Fundamental Groups of Split Real Kac-Moody Groups and Generalized Real Flag Manifolds

Abstract

We determine the fundamental groups of symmetrizable algebraically simply connected split real Kac-Moody groups endowed with the Kac-Peterson topology. In analogy to the finite-dimensional situation, because of the Iwasawa decomposition G = KAU+, the embedding K ,↪ G is a weak homotopy equivalence, in particular π1(G) = π1(K). It thus suffices to determine π1(K), which we achieve by investigating the fundamental groups of generalized ag varieties. Our results apply in all cases in which the Bruhat decomposition of the generalized ag variety is a CW decomposition- in particular, we cover the complete symmetrizable situation; furthermore, the results concerning only the structure of π1(K) actually also hold in the nonsymmetrizable two-spherical case.

A master exceptional field theory

“…The first steps for the inclusion of fermions into the duality equations were undertaken in [13], building on the finite-dimensional spinor representations of K(E 11 ), the double cover of K(E 11 ) [73], that were found in [10,12]. While we have proposed a supersymmetric version of the duality equation extended by fermion bilinears in [13], adding a Rarita-Schwinger-like term to the pseudo-Lagrangian and employing a Noether procedure appears to be an interesting challenge.…”

“…Its intersection with gl (11) discussed above is so(1, 10) and therefore K(e 11 ) should be thought of as an infinite generalisation of the Lorentz algebra. 10 We shall denote the corresponding group by K(E 11 ) and it is known that it has a two-fold cover K(E 11 ) [73], generalising the spin groups and that has finite-dimensional spinor representations [10,13]. We shall not consider fermions in this paper.…”

Generalised holonomies and K(E$_9$)

“…It was found in [15][16][17][18] that the involutory subalgebra K(e 10 ) of the Kac-Moody Lie algebra e 10 admits a 32-dimensional representation, dubbed spin- 1 2 for obvious reasons, as well as a 320-component representation that corresponds to the spin- 3 2 gravitino. These representations have the property that they lift correctly to the spin cover K(E 10 ) of the corresponding group (see [19] for a discussion of the spin cover). Moreover, the representations have the property that they possess the correct branching to the other maximal supergravity theories, including the chiral fermions of type IIB, which cannot be obtained by usual dimensional reduction [20], and thus provide a common origin for both IIA and IIB fermions.…”