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quadratic space (Definition)

A quadratic space (over a field) is a vector space $ V$ equipped with a quadratic form $ Q$ on $ V$. It is denoted by $ (V,Q)$. 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 $ (V_1,Q_1)$ and $ (V_2,Q_2)$ are said to be isomorphic if there exists an isomorphic linear transformation $ T:V_1\to V_2$ such that for any $ v\in V_1$, $ Q_1(v)=Q_2(Tv)$. Since $ T$ is easily seen to be an isometry between $ V_1$ and $ V_2$ (over the symmetric bilinear forms induced by $ Q_1$ and $ Q_2$ respectively), we also say that $ (V_1,Q_1)$ and $ (V_2,Q_2)$ 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 $ F$ be a field and $ a,b\in \dot{F}:=F-\lbrace 0\rbrace$. Let $ H$ be the algebra over $ F$ generated by $ i,j$ with the following defining relations:

  1. $ i^2=a$,
  2. $ j^2=b$, and
  3. $ ij=-ji$.
Then $ \lbrace 1,i,j,k\rbrace$, where $ k:=ij$, forms a basis for the vector space $ H$ over $ F$. For a direct proof, first note $ (ij)^2=(ij)(ij)=i(ji)j=i(-ij)j=-ab\neq 0$, so that $ k\in\dot{F}$. It's also not hard to show that $ k$ anti-commutes with both $ i,j$: $ ik=-ki$ and $ jk=-kj$. Now, suppose $ 0=r+si+tj+uk$. Multiplying both sides of the equation on the right by $ i$ gives $ 0=ri+sa+tji+uki$. Multiplying both sides on the left by $ i$ gives $ 0=ri+sa+tij+uik$. Adding the two results and reduce, we have $ 0=ri+sa$. Multiplying this again by $ i$ gives us $ 0=ra+sai$, or $ 0=r+si$. Similarly, one shows that $ 0=r+tj$, so that $ si=tj$. This leads to two equations, $ sa=tij$ and $ sa=tji$, if one multiplies it on the left and right by $ i$. Adding the results then dividing by 2 gives $ sa=0$. Since $ a\ne 0$, $ s=0$. Therefore, $ 0=r+si=r$. Same argument shows that $ t=u=0$ as well.

Next, for any element $ \alpha=r+si+tj+uk\in H$, define its conjugate $ \overline{\alpha}$ by $ r-si-tj-uk$. Note that $ \alpha=\overline{\alpha}$ iff $ \alpha\in F$. Also, it's not hard to see that

  • $ \overline{\overline{\alpha}}=\alpha$,
  • $ \overline{\alpha+\beta}=\overline{\alpha}+\overline{\beta}$,
  • $ \overline{\alpha\beta}=\overline{\beta}\overline{\alpha}$,

We next define the norm $ N$ on $ H$ by $ N(\alpha)=\alpha\overline{\alpha}$. Since $ \overline{N(\alpha)}=\overline{\alpha\overline{\alpha}}= \overline{\overline{\alpha}}\ \overline{\alpha}=\alpha\overline{\alpha}=N(\alpha)$, $ N(\alpha)\in F$. It's easy to see that $ N(r\alpha)=r^2N(\alpha)$ for any $ r\in F$.

Finally, if we define the trace $ T$ on $ H$ by $ T(\alpha)=\alpha+ \overline{\alpha}$, we have that $ N(\alpha+\beta)-N(\alpha)-N(\beta)=T(\alpha \overline{\beta})$ is bilinear (linear each in $ \alpha$ and $ \beta$).

Therefore, $ N$ defines a quadratic form on $ H$ ($ N$ is commonly called a norm form), and $ H$ is thus a quadratic space over $ F$. $ H$ is denoted by

$\displaystyle \Big( \frac{a,b}{F} \Big).$
It can be shown that $ H$ is a central simple algebra over $ F$. Since $ H$ is four dimensional over $ F$, it is a quaternion algebra. It is a direct generalization of the quaternions $ \mathbb{H}$ over the reals
$\displaystyle \Big( \frac{-1,-1}{\mathbb{R}} \Big).$
In fact, every quaternion algebra (over a field $ F$) is of the form $ \displaystyle{\Big( \frac{a,b}{F} \Big)}$ for some $ a,b\in F$.



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See Also: quadratic form, quaternion algebra

Other names:  non-degenerate quadratic space
Also defines:  norm form, isomorphic quadratic spaces, isometric quadratic spaces, generalized quaternion algebra, regular quadratic space
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Cross-references: reals, quaternions, quaternion algebra, central simple algebra, trace, easy to see, norm, iff, conjugate, argument, right, equation, sides, proof, basis, defining relations, generated by, algebra, regular quadratic form, isometric, symmetric bilinear forms, isometry, linear transformation, isomorphic, induced, bilinear form, dimension, quadratic form, vector space, field
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This is version 11 of quadratic space, born on 2005-02-25, modified 2007-12-18.
Object id is 6827, canonical name is QuadraticSpace.
Accessed 5396 times total.

Classification:
AMS MSC: 15A63 (Linear and multilinear algebra; matrix theory :: Quadratic and bilinear forms, inner products)
 11E88 (Number theory :: Forms and linear algebraic groups :: Quadratic spaces; Clifford algebras)
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