The mod 3 homology of the space of loops on the exceptional Lie groups and the adjoint action

Abstract

Let p be a prime number and G be a compact, connected, simply connected and simple Lie group. Let ΩG be the loop space of G. Bott showed H∗(ΩG; Z/p) is a finitely generated bicommutative Hopf algebra concentrated in even degrees, and determined it for classical groups G ([1]). Here, let G be an exceptional Lie group, that is, G = G2, F4, E6, E7, E8. In [2], K.Kozima and A.Kono determined H∗(ΩG; Z/2) as a Hopf algebra over A2, where Ap is the mod p Steenrod Algebra and acts on it dually. Let Ad : G × G → G and ad : G × ΩG → ΩG be the adjoint actions of G on G and ΩG respectively. In [3], the cohomology maps of these adjoint actions are studied and it is shown that H∗(ad; Z/p) = H∗(p2; Z/p) where p2 is the second projection if and only if H ∗(G; Z) is p-torsion free. For p = 2, 3 and 5, some exceptional Lie groups have p-torsions on its homology.