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Lorentzian characterization of genus

Lorentzian characterization of genus

In the first case, suppose $|b| < |a|.$ Then $ | 2 a b| = \xi^2 \leq 2 a^2,$ so in fact $$ \left( \frac{\xi}{a} \right)^2 < 2.$$ We choose the root $ \tilde{\lambda} = \left( 0, 1, 1 \right).$ Then $z \cdot \tilde{\lambda} = b + a,$ and $$ s_{\tilde{\lambda}} (z) = z - ( \tilde{\lambda} \cdot z) \tilde{\lambda} = (\xi, -b,-a).$$

In the first case, suppose $|b| < |a|.$ Then $ | 2 a b| = \xi^2 < 2 a^2,$ so in fact $$ \left( \frac{\xi}{a} \right)^2 < 2.$$ We choose the root $ \tilde{\lambda} = \left( 0, 1, 1 \right).$ Then $z \cdot \tilde{\lambda} = b + a,$ and $$ s_{\tilde{\lambda}} (z) = z - ( \tilde{\lambda} \cdot z) \tilde{\lambda} = (\xi, -b,-a).$$

This is the more important case of "even" lattices, where all inner products are integral and all vector norms are even. We follow pages 131-134 in Wolfgang Ebeling, Lattices and Codes, available for sale at LINK \$45.00 for paperback.

This is the more important case of "even" lattices, where all inner products are integral and all vector norms are even. We follow pages 131-134 in Wolfgang Ebeling, Lattices and Codes, available for sale at LINK \\$45.00 for paperback.