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This is an expository paper. Its purpose is to explain the linear algebra that underlies Donaldson-Thomas theory and the geometry of Riemannian manifolds with holonomy in $G_2$ and ${rm Spin}(7)$.
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We prove simplicity, and compute $delta$-derivations and symmetric associative forms of algebras in the title.
Given a grading $Gamma: A=oplus_{gin G}A_g$ on a nonassociative algebra $A$ by an abelian group $G$, we have two subgroups of the group of automorphisms of $A$: the automorphisms that stabilize each homogeneous component $A_g$ (as a subspace) and the automorphisms that permute the components. By the Weyl group of $Gamma$ we mean the quotient of the latter subgroup by the former. In the case of a Cartan decomposition of a semisimple complex Lie algebra, this is the automorphism group of the root system, i.e., the so-called extended Weyl group. A grading is called fine if it cannot be refined. We compute the Weyl groups of all fine gradings on matrix algebras, octonions and the Albert algebra over an algebraically closed field (of characteristic different from 2 in the case of the Albert algebra).
The purpose of these old notes (written in 1998 during a research project on holonomy of pseudo-Riemannian manifolds of type (10,1)) is to determine the orbit structure of the groups Spin(p,q) acting on their spinor spaces for the values (p,q) = (8,0), (9,0), (9,1), (10,0), (10,1), and (10,2). Im making them available on the arXiv because I continue to get requests for them as well as questions about how they can be cited.
We interpret an open orbit in a 32-dimensional representation space of Spin(9,1) x SL(2,R) as a substitute for the non-existent group of invertible 2x2 matrices over the octonions and study various natural homogeneous subspaces. The approach is via twistor geometry in eight dimensions.
We discuss some properties of Jacobi fields that do not involve assumptions on the curvature endomorphism. We compare indices of different spaces of Jacobi fields and give some applications to Riemannian geometry.
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