A quadratic space (over a field) is a vector space equipped with a quadratic form on . It is denoted by . The dimension of the quadratic space is the dimension of the underlying vector space. Any vector space admitting a bilinear form has an induced quadratic form and thus is a quadratic space.
Two quadratic spaces and are said to be isomorphic if there exists an isomorphic linear transformation such that for any , . Since is easily seen to be an isometry between and (over the symmetric bilinear forms induced by and respectively), we also say that and are isometric.
A quadratic space equipped with a regular quadratic form is called a regular quadratic space.
Example of a Qudratic Space. The Generalized Quaternion Algebra.
Let be a field and . Let be the algebra over generated by with the following defining relations:
Then , where , forms a basis for the vector space over . For a direct proof, first note , so that . It’s also not hard to show that anti-commutes with both : and . Now, suppose . Multiplying both sides of the equation on the right by gives . Multiplying both sides on the left by gives . Adding the two results and reduce, we have . Multiplying this again by gives us , or . Similarly, one shows that , so that . This leads to two equations, and , if one multiplies it on the left and right by . Adding the results then dividing by 2 gives . Since , . Therefore, . Same argument shows that as well.
We next define the norm on by . Since , . It’s easy to see that for any .
Finally, if we define the trace on by , we have that is bilinear (linear each in and ).
Therefore, defines a quadratic form on ( is commonly called a norm form), and is thus a quadratic space over . is denoted by
In fact, every quaternion algebra (over a field ) is of the form for some .
|Date of creation||2013-03-22 15:05:55|
|Last modified on||2013-03-22 15:05:55|
|Last modified by||CWoo (3771)|
|Synonym||non-degenerate quadratic space|
|Defines||isomorphic quadratic spaces|
|Defines||isometric quadratic spaces|
|Defines||generalized quaternion algebra|
|Defines||regular quadratic space|