. The space E(2;G) was introduced by Adem, F. Cohen and Torres-Giese, and subsequently studied by Adem and Gómez, and other authors. In this short note, we prove that G is abelian if and only if πi(E(2;G)) = 0 for i = 1; 2; 4. This is a Lie group analogue of the fact that the poset of cosets of abelian subgroups of a discrete group is simply connected if and only if the group is abelian.

The topological nilpotence degree of a Noetherian unstable algebra

We investigate the topological nilpotence degree, in the sense of Henn–Lannes–Schwartz, of a connected Noetherian unstable algebra R. When R is the mod p cohomology ring of a compact Lie group, Kuhn showed how this invariant is controlled by centralizers of elementary abelian p-subgroups. By replacing centralizers of elementary abelian p-subgroups with components of Lannes’ T-functor, and utilizing the techniques of unstable algebras over the Steenrod algebra, we are able to generalize Kuhn’s result to a large class of connected Noetherian unstable algebras. We show how this generalizes Kuhn’s result to more general classes of groups, such as groups of finite virtual cohomological dimension, profinite groups, and Kac–Moody groups. In fact, our results apply much more generally, for example, we establish results for p-local compact groups in the sense of Broto–Levi–Oliver, for connected H-spaces with Noetherian mod p cohomology, and for the Borel equivariant cohomology of a compact Lie group acting on a manifold. Along the way we establish several results of independent interest. For example, we formulate and prove a version of Carlson’s depth conjecture in the case of a Noetherian unstable algebra of minimal depth.

Inverse Problem for the Yang–Mills Equations

We show that a connection can be recovered up to gauge from source-to-solution type data associated with the Yang–Mills equations in Minkowski space $${\mathbb {R}}^{1+3}$$ R 1 + 3 . Our proof analyzes the principal symbols of waves generated by suitable nonlinear interactions and reduces the inversion to a broken non-abelian light ray transform. The principal symbol analysis of the interaction is based on a delicate calculation that involves the structure of the Lie algebra under consideration and the final result holds for any compact Lie group.

Current Algebra, a U(1) Gauge Theory and the Wess-Zumino-Witten Model

In this note we realise current algebra with anomalous terms in terms of a U(1) gauge theory, in the space of maps M, from S1 into a compact Lie group corresponding to the current algebra. The Wilson loop around a closed curve in M is shown to be the Wess-Zumino-Witten term. This discussion enables a simple understanding of the non-Abelian anomaly in the Schrödinger picture.

RATIONAL LOCAL SYSTEMS AND CONNECTED FINITE LOOP SPACES

Greenlees has conjectured that the rational stable equivariant homotopy category of a compact Lie group always has an algebraic model. Based on this idea, we show that the category of rational local systems on a connected finite loop space always has a simple algebraic model. When the loop space arises from a connected compact Lie group, this recovers a special case of a result of Pol and Williamson about rational cofree G-spectra. More generally, we show that if K is a closed subgroup of a compact Lie group G such that the Weyl group W G K is connected, then a certain category of rational G-spectra “at K” has an algebraic model. For example, when K is the trivial group, this is just the category of rational cofree G-spectra, and this recovers the aforementioned result. Throughout, we pay careful attention to the role of torsion and complete categories.