Homotopy nilpotency of some homogeneous spaces

Let $${\mathbb {K}}={\mathbb {R}},\,{\mathbb {C}}$$ K = R , C , the field of reals or complex numbers and $${\mathbb {H}}$$ H , the skew $${\mathbb {R}}$$ R -algebra of quaternions. We study the homotopy nilpotency of the loop spaces $$\Omega (G_{n,m}({\mathbb {K}}))$$ Ω ( G n , m ( K ) ) , $$\Omega (F_{n;n_1,\ldots ,n_k}({\mathbb {K}}))$$ Ω ( F n ; n 1 , … , n k ( K ) ) , and $$\Omega (V_{n,m}({\mathbb {K}}))$$ Ω ( V n , m ( K ) ) of Grassmann $$G_{n,m}({\mathbb {K}})$$ G n , m ( K ) , flag $$F_{n;n_1,\ldots ,n_k}({\mathbb {K}})$$ F n ; n 1 , … , n k ( K ) and Stiefel $$V_{n,m}({\mathbb {K}})$$ V n , m ( K ) manifolds. Additionally, homotopy nilpotency classes of p-localized $$\Omega (G^+_{n,m}({\mathbb {K}})_{(p)})$$ Ω ( G n , m + ( K ) ( p ) ) and $$\Omega (V_{n,m}({\mathbb {K}})_{(p)})$$ Ω ( V n , m ( K ) ( p ) ) for certain primes p are estimated, where $$G^+_{n,m}({\mathbb {K}})_{(p)}$$ G n , m + ( K ) ( p ) is the oriented Grassmann manifolds. Further, the homotopy nilpotency classes of loop spaces of localized homogeneous spaces given as quotients of exceptional Lie groups are investigated as well.

NOTE ON SAMELSON PRODUCTS IN EXCEPTIONAL LIE GROUPS

We determine the (non-)triviality of Samelson products of inclusions of factors of the mod p decomposition of $G_{(p)}$ for $(G,p)=(E_7,5),(E_7,7),(E_8,7)$ . This completes the determination of the (non-)triviality of those Samelson products in p-localized exceptional Lie groups when G has p-torsion-free homology.

Discretely decomposable restrictions of (𝔤,K)-modules for Klein four symmetric pairs of exceptional Lie groups of Hermitian type

Let [Formula: see text] be a Klein four symmetric pair. The author wants to classify all the Klein four symmetric pairs [Formula: see text] such that there exists at least one nontrivial unitarizable simple [Formula: see text]-module [Formula: see text] that is discretely decomposable as a [Formula: see text]-module. In this paper, three assumptions will be made. First, [Formula: see text] is an exceptional Lie group of Hermitian type, i.e. [Formula: see text] or [Formula: see text]. Second, [Formula: see text] is noncompact. Third, there exists an element [Formula: see text] corresponding to a symmetric pair of anti-holomorphic type such that [Formula: see text] is discretely decomposable as a [Formula: see text]-module.

Deducing the symmetry of the standard model from the automorphism and structure groups of the exceptional Jordan algebra

We continue the study undertaken in Ref. 16 of the exceptional Jordan algebra [Formula: see text] as (part of) the finite-dimensional quantum algebra in an almost classical space–time approach to particle physics. Along with reviewing known properties of [Formula: see text] and of the associated exceptional Lie groups we argue that the symmetry of the model can be deduced from the Borel–de Siebenthal theory of maximal connected subgroups of simple compact Lie groups.