Witt decomposition

of a vector space

A decomposition of the space into a direct sum of three subspaces having certain properties. More exactly, let $ V $ be a vector space over a field $ k $ with characteristic different from 2, provided with a metric structure by means of a symmetric or skew-symmetric bilinear form $ f $. The direct decomposition

$$ V = N _ {1} + N _ {2} + D $$

is said to be a Witt decomposition of $ V $ if $ N _ {1} $ and $ N _ {2} $ are totally isotropic, while $ D $ is anisotropic and is orthogonal to $ N _ {1} + N _ {2} $ with respect to $ f $. The Witt decomposition plays an important role in the study of the structure of the form $ f $ and in problems of classification of bilinear forms.

Let $ f $ be a non-degenerate bilinear form and let $ V $ be finite-dimensional. Then any maximal totally isotropic subspace in $ V $ may be included in a Witt decomposition of $ V $ as $ N _ {1} $ or $ N _ {2} $. For any Witt decomposition $ \mathop{\rm dim} N _ {1} = \mathop{\rm dim} N _ {2} $, and for any basis $ v _ {1} ^ {(} 1) \dots v _ {n} ^ {(} 1) $ in $ N _ {1} $, there exists a basis $ v _ {1} ^ {(} 2) \dots v _ {n} ^ {(} 2) $ in $ N _ {2} $ such that $ f ( v _ {i} ^ {(} 1) , v _ {j} ^ {(} 2) ) = \delta _ {ij} $( $ \delta _ {ij} $ are the Kronecker symbols). For any two Witt decompositions

$$ V = N _ {1} + N _ {2} + D = \ N _ {1} ^ { \prime } + N _ {2} ^ { \prime } + D ^ \prime $$

the condition $ \mathop{\rm dim} N _ {i} = \mathop{\rm dim} N _ {i} ^ { \prime } $, $ i = 1, 2, $ is necessary and sufficient for the existence of a metric automorphism $ \phi $ of $ V $ such that

$$ \phi ( N _ {1} ) = N _ {1} ^ { \prime } ,\ \ \phi ( N _ {2} ) = N _ {2} ^ { \prime } ,\ \ \phi ( D) = D ^ \prime . $$

A non-degenerate symmetric or skew-symmetric bilinear form $ f $ on $ V $ is said to be neutral if $ V $ is finite-dimensional and has a Witt decomposition with $ D= 0 $. In this case the symmetric form is said to be a hyperbolic form, while $ V $ is said to be a hyperbolic space. An orthogonal direct sum of neutral forms is neutral. The matrix of a neutral form (in the basis $ v _ {1} ^ {(} 1) \dots v _ {n} ^ {(} 1) , v _ {1} ^ {(} 2) \dots v _ {n} ^ {(} 2) $ of the space $ V = N _ {1} + N _ {2} $ described above) looks like

$$ \left \| \frac{0}{\epsilon E _ {n} } \left | \frac{E _ {n} }{0} \right . \right \| , $$

where $ E _ {n} $ is the identity matrix of order $ n $, while $ \epsilon = 1 $ for a symmetric form and $ \epsilon = - 1 $ for a skew-symmetric form. Two neutral forms are isometric if and only if they have the same rank. The class of neutral symmetric bilinear forms is the zero (i.e. the neutral element for addition) in the Witt ring of the field $ k $. Neutral forms and only such forms have Witt index $ ( \mathop{\rm dim} V ) / 2 $. A skew-symmetric form on a finite-dimensional space is neutral.

If $ f $ is a non-degenerate symmetric bilinear form on a finite-dimensional space $ V $ and $ V = N _ {1} + N _ {2} + D $ is a Witt decomposition in which $ \mathop{\rm dim} N _ {1} = \mathop{\rm dim} N _ {2} $ is equal to the Witt index of $ f $, the restriction of $ f $ to $ D $ is a definite, or anisotropic, bilinear form, i.e. is such that $ f( v, v) \neq 0 $ for all non-zero $ v \in D $. This form is independent (apart from an isometry) of the choice of the Witt decomposition of $ V $. In the set of definite bilinear forms it is possible to introduce an addition, converting it into an Abelian group — the Witt group of $ k $( cf. Witt ring).

Let $ v _ {1} ^ {(} i) \dots v _ {n} ^ {(} i) $ be bases in $ N _ {i} $, $ i = 1, 2 $, such that $ f( v _ {i} ^ {(} 1) , v _ {j} ^ {(} 2) ) = \delta _ {ij} $; the union of these bases with an arbitrary basis in $ D $ yields a basis in $ V $ in which the matrix of $ f $ looks like

$$ \left \| For symmetric bilinear forms there exists an orthogonal basis in $ V $, i.e. a basis in which the matrix of the form is diagonal. If the field $ k $ is algebraically closed, there even exists an orthonormal basis (a basis in which the matrix of the form is the identity), and for this reason two non-degenerate symmetric bilinear forms of finite rank over $ k $ are isometric if and only if they have the same [[Rank|rank]]. In the general case the classification of such forms substantially depends on the arithmetical properties of the field $ k $. The study and classification of degenerate symmetric and skew-symmetric forms can be reduced to the study of non-degenerate forms (the restriction of the form to a subspace which is complementary to the kernel of the form). All what has been said above permits a generalization to the case of $ \epsilon $- Hermitian forms over a skew-field with property $ ( T) $( cf. [[Witt theorem|Witt theorem]]), and also to the case of symmetric bilinear forms associated with quadratic forms, without restrictions on the characteristic of the field. ===='"`UNIQ--h-0--QINU`"'References==== <table><tr><td valign="top">[1]</td> <td valign="top"> N. Bourbaki, "Algebra" , ''Elements of mathematics'' , '''1''' , Addison-Wesley (1974) pp. Chapts. 1–2 (Translated from French) [https://mathscinet.ams.org/mathscinet/article?mr=0354207 MR0354207] [https://zbmath.org/?q=an%3A0281.00006 Zbl 0281.00006] </td></tr><tr><td valign="top">[2]</td> <td valign="top"> S. Lang, "Algebra" , Addison-Wesley (1974) [https://mathscinet.ams.org/mathscinet/article?mr=0783636 MR0783636] [https://zbmath.org/?q=an%3A0712.00001 Zbl 0712.00001] </td></tr><tr><td valign="top">[3]</td> <td valign="top"> E. Artin, "Geometric algebra" , Interscience (1957) [https://mathscinet.ams.org/mathscinet/article?mr=1529733 MR1529733] [https://mathscinet.ams.org/mathscinet/article?mr=0082463 MR0082463] [https://zbmath.org/?q=an%3A0077.02101 Zbl 0077.02101] </td></tr><tr><td valign="top">[4]</td> <td valign="top"> J.A. Dieudonné, "La géométrie des groups classiques" , Springer (1955) [https://zbmath.org/?q=an%3A0221.20056 Zbl 0221.20056] </td></tr></table> ===='"`UNIQ--h-1--QINU`"'Comments==== A vector space with a neutral non-degenerate bilinear form on it is called split or metabolic. A different form of the Witt decomposition theorem gives a decomposition of a quadratic space $ ( V, q) $( i.e. a vector space $ V $ with a quadratic form $ q $ on it) into an orthogonal sum $$ \tag{* } ( V, q) = ( V _ {t} , q _ {t} ) \oplus ( V _ {h} , q _ {h} )\oplus ( V _ {a} , q _ {a} ), $$ with $ ( V _ {t} , q _ {t} ) $ totally isotropic, $ ( V _ {h} , q _ {h} ) $ hyperbolic and $ ( V _ {a} , q _ {a} ) $ anisotropic. Moreover, the isometry classes of $ ( V _ {t} , q _ {t} ) $, $ ( V _ {h} , q _ {h} ) $ and $ ( V _ {a} , q _ {a} ) $ are uniquely determined by that of $ ( V, q) $. In this decomposition, $ ( V _ {t} , q _ {t} ) $ is the radical of $ V $, $ V _ {t} = \mathop{\rm rad} ( V)= \{ {v \in V } : {B( v, w)= 0 \textrm{ for all } w \in V } \} $, where $ B $ is the symmetric bilinear form on $ V $ associated to $ q $: $$ B( v, w) = \frac{1}{2}

\{ q( v+ w)- q( v)- q( w) \} .

$$

The uniqueness of the factors in the Witt decomposition (*) follows from the Witt cancellation theorem, which says that if $ q\oplus q _ {1} $ is isometric to $ q\oplus q _ {2} $, then $ q _ {1} $ and $ q _ {2} $ are isometric.

References

[a1]	J. Milnor, D. Husemoller, "Symmetric bilinear forms" , Springer (1973) pp. 16 MR0506372 Zbl 0292.10016
[a2]	T.Y. Lam, "The algebraic theory of quadratic forms" , Benjamin (1973) MR0396410 Zbl 0259.10019