Timeline for How to determine $O(L)$ is finite or not?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Nov 3, 2013 at 3:24 | comment | added | user2013 | You are right. $2$ should be $k$ in the comment above. Thank you for the clarification. | |
| Nov 3, 2013 at 1:54 | history | edited | WKC | CC BY-SA 3.0 |
I correct the statement to "....it is definite or it is on the hyperbolic plane..."
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| Nov 3, 2013 at 1:53 | comment | added | WKC | I see what you mean (isn't your conditions on $e$ and $f$ should be then $e^2 = f^2 = (e,f) - k = 0$?). It is my fault--Kneser's statement should be ``...it is definite or it is on the hyerbolic plane..." | |
| Nov 3, 2013 at 1:35 | comment | added | user2013 | I thought any automorphism of $U(k)$ is induced by that of $U$. Am I wrong? | |
| Nov 3, 2013 at 1:33 | comment | added | user2013 | Sorry for the confusion. I denote by $U(k)$ the hyperbolic lattice mulptiplied by $k$. So there are basis $e,f$ with $e^2=f^2=(e,f)-2=0$. | |
| Nov 3, 2013 at 0:36 | comment | added | WKC | I assume that $U(k)$ is the orthogonal sum of $k$ copies of the hyperbolic plane. When $k \geq 2$, $O(U(k))$ is infinite. Suppose $k = 2$, and $e,f$ and $x,y$ are two orthogonal pairs of hyperbolic pairs. Then for any integer $a$, the linear map such that $e\mapsto e + ax$, $f \mapsto f + ax$, $x \mapsto -x$, and $y \mapsto ae + af + a^2x - y$, is an isometry of $U(2)$. | |
| Nov 2, 2013 at 22:23 | comment | added | user2013 | Isn't $O(U(K)))$ finite? | |
| Nov 1, 2013 at 2:16 | comment | added | WKC | Hyperbolic plane is the rank 2 lattice with Gram matrix $\begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}$. This is probably the $U$ in the original question. | |
| Oct 31, 2013 at 20:14 | comment | added | user2013 | Do you mean $U(k)$ by the hyperbolic plane? | |
| Oct 26, 2013 at 2:09 | history | answered | WKC | CC BY-SA 3.0 |