E8-form (Ex)

Latest revision as of 21:23, 25 August 2013

Let $<wikitex>; Let $E_8$ denote the negative definite even form with signature $-8$ and rank $: $$E_8 = \left( \begin{array}{cccccccc} -2 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \ 0 & -2 & -1 & 0 & 0 & 0 & 0 & 0 \ 0 & -1 & -2 & -1 & 0 & 0 & 0 & 0 \ -1 & 0 & -1 & -2 & -1 & 0 & 0 & 0 \ 0 & 0 & 0 & -1 & -2 & -1 & 0 & 0 \ 0 & 0 & 0 & 0 & -1 & -2 & -1 & 0 \ 0 & 0 & 0 & 0 & 0 & -1 & -2 & -1 \ 0 & 0 & 0 & 0 & 0 & 0 & -1 & -2 \end{array}\right):\Z^8 \to (\Z^8)^*. $$ Consider the forms $E_8\oplus \langle -1\rangle$ and $\langle -9\rangle$. Verify that these two forms are both odd, of the same rank and signature but yet still inequivalent (non-isometric). </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]]E_8$ denote the negative definite even form with signature $-8$ and rank $8$:

$$\displaystyle E_8 = \left( \begin{array}{cccccccc} -2 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & -2 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & -2 & -1 & 0 & 0 & 0 & 0 \\ -1 & 0 & -1 & -2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & -2 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & -2 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & -2 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & -2 \end{array}\right):\Z^8 \to (\Z^8)^*.$$

Consider the forms $E_8\oplus \langle -1\rangle$ and $\langle -9\rangle$. Verify that these two forms are both odd, of the same rank and signature but yet still inequivalent (non-isometric).

References