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.



General Orthogonal Group and its properties
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 $k$.


