# Gradings of Lie algebras, magical spin geometries and matrix factorizations

@article{Abuaf2019GradingsOL, title={Gradings of Lie algebras, magical spin geometries and matrix factorizations}, author={Roland Abuaf and Laurent Manivel}, journal={arXiv: Algebraic Geometry}, year={2019} }

We describe a remarkable rank fourtenn matrix factorization of the octic Spin(14)-invariant polynomial on either of its half-spin representations. We observe that this representation can be, in a suitable sense, identified with a tensor product of two octonion algebras. Moreover the matrix factorisation can be deduced from a particular Z-grading of the exceptional Lie algebra $\mathfrak{e}_8$. Intriguingly, the whole story can be extended to the whole Freudenthal-Tits magic square and yields… CONTINUE READING

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