8 research outputs found
Higher rank homogeneous Clifford structures
We give an upper bound for the rank of homogeneous (even) Clifford
structures on compact manifolds of non-vanishing Euler characteristic. More
precisely, we show that if with odd, then for
, for , for and for .
Moreover, we describe the four limiting cases and show that there is exactly
one solution in each case.Comment: 20 pages, final versio
Invariant four-forms and symmetric pairs
We give criteria for real, complex and quaternionic representations to define
s-representations, focusing on exceptional Lie algebras defined by spin
representations. As applications, we obtain the classification of complex
representations whose second exterior power is irreducible or has an
irreducible summand of co-dimension one, and we give a conceptual
computation-free argument for the construction of the exceptional Lie algebras
of compact type.Comment: 16 pages [v2: references added, last section expanded
The even Clifford structure of the fourth Severi variety
The Hermitian symmetric space appears in the classification
of complete simply connected Riemannian manifolds carrying a parallel even
Clifford structure. This means the existence of a real oriented Euclidean
vector bundle over it together with an algebra bundle morphism
mapping
into skew-symmetric endomorphisms, and the existence of a metric connection on
compatible with . We give an explicit description of such a vector
bundle as a sub-bundle of . From this we construct a
canonical differential 8-form on , associated with its holonomy
, that represents
a generator of its cohomology ring. We relate it with a Schubert cycle
structure by looking at as the smooth projective variety
known as the fourth Severi variety
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Invariant four-forms and symmetric pairs
We give criteria for real, complex and quaternionic representations to define
s-representations, focusing on exceptional Lie algebras defined by spin representations. As
applications, we obtain the classification of complex representations whose second exterior
power is irreducible or has an irreducible summand of co-dimension one, and we give a
conceptual computation-free argument for the construction of the exceptional Lie algebras
of compact type