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I know that exist a Lie Group Called the Orthogonal Group $O(n)$. That correspond to all matrix of $n \times n$ in the real numbers such that the columns are a orthogonal basis for $\mathbb{R}^n$. Is posible to construct a "General Orthogonal Group" over a field $k$ of characteristic zero? It won't be a Lie Group, but maybe it is posible to give it a topology induced by the Zariski topology on $k^{n\times n}$ and it have some nice topological properties.

Did anyone know about something like this?

Thanks in advance.

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    $\begingroup$ There are many generalizations of the orthogonal group, see en.wikipedia.org/wiki/Orthogonal_group. $\endgroup$
    – Dietrich Burde
    Commented May 22, 2013 at 19:52
  • $\begingroup$ To amplify Dietrich's comment, note that over $\mathbf{R}$, one sees a variety of kinds of "orthogonal groups", corresponding to different signatures $(p, n-p)$ for $0 \le p \le n$. Since negating a quadratic form has no effect on the orthogonal group, we may as well restrict to $n/2 \le p \le n$. The case $p = n$ (positive-definite) is then the only one for which the orthogonal group does not contain an "algebraic" copy of the multiplicative group of the field as a Zariski-closed subgroup. Over other fields there can be a lot more such examples (depending on the field). $\endgroup$
    – user29283
    Commented May 23, 2013 at 1:41

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Yes, the orthogonal group makes sense over any field $k$. It is an linear algebraic group. In fact the theory of linear algebraic groups generalizes that of linear Lie groups over the real or complex numbers to give something that makes sense over an arbitrary field $k$.

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  • $\begingroup$ Do you have a paper or a book where i can read about Orthogonal group in fields of characteristic zero? Thanks in advance $\endgroup$
    – Joaquín Moraga
    Commented May 22, 2013 at 19:59
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    $\begingroup$ There are nice books on classical groups (like orthogonal groups). I found a short bibliography here: maths.qmul.ac.uk/~pjc/class_gps/cgbib.pdf. $\endgroup$
    – Dietrich Burde
    Commented May 22, 2013 at 20:09
  • $\begingroup$ see also Larry Grove's book: Classical Groups and Geometric Algebra $\endgroup$
    – Name
    Commented May 22, 2013 at 20:24

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