Stability of the E8-form (Ex)
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$\langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle -1 \rangle \cong ({\mathbb Z}^8,\lambda) \oplus \langle -1 \rangle$ | $\langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle -1 \rangle \cong ({\mathbb Z}^8,\lambda) \oplus \langle -1 \rangle$ | ||
| − | where $\lambda$ is unimodular, even, and positive definite. (There is a unique such form in dimension 8: extra credit is given for an isometry $({\mathbb Z}^8,\lambda) \cong | + | where $\lambda$ is unimodular, even, and positive definite. (There is a unique such form in dimension 8: extra credit is given for an isometry $({\mathbb Z}^8,\lambda) \cong E_8$). |
Revision as of 06:06, 8 January 2019
Show there exists an isometry of unimodular forms
where 
is unimodular, even, and positive definite. (There is a unique such form in dimension 8: extra credit is given for an isometry 
).
