The classification of p-compact groups for p odd

Abstract

A p-compact group, as defined by Dwyer and Wilkerson, is a purely homotopically defined p-local analog of a compact Lie group. It has long been the hope, and later the conjecture, that these objects should have a classification similar to the classification of compact Lie groups. In this paper we finish the proof of this conjecture, for p an odd prime, proving that there is a one-to-one correspondence between connected p-compact groups and finite reflection groups over the p-adic integers. We do this by providing the last, and rather intricate, piece, namely that the exceptional compact Lie groups are uniquely determined as p-compact groups by their Weyl groups seen as finite reflection groups over the p-adic integers. Our approach in fact gives a largely self-contained proof of the entire classification theorem for p odd.

Fine gradings on $\mathfrak{e}_6$

“…We can extend the results in Lemma 3 a little bit. If f ∈ N 0 (T ) verifies that its projection π(f ) has order r ′ (in general, r ′ divides the order of f ), then H f ⊂ {t ∈ T | t r ′ = 1 G } and the torus 4 , what happens is ( σ 3 s) 8 ∈ S 3 = id but id = ( σ 3 s) 4 ∈ T 3 for every s in the corresponding torus.…”

“…As F 2 is of type 3D, the subalgebra fix F 2 is of type d 4 summed with a two-dimensional center. Now F 1 T ξ,ξ preserves this subalgebra and its derived subalgebra, that is, d 4 , producing a Z 3 -grading on d 4 . This implies that the restriction F 1 T ξ,ξ | d4 must fix a subalgebra of some of the types {a 2 , g 2 , 3a 1 +Z, a 3 +Z}, of dimensions {8, 14, 10, 16} respectively.…”

“…and dim L (1,ī,j) = 1 for the remaining i, j, and the type of the restriction to N1 is (12,4). It is not difficult to work with the other two components to arrive at the conclusion that both L2 and L3 break into 12 onedimensional components and 4 two-dimensional components, therefore the type of the grading induced by Q 14 in e 6 is (12, 3) + 3(12, 4) = (48, 15).…”

“…• {t x,y,z,u | x, y, z, u ∈ F * } ∼ = (F * ) 4 , which produces fine gradings on J and Der J of types (24, 0, 1) and (48, 0, 0, 1) respectively.…”

“…Afterwards, in Section 5 we describe all the possible fine gradings, up to equivalence, produced by a group of automorphisms not all of them inner. We consider the gradings obtained extending gradings on f 4 and the ones obtained extending gradings on c 4 , and last, we show an example of (outer) grading which is not in any of the two previous situations. Finally Section 6 is devoted to prove that there are no more fine gradings that seen before.…”