hyperbolic plane in quadratic spaces
A non-singular (http://planetmath.org/NonDegenerateQuadraticForm) isotropic quadratic space of dimension 2 (over a field) is called a hyperbolic plane. In other words, is a 2-dimensional vector space over a field equipped with a quadratic form such that there exists a non-zero vector with .
Examples. Fix the ground field to be , and be the two-dimensional vector space over with the standard basis and .
Let . Then for all . is a hyperbolic plane. When is written in matrix form, we have
Let . Then for all . is a hyperbolic plane. As above, can be written in matrix form:
It’s not hard to see that the two examples above are equivalent quadratic forms. To transform from the first form to the second, for instance, follow the linear substitutions and , or in matrix form:
In fact, we have the following
From the first example above, we see that the quadratic space with the quadratic form is a hyperbolic plane. Conversely, if we can show that any hyperbolic plane is isometric the example (with the ground field switched from to ), we are done.
Pick a non-zero vector and suppose it is isotropic: . Pick another vector so forms a basis for . Let be the symmetric bilinear form associated with . If , then for any with , , contradicting the fact that is non-singular. So . By dividing by , we may assume that .
Suppose . Then the matrix associated with the quadratic form corresponding to the basis is
If , then the trick is to replace with an isotropic vector so that the bottom right cell is also 0. Let . It’s easy to verify that . As a result, the isometry required has the matrix form
Thus we may speak of the hyperbolic plane over a field without any ambiguity, and we may identify the hyperbolic plane with either of the two quadratic forms or . Its notation, corresponding to the second of the forms, is , or simply .
A hyperbolic space is a finite dimensional orthogonal direct sum of hyperbolic planes. It is always even dimensional and has the notation or simply , where is the dimensional of the hyperbolic space.
Instead of being associated with a quadratic form, a hyperbolic plane is sometimes defined in terms of an alternating form. In any case, the two definitions of a hyperbolic plane coincide if the ground field has characteristic 2.
|Title||hyperbolic plane in quadratic spaces|
|Date of creation||2013-03-22 15:41:47|
|Last modified on||2013-03-22 15:41:47|
|Last modified by||CWoo (3771)|